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<h1>GAP - Reference Manual</h1>

<p>Release 4.15.1, 2025-10-18</p>

</div>
<p><b>The GAP Group
    
    
  </b>
<br />Email: <span class="URL"><a href="mailto:support@gap-system.org">support@gap-system.org</a></span>
<br />Homepage: <span class="URL"><a href="https://www.gap-system.org">https://www.gap-system.org</a></span>
</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>Copyright © (1987-2025) for the core part of the <strong class="pkg">GAP</strong> system by the <strong class="pkg">GAP</strong> Group.</p>

<p>Most parts of this distribution, including the core part of the <strong class="pkg">GAP</strong> system are distributed under the terms of the GNU General Public License Version 2, see <span class="URL"><a href="https://www.gnu.org/licenses/old-licenses/gpl-2.0.html">https://www.gnu.org/licenses/old-licenses/gpl-2.0.html</a></span> or the <code class="file">LICENSE</code> file in the root directory of the <strong class="pkg">GAP</strong> installation.</p>

<p>More detailed information about copyright and licenses of parts of this distribution can be found in Section <a href="chap1.html#X7950EFA183E3F666"><span class="RefLink">1.4</span></a> of this manual.</p>

<p><strong class="pkg">GAP</strong> has been developed over a long time and has many authors and contributors. More detailed information can be found in Section <a href="chap1.html#X877A62A1781C2147"><span class="RefLink">1.2</span></a> of this manual.</p>

<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>

<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1.html#X874E1D45845007FE">1 <span class="Heading">Preface</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X863F306C7D32F4B0">1.1 <span class="Heading">The <strong class="pkg">GAP</strong> System</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X877A62A1781C2147">1.2 <span class="Heading">Authors and Maintainers</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X82A988D47DFAFCFA">1.3 <span class="Heading">Acknowledgements</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7950EFA183E3F666">1.4 <span class="Heading">Copyright and License</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7BF552C07E2F8F7C">1.5 <span class="Heading">Further Information about <strong class="pkg">GAP</strong></span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2.html#X8755A2C67B197C63">2 <span class="Heading">The Help System</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7E2C53D2844DD8C3">2.1 <span class="Heading">Invoking the Help</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7BE8068878B7D7D1">2.2 <span class="Heading">Browsing through the Sections</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X863FF9087EDA8DF9">2.3 <span class="Heading">Changing the Help Viewer</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X87C1BFB2826488B0">2.3-1 SetHelpViewer</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X84AFFC817B282359">2.4 <span class="Heading">The Pager Command</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7ED03E41792C3840">2.4-1 Pager</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap3.html#X79CCD3A6821E5A37">3 <span class="Heading">Running GAP</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X782751D5858A6EAF">3.1 <span class="Heading">Command Line Options</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7FD66F977A3B02DF">3.2 <span class="Heading">The gap.ini and gaprc files</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X87DF11C885E73583">3.2-1 <span class="Heading">The gap.ini file</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84D4CF587D437C00">3.2-2 <span class="Heading">The gaprc file</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7B0AD104839B6C3C">3.2-3 <span class="Heading">Configuring User preferences</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7F1DF6757B248014">3.2-4 DeclareUserPreference</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X870A11E7864F9CA7">3.2-5 <span class="Heading">User Preferences Defined by <strong class="pkg">GAP</strong></span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7CB282757ACB1C09">3.3 <span class="Heading">Saving and Loading a Workspace</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X876544A57C73C488">3.3-1 SaveWorkspace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X83BF07587F2CC6CD">3.4 <span class="Heading">Testing for the System Architecture</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C825AF087A27884">3.4-1 ARCH_IS_UNIX</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X82A6893A7EC8FA72">3.4-2 ARCH_IS_MAC_OS_X</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A14B659847B8627">3.4-3 ARCH_IS_WINDOWS</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X87E7CC3B8395BBB3">3.4-4 ARCH_IS_WSL</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X8719B2118511645F">3.5 <span class="Heading">Global Values that Control the <strong class="pkg">GAP</strong> Session</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8354754E7935F935">3.5-1 GAPInfo</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X818F2DDC863C381E">3.6 <span class="Heading">Coloring the Prompt and Input</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84F3481C8466C7FC">3.6-1 ColorPrompt</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap4.html#X7FE7C0C17E1ED118">4 <span class="Heading">The Programming Language</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7B5FF6827DFBDF20">4.1 <span class="Heading">Language Overview</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X80A85A707B6F4BE7">4.2 <span class="Heading">Lexical Structure</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7E90E6607F4E4943">4.3 <span class="Heading">Symbols</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7C53CEFC8641B919">4.4 <span class="Heading">Whitespaces</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X87506BDC7D5F789E">4.5 <span class="Heading">Keywords</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X860313A179A5163F">4.6 <span class="Heading">Identifiers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85CF993B7D19F2C4">4.6-1 IsValidIdentifier</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X839A7F8E84BBCA57">4.6-2 <span class="Heading">Conventions about Identifiers</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7BAFE9C1817253C6">4.7 <span class="Heading">Expressions</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7A4C2D0E7E286B4F">4.8 <span class="Heading">Variables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X842B89D4860FD5DB">4.8-1 IsBound</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7BABB3E77F52626C">4.8-2 Unbind</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X816FBEEA85782EC2">4.9 <span class="Heading">More About Global Variables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7CD3523B84744EB2">4.9-1 IsReadOnlyGlobal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X850CE44478254F27">4.9-2 MakeReadOnlyGlobal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X832AAF13861968BE">4.9-3 MakeReadWriteGlobal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X847706237E72418F">4.9-4 MakeConstantGlobal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X84BB4B1E872849FF">4.9-5 ValueGlobal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X823D4BC378395B32">4.9-6 IsBoundGlobal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X829A5F0E811F77D3">4.9-7 UnbindGlobal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D39D3E17CF49F5B">4.9-8 BindGlobal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X876A6EB68745A510">4.9-9 NamesGVars</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E604AF579A7BC92">4.9-10 NamesSystemGVars</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X870169447AF490D8">4.9-11 NamesUserGVars</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7DF8774F7D542298">4.10 <span class="Heading">Namespaces for <strong class="pkg">GAP</strong> packages</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X815F71EA7BC0EB6F">4.11 <span class="Heading">Function</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X78C70489791FDF43">4.12 <span class="Heading">Function Calls</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X80B93A9C7E0A57F4">4.12-1 <span class="Heading">Function Call With Arguments</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X867D54987EF86D1D">4.12-2 <span class="Heading">Function Call With Options</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7A274A1F8553B7E6">4.13 <span class="Heading">Comparisons</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7B66C8707B5DE10A">4.14 <span class="Heading">Arithmetic Operators</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8543285D87361BE6">4.15 <span class="Heading">Statements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E6A50307F4D3FAE">4.15-1 <span class="Heading">Assignments</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X825803DE78251DA6">4.15-2 <span class="Heading">Procedure Calls</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X875000188622700D">4.15-3 <span class="Heading">If</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87AA46408783383F">4.15-4 <span class="Heading">While</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8295CBF47FAA05C9">4.15-5 <span class="Heading">Repeat</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X78783E777867638A">4.15-6 <span class="Heading">For</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7B60C6127E183021">4.15-7 <span class="Heading">Break</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7CCBA2247AA366BD">4.15-8 <span class="Heading">Continue</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X812C6ABC7A182E9E">4.15-9 <span class="Heading">Return (With or without Value)</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8732D9257FFCEA1B">4.16 <span class="Heading">Syntax Trees</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X81558D66810BEA67">4.16-1 SyntaxTree</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap5.html#X86FA580F8055B274">5 <span class="Heading">Functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X870553EF8605792F">5.1 <span class="Heading">Information about a function</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X79C3BDC4781FA0FD">5.1-1 NameFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X877F03F77FD74C98">5.1-2 NumberArgumentsFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X818BAB817A4FB346">5.1-3 NamesLocalVariablesFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X80E108C57F90FAA3">5.1-4 FilenameFunc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7FF7643781D2C194">5.1-5 StartlineFunc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X844F95767C74834F">5.1-6 LocationFunc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X845A929B83D46E01">5.1-7 PageSource</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X851B58408520700D">5.2 <span class="Heading">Calling a function with a list argument that is interpreted as
several arguments</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7CF4DDB97D65AE52">5.2-1 CallFuncList</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X83066E5A80B5FB71">5.3 <span class="Heading">Wrapping a function, so the values produced are cached</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X817ED3B280A64601">5.3-1 MemoizePosIntFunction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7EB0A85F7D128BE0">5.4 <span class="Heading">Functions that do nothing</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7DB422A2876CCC4D">5.4-1 ReturnTrue</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7C131FB17D7518FC">5.4-2 ReturnFalse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7A0994DE7C258E55">5.4-3 ReturnFail</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X818EA8C47B46A634">5.4-4 ReturnNothing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X8788D7D780FCE169">5.4-5 ReturnFirst</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X810325697BDEF899">5.4-6 IdFunc</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X80FE39D27CE3DE1B">5.5 <span class="Heading">Function Types</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X85E40340806C2B8C">5.5-1 IsFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X87838FE885A9AAF9">5.5-2 FunctionsFamily</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X81F732457F7BC851">5.6 <span class="Heading">Naming Conventions</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7A1721CD79F08E71">5.7 <span class="Heading">Code annotations (pragmas)</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap6.html#X7DB71A2A841CADA5">6 <span class="Heading">Main Loop and Break Loop</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X81667F568237B232">6.1 <span class="Heading">Main Loop</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X866092F281910B74">6.2 <span class="Heading">Special Rules for Input Lines</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X8074A8387C9DB9A8">6.3 <span class="Heading">View and Print</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X8082880F824292E9">6.3-1 <span class="Heading">Default delegations in the library</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X87D445D37B31DADB">6.3-2 <span class="Heading">Recommendations for the implementation</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X851902C583B84CDC">6.3-3 View</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7AFA64D97A1F39A3">6.3-4 Print</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X815BF22186FD43C9">6.3-5 ViewObj</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X83A5C59278E13248">6.3-6 Display</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X87E546E27A1F1FAB">6.3-7 SetNameObject</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X8593B49F8705B486">6.4 <span class="Heading">Break Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X83033EEB81CF4F49">6.4-1 <span class="Heading">quit from a break loop</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7A388B808167FE09">6.4-2 <span class="Heading">return from a break loop</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X82EBF01181C3C859">6.4-3 OnBreak</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X80711C807C99C220">6.4-4 OnBreakMessage</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7A7FFA2B7C1EF5A3">6.4-5 Where</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7EE5CF2C8419F061">6.5 <span class="Heading">Variable Access in a Break Loop</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X79E66DA2875303B0">6.5-1 <span class="Heading">DownEnv and UpEnv</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7BC8D2E37ADE9062">6.6 <span class="Heading">Error and ErrorCount</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7E7AD8D87EBA1A08">6.6-1 Error</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7A5C000D7E4984DD">6.6-2 ErrorNoReturn</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X86A11BCC7FECEEA4">6.6-3 ErrorCount</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X83704B1080FD9B40">6.7 <span class="Heading">Leaving GAP</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7ECC75048583853B">6.7-1 QUIT</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X838B50A9790DE55B">6.7-2 GapExitCode</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7AB1567987922580">6.7-3 QuitGap</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X85A8DD6B7A20DD89">6.7-4 ForceQuitGap</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7A2C380986F46FEE">6.7-5 InstallAtExit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X843C07A4869EAA1D">6.7-6 SaveOnExitFile</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X82234FD181899530">6.8 <span class="Heading">Line Editing</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7AD8D65F7BA1C3E0">6.9 <span class="Heading">Editing using the <code class="code">readline</code> library</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7C38F9E0783D9442">6.9-1 <span class="Heading">Readline customization</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X846C3DED84AD7593">6.9-2 <span class="Heading">The command line history</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7C1F4D04861C1197">6.9-3 SaveCommandLineHistory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X87D4EA197A263FB7">6.9-4 <span class="Heading">Writing your own command line editing functions</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7D8E1CF47E97A764">6.10 <span class="Heading">Editing Files</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X82E5859C8113BA4D">6.10-1 Edit</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7B67FF1E87FE67D1">6.11 <span class="Heading">Editor Support</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X83279E897ACCFFFA">6.12 <span class="Heading">Changing the Screen Size</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X8723E0A1837894F3">6.12-1 SizeScreen</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X87847E5087D6F47D">6.13 <span class="Heading">Teaching Mode</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7BE2515F82425404">6.13-1 TeachingMode</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap7.html#X8345F6817DFD6394">7 <span class="Heading">Debugging and Profiling Facilities</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X83C45B0A797AAF96">7.1 <span class="Heading">Recovery from NoMethodFound-Errors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X86B5FEC67A9394DC">7.1-1 ShowArguments</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X834BD9928773DCC1">7.1-2 ShowArgument</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7D25D904800D5CBA">7.1-3 ShowDetails</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7F6996CA872478B8">7.1-4 ShowMethods</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7E5E2E7B85029E34">7.1-5 ShowOtherMethods</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7FDA1D4B87BD25A8">7.2 <span class="Heading">Inspecting Applicable Methods</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X80848FF486BD6F9F">7.2-1 ApplicableMethod</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7D43A2D885B37739">7.3 <span class="Heading">Tracing Methods</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X80B044017C9E4137">7.3-1 TraceMethods</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7D34CADB813A4AF1">7.3-2 TraceAllMethods</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7EB04D387C53E4C1">7.3-3 UntraceMethods</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7B3018AA82D55949">7.3-4 UntraceAllMethods</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X81078D3387A38E31">7.3-5 TraceImmediateMethods</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X81B000CF86BA1534">7.3-6 TraceInternalMethods</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7A9C902479CB6F7C">7.4 <span class="Heading">Info Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7AA1A1CF79F20790">7.4-1 NewInfoClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7B3709C584B3DA1E">7.4-2 DeclareInfoClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7A43B9E68765EE9E">7.4-3 SetInfoLevel</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7B2ADC37783104B9">7.4-4 InfoLevel</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7BA636EF80A1435A">7.4-5 ShowUsedInfoClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X864E4B6886E2697D">7.4-6 Info</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X800234B5815CAC97">7.4-7 <span class="Heading">Customizing <code class="func">Info</code> (<a href="chap7.html#X864E4B6886E2697D"><span class="RefLink">7.4-6</span></a>) statements</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7A28F77C82D6A3E0">7.4-8 InfoWarning</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X86425F067FC63A4C">7.5 <span class="Heading">Assertions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7C7596418423660B">7.5-1 SetAssertionLevel</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X876C83707F13A0FD">7.5-2 AssertionLevel</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X830E443284780FB9">7.5-3 Assert</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X792BA9A67E64CDED">7.6 <span class="Heading">Timing</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X80355C9282B35673">7.6-1 Runtimes</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7E32B27F81870D24">7.6-2 Runtime</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X844E1CFE80F41760">7.6-3 NanosecondsSinceEpoch</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7C0F91F982189624">7.6-4 time</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7B543F357C7202CF">7.6-5 Sleep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X844CB04081A771FB">7.7 <span class="Heading">Tracking Memory Usage</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8077B50B844C4EFC">7.7-1 TotalMemoryAllocated</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8156D7208591460F">7.7-2 memory_allocated</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7FDF923D7D2937A1">7.8 <span class="Heading">Profiling</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7939F6F182FDA5F1">7.8-1 <span class="Heading">Function Profiling</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X79D6CB927BBEB940">7.8-2 ProfileGlobalFunctions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7C893F68841B990B">7.8-3 ProfileOperations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X79D41E977DCA2BEE">7.8-4 ProfileOperationsAndMethods</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X81E8A8627C34FD3B">7.8-5 ProfileFunctions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X79D394EC7BE8D008">7.8-6 UnprofileFunctions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X787AC3BE7F991344">7.8-7 ProfileMethods</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X87A05F977F033693">7.8-8 UnprofileMethods</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X80FEA6A08775A48E">7.8-9 DisplayProfile</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7DAF9AB9793AE203">7.8-10 ClearProfile</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7C5CE32579891120">7.8-11 <span class="Heading">An Example of Function Profiling</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X812F9CE0817110EA">7.8-12 <span class="Heading">Line By Line Profiling</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7E9C65B17B8EF993">7.8-13 <span class="Heading">Line by Line profiling example</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X86557887796F66FA">7.8-14 ProfileLineByLine</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X87CC48807DB4C008">7.8-15 CoverageLineByLine</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7C5DED9C7CC77504">7.8-16 UnprofileLineByLine</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7B705B2D8670A9C5">7.8-17 UncoverageLineByLine</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7823C83D79B36D3B">7.8-18 IsLineByLineProfileActive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X83D8A42B7BB92F5B">7.8-19 DisplayCacheStats</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X79C58704838232CC">7.8-20 ClearCacheStats</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7EE874867C0BEEDD">7.9 <span class="Heading">Information about the version used</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X801051CC86594630">7.10 <span class="Heading">Test Files</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8213757B7ACC76E6">7.10-1 <span class="Heading">Starting and stopping test</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X87712F9D8732193C">7.10-2 Test</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X87AF67528799481F">7.10-3 TestDirectory</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X85FF55448787CCA0">7.11 <span class="Heading">Debugging Recursion</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7D8968FC7E24A4E5">7.11-1 SetRecursionTrapInterval</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X85679F17791D9B63">7.12 <span class="Heading">Global Memory Information</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7F1F741D7F0899D1">7.12-1 <span class="Heading">Garbage Collection</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7848AB367F3A1221">7.12-2 CollectGarbage</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X836977DE80416F3D">7.12-3 GasmanStatistics</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X85327FA5872E0356">7.12-4 GasmanMessageStatus</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X80C683247E94769F">7.12-5 GasmanLimits</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap8.html#X7FD84061873F72A2">8 <span class="Heading">Options Stack</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X794C5B5A80203CF9">8.1 <span class="Heading">Functions Dealing with the Options Stack</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7D4939FF7FB37FBE">8.1-1 PushOptions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7818A5278679FD43">8.1-2 PopOptions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X83D1190984DA3B85">8.1-3 ResetOptionsStack</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X78D87D1081BF99FE">8.1-4 OnQuit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7F9373AD7DB88D1F">8.1-5 ValueOption</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7EDA4EB67D43FE33">8.1-6 DisplayOptionsStack</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X832F41187B150C19">8.1-7 InfoOptions</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X7BB781647CAAE9B4">8.2 <span class="Heading">Options Stack – an Example</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap9.html#X82BCD4297920C903">9 <span class="Heading">Files and Filenames</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X83D8AAA484EE95D9">9.1 <span class="Heading">Portability</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X87D278437A916905">9.1-1 LastSystemError</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X7A4973627A5DB27D">9.2 <span class="Heading">GAP Root Directories</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X8223D52E78AF4420">9.3 <span class="Heading">GAP Package Directories</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X85030B35865A1080">9.4 <span class="Heading">Directories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X82B3E24683942597">9.4-1 IsDirectory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X86A71E927EEC7EAD">9.4-2 Directory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8222B1A886E6195E">9.4-3 DirectoryTemporary</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7BAD8036849E8430">9.4-4 DirectoryCurrent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X81DDD2E87F68E086">9.4-5 ChangeDirectoryCurrent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X87ED469A85343A3C">9.4-6 DirectoriesLibrary</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X808E2C187DD984B4">9.4-7 DirectoriesSystemPrograms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7B225E5282534EDA">9.4-8 DirectoryContents</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X86F4A32C83B82369">9.4-9 DirectoryDesktop</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7B0D818A808A3481">9.4-10 DirectoryHome</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X8545E03E7D651456">9.5 <span class="Heading">File Names</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7E352E1F87060602">9.5-1 <span class="Heading">Filename</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X86C7683E7A2A2146">9.5-2 PathSystemProgram</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X85EC7D9087C481B0">9.6 <span class="Heading">Special Filenames</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X87271FEF86A6A0F9">9.7 <span class="Heading">File Access</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8269697A7B927AF1">9.7-1 IsExistingFile</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7E156EC886E11BBC">9.7-2 IsReadableFile</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8412F485796B25F5">9.7-3 IsWritableFile</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X83A1AAD58435FC4C">9.7-4 IsExecutableFile</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7D1BE00F83C4EEE8">9.7-5 IsDirectoryPath</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X81A0A4FF842B039B">9.8 <span class="Heading">File Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8373AC6B7D5F9167">9.8-1 Read</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7824CB7D7D4BAFBC">9.8-2 ReadAsFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X86956C577FFEE1F9">9.8-3 <span class="Heading">PrintTo and AppendTo</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X79813A6686894960">9.8-4 <span class="Heading">LogTo</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7CAB119378B075B7">9.8-5 <span class="Heading">InputLogTo</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7A5591D87EAFA6CC">9.8-6 <span class="Heading">OutputLogTo</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8241CEAD80415BB9">9.8-7 CrcFile</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7E63ACA38142BE96">9.8-8 RemoveFile</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X83F3B0337C7EA5CC">9.8-9 UserHomeExpand</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X79EE267A7FAF28A6">9.8-10 Reread</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap10.html#X839725177BF8B5B4">10 <span class="Heading">Streams</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X7F89070B7CF52DE0">10.1 <span class="Heading">Categories for Streams and the StreamsFamily</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7E974B96785E91A8">10.1-1 IsStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7FE4096F8497B7F2">10.1-2 IsClosedStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7FB4391283847C3A">10.1-3 IsInputStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7C8956BB7FE2A89C">10.1-4 IsInputTextStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7DCD6ADC86CF2472">10.1-5 IsInputTextNone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7D357CA07E7B1E78">10.1-6 IsOutputStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X8248B8A4844CB8AB">10.1-7 IsOutputTextStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7C89CDD47E33E741">10.1-8 IsOutputTextNone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7F0F9DD47DE16DAB">10.1-9 StreamsFamily</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X8461F4DF7FC20C4B">10.2 <span class="Heading">Operations applicable to All Streams</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X786E5520803FDE00">10.2-1 CloseStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7F0459287E717456">10.2-2 FileDescriptorOfStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X87BC257A78F96828">10.2-3 UNIXSelect</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X7D1D33A587BFD93D">10.3 <span class="Heading">Operations for Input Streams</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7A5DC83D7E295568">10.3-1 Read</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7D62F2877F0E45A7">10.3-2 ReadAsFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X79E1E6A57AE58BB8">10.3-3 ReadByte</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7D2CA44C7D110C4F">10.3-4 ReadLine</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X85C603D7867430D0">10.3-5 ReadAll</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X81D4FB097F631A79">10.3-6 IsEndOfStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7B646FA3860521D1">10.3-7 PositionStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7A777E1186EB330B">10.3-8 RewindStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7A60AD8C7E0D0507">10.3-9 SeekPositionStream</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X7F454EB286947C85">10.4 <span class="Heading">Operations for Output Streams</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7D37C7A07E9C319C">10.4-1 WriteByte</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X79FA85498596CC99">10.4-2 WriteLine</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X78C113917936058D">10.4-3 WriteAll</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7F4E090C86AACCF7">10.4-4 <span class="Heading">PrintTo and AppendTo (for streams)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7BF4E44C7D51E085">10.4-5 LogTo</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7B843516796B2A18">10.4-6 InputLogTo</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X834A6DD17B0E2062">10.4-7 OutputLogTo</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X8663FCD57E8BC390">10.4-8 SetPrintFormattingStatus</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X80B5F2E4856D8980">10.5 <span class="Heading">File Streams</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X8343D04981128784">10.5-1 InputTextFile</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X83F53291822B7126">10.5-2 OutputTextFile</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X808348977A05477A">10.6 <span class="Heading">User Streams</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X83531E4C7C53544F">10.6-1 InputTextUser</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X83E5FC9487766297">10.6-2 OutputTextUser</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7DAF5B7085F4F893">10.6-3 InputFromUser</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X8028E1D87CE2F059">10.7 <span class="Heading">String Streams</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7ABABCDF7ED81F7F">10.7-1 InputTextString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7FEDA5167979B74D">10.7-2 OutputTextString</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X8563EF8387236417">10.8 <span class="Heading">Input-Output Streams</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X82822D3D8339F635">10.8-1 IsInputOutputStream</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X820799A3824684AC">10.8-2 InputOutputLocalProcess</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7CDF48447E823977">10.8-3 ReadAllLine</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X8724699C7D67BA47">10.9 <span class="Heading">Dummy Streams</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7C732324806716C6">10.9-1 InputTextNone</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7CC5C1FC81715E38">10.9-2 OutputTextNone</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X7CB5832F8721ADF3">10.10 <span class="Heading">Handling of Streams in the Background</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X81FB42517E3EA96D">10.10-1 InstallCharReadHookFunc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X8492474C7A0B10AD">10.10-2 UnInstallCharReadHookFunc</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X848DD7DC79363341">10.11 <span class="Heading">Comma separated files</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X86FDC1EF82CAD2DA">10.11-1 ReadCSV</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X8779DAC585E05A47">10.11-2 PrintCSV</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10.html#X87396F857ADA3F97">10.12 <span class="Heading">Opening files in the Operating System</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X86B98E287AD42BE8">10.12-1 OpenExternal</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap11.html#X7882133B7BDD51BC">11 <span class="Heading">Processes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X8390266186E61CCE">11.1 <span class="Heading">Process and Exec</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X7B09033178D1107A">11.1-1 Process</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X81402C91833986FC">11.1-2 Exec</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap12.html#X86710F997832ABA4">12 <span class="Heading">Objects and Elements</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X78497E777FB3E402">12.1 <span class="Heading">Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X7B130AC98415CAFB">12.1-1 IsObject</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X780C66027A49D110">12.2 <span class="Heading">Elements as equivalence classes</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X83BE0C20875DD285">12.3 <span class="Heading">Sets</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X7BAF69417BB925F6">12.4 <span class="Heading">Domains</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X84545F3985C60F5B">12.5 <span class="Heading">Identical Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X7961183378DFB902">12.5-1 IsIdenticalObj</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X811976EC78EC5E29">12.5-2 IsNotIdenticalObj</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X7F0C119682196D65">12.6 <span class="Heading">Mutability and Copyability</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X811EFD727EBD1ADC">12.6-1 IsCopyable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X7999AD1D7A4F1F46">12.6-2 IsMutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X7F0ABF2C870B0CBB">12.6-3 Immutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X80CE136D804097C7">12.6-4 MakeImmutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X7FBA5F4D7C6872BD">12.6-5 <span class="Heading">Mutability of Iterators</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X7ADB82997A16E853">12.6-6 <span class="Heading">Mutability of Results of Arithmetic Operations</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X786B942B82D684BD">12.7 <span class="Heading">Duplication of Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X846BC7107C352031">12.7-1 ShallowCopy</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X7C1E70587EBDD2CB">12.7-2 StructuralCopy</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X86E7193D848C53FC">12.8 <span class="Heading">Other Operations Applicable to any Object</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X85D6D47B83BD02A1">12.8-1 SetName</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X7F14EF9D81432113">12.8-2 Name</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X871562FD7F982C12">12.8-3 InfoText</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X7F6C5C3287E8B816">12.8-4 IsInternallyConsistent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X7F4D216B7DF7BE9D">12.8-5 MemoryUsage</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap13.html#X7E8202627B421DB1">13 <span class="Heading">Types of Objects</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X846063757EC05986">13.1 <span class="Heading">Families</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7CF70EAC84284919">13.1-1 FamilyObj</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7FB4123E7E22137D">13.1-2 NewFamily</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X84EFA4C07D4277BB">13.2 <span class="Heading">Filters</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X82E62B997C05E05E">13.2-1 RankFilter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7A78ECC67E2C9D78">13.2-2 NamesFilter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7F6645D87DD26CF0">13.2-3 FilterByName</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7F9568A67F3840DE">13.2-4 ShowImpliedFilters</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X836FAA18861BE387">13.2-5 FiltersType</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X7CC6903E78F24167">13.3 <span class="Heading">Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X792A23BF82BDF66B">13.3-1 IsCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X85C6EB707A406A5A">13.3-2 CategoriesOfObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X85D07C3E7F4D4043">13.3-3 CategoryByName</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X87F68F887B44DBBD">13.3-4 NewCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X879DE2A17A6C6E92">13.3-5 DeclareCategory</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X787BACEE7937EF01">13.3-6 CategoryFamily</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X8698205F8648EB33">13.4 <span class="Heading">Representation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X805F1C3B7C730062">13.4-1 <span class="Heading">Basic Representations of Objects</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X86D42C7783ACA5F4">13.4-2 IsRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7BBE93BE7977750F">13.4-3 RepresentationsOfObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7CC8106F809E15CF">13.4-4 NewRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7C81FB2682AE54CD">13.4-5 DeclareRepresentation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X7C701DBF7BAE649A">13.5 <span class="Heading">Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7973C8F4782D15A1">13.5-1 IsAttribute</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7F7960338163AA88">13.5-2 KnownAttributesOfObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7B9654807858A3B0">13.5-3 NewAttribute</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7A00FC8A7A677A56">13.5-4 DeclareAttribute</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7A951C33839AF2C1">13.5-5 IsAttributeStoringRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X79DE5208877AE42A">13.6 <span class="Heading">Setter and Tester for Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X87D5B5AC7DAF932D">13.6-1 Tester</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7FD8952C841D2B1F">13.6-2 Setter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X8529F8A17884A32C">13.6-3 AttributeValueNotSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X79120CE37BB69D11">13.6-4 InfoAttributes</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7851E2DB79656DB0">13.6-5 DisableAttributeValueStoring</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7E5DACBE7A9A9AD1">13.6-6 EnableAttributeValueStoring</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X871597447BB998A1">13.7 <span class="Heading">Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X81F1C3EE83003FA0">13.7-1 IsProperty</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7E51C08286E03E7F">13.7-2 KnownPropertiesOfObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X86711BC77B62EB02">13.7-3 KnownTruePropertiesOfObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7F2D6FD979FE23DD">13.7-4 NewProperty</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7F4602F082682A04">13.7-5 DeclareProperty</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X7997705185C7E720">13.8 <span class="Heading">Other Filters</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X821635DA7821ED74">13.8-1 NewFilter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X846EA18A7D36626C">13.8-2 DeclareFilter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7C92D53E7920CE02">13.8-3 SetFilterObj</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X8117FD03870FB02E">13.8-4 ResetFilterObj</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X7E340B8C833BC440">13.9 <span class="Heading">Types</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7D3E6B6482BE5B16">13.9-1 TypeObj</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X85A60A7F8083C1C4">13.9-2 DataType</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7CE39E9478AEC826">13.9-3 NewType</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap14.html#X853DF11B80068ED5">14 <span class="Heading">Integers</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X838230CE810107A3">14.1 <span class="Heading">Integers: Global Variables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7E20D82B79DE5129">14.1-1 Integers</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X818683B17F8C97F3">14.1-2 IsIntegers</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X80CF510B8080C7CA">14.2 <span class="Heading">Elementary Operations for Integers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X87AEADF07DC8303B">14.2-1 IsInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X82A854757DFA9C76">14.2-2 IsPosInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X87CA734380B5F68C">14.2-3 Int</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X87DD1EEE7EF18036">14.2-4 IsEvenInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X8621BA927CD12EFB">14.2-5 IsOddInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X782095927FB9F1DB">14.2-6 AbsInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X842614817FE48D62">14.2-7 SignInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X8197C4E882BAF14E">14.2-8 LogInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X83D9B5C87EEA2A77">14.2-9 RootInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7F98A0CE7B9FD366">14.2-10 SmallestRootInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X83B998E486893FED">14.2-11 IsSquareInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X862D1BD786EFFDA9">14.2-12 ListOfDigits</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X8185784B7E228DEA">14.2-13 Random</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X7A9FD25D81D88D1B">14.3 <span class="Heading">Quotients and Remainders</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X849D0F807F697D35">14.3-1 QuoInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X795170A385AC8FEE">14.3-2 BestQuoInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X805ADD5A826D844D">14.3-3 RemInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7A4FEFCA8128E3C3">14.3-4 GcdInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X8775930486BD0C5B">14.3-5 Gcdex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7B33143E78A8DDE3">14.3-6 LcmInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X79B466E984CD52D4">14.3-7 CoefficientsQadic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X83124F86839DC7E6">14.3-8 CoefficientsMultiadic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X84A1900E82902B5F">14.3-9 ChineseRem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7E404B1183DBC82A">14.3-10 PowerModInt</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X82005E587F0CB02A">14.4 <span class="Heading">Prime Integers and Factorization</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X86F5E4CD82FEB9F4">14.4-1 Primes</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X78FDA4437EDCA70C">14.4-2 IsPrimeInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7CD977B17B4A7A4B">14.4-3 PrimalityProof</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X8443125D7FD6F2A6">14.4-4 IsPrimePowerInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X78744C367A94C69F">14.4-5 NextPrimeInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X819060E17E83728A">14.4-6 PrevPrimeInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X82C989DB84744B36">14.4-7 FactorsInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X80E7A5D381C64CC9">14.4-8 PrimeDivisors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X786FF92C7C54BF97">14.4-9 PartialFactorization</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X803D431087B6FF28">14.4-10 PrintFactorsInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X82148B347E294C87">14.4-11 PrimePowersInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X809E0E1B83AF7695">14.4-12 DivisorsInt</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X864BF040862409FC">14.5 <span class="Heading">Residue Class Rings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X87B1210B8581D5B2"><code>14.5-1 \mod</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X79CE76AD82B3E2B2">14.5-2 ZmodnZ</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X838F36507D985EDA">14.5-3 ZmodnZObj</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7D0107DD79753901">14.5-4 IsZmodnZObj</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X7904B6D681EBF091">14.6 <span class="Heading">Check Digits</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X82BABA8F868BD425">14.6-1 CheckDigitISBN</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X85F1A6A5870485B9">14.6-2 CheckDigitTestFunction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X85361FAE8088C006">14.7 <span class="Heading">Random Sources</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X82E31A697E389F1D">14.7-1 IsRandomSource</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X821004F286282D49">14.7-2 Random</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X86FFFBC9790F9742">14.7-3 <span class="Heading">State and Reset for Random Sources</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7AC96008820FAF1F">14.7-4 <span class="Heading">Kinds of Random Sources</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7CB0B5BC82F8FD8F">14.7-5 RandomSource</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X8653AE447D94C1DC">14.7-6 <span class="Heading">Implementing new kinds of random sources</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X7A0311DF78DB4FD8">14.8 <span class="Heading">Bitfields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X85C7BD9E7FCC6C10">14.8-1 MakeBitfields</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X8068CE3781F4003C">14.8-2 BuildBitfields</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap15.html#X7FB995737B7ED8A2">15 <span class="Heading">Number Theory</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X7845C1F97A1742C7">15.1 <span class="Heading">InfoNumtheor (Info Class)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X796F0DFE7D5D211C">15.1-1 InfoNumtheor</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X823386567DAC22E6">15.2 <span class="Heading">Prime Residues</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X7FA3F5347B7004BA">15.2-1 PrimeResidues</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X85A0C67982D9057A">15.2-2 Phi</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X85296F3087611B03">15.2-3 Lambda</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X7D191CF67E5018BE">15.2-4 GeneratorsPrimeResidues</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X83103A5385821BAE">15.3 <span class="Heading">Primitive Roots and Discrete Logarithms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X82373F3D8277EE9E">15.3-1 OrderMod</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X81AD9C7779A7BA89">15.3-2 LogMod</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X84A138947E8C49A8">15.3-3 DLog</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X82440BB9812FF148">15.3-4 PrimitiveRootMod</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X790466C07BD90E20">15.3-5 IsPrimitiveRootMod</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X7F9069D77AC48054">15.4 <span class="Heading">Roots Modulo Integers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X83449DBC80495971">15.4-1 Jacobi</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X81464ABF7F10E544">15.4-2 Legendre</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X83E3ED577B7A04ED">15.4-3 RootMod</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X84D3F03B862841F8">15.4-4 RootsMod</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X81F856E682A8ECBA">15.4-5 RootsUnityMod</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X7B3A5A0378A32F83">15.5 <span class="Heading">Multiplicative Arithmetic Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X823707DF821E79A0">15.5-1 Sigma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X798C62847EE0372E">15.5-2 Tau</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X79C1DA36827C2959">15.5-3 MoebiusMu</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X7B2E061C835159B9">15.6 <span class="Heading">Continued Fractions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X874C161B83416092">15.6-1 ContinuedFractionExpansionOfRoot</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X8059667580A039A6">15.6-2 ContinuedFractionApproximationOfRoot</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X7C5563A37D566DA5">15.7 <span class="Heading">Miscellaneous</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X8243EAA586D78ED4">15.7-1 PValuation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap15.html#X85E1EFC484F648A4">15.7-2 TwoSquares</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap16.html#X7BDA99EE7CEADA7C">16 <span class="Heading">Combinatorics</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap16.html#X800E48927D5C83F5">16.1 <span class="Heading">Combinatorial Numbers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X87665F748594BF29">16.1-1 Factorial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7A9AF5F58682819D">16.1-2 Binomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7DC5667580522BDA">16.1-3 Bell</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X792FF6EA786A5C2B">16.1-4 Bernoulli</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X85037456785BB33C">16.1-5 Stirling1</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7C93E14D7BC360F0">16.1-6 Stirling2</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap16.html#X81B4696585C38147">16.2 <span class="Heading">Combinations, Arrangements and Tuples</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X8770F16D794C0ADB">16.2-1 Combinations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X78DD5C0D81057540">16.2-2 <span class="Heading">Iterator and enumerator of combinations</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X82A6E98C85714FD0">16.2-3 NrCombinations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7837B3357C7566C8">16.2-4 Arrangements</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7DE1ABD47D19F140">16.2-5 NrArrangements</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X81601C6786120DDC">16.2-6 UnorderedTuples</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7959281584C42C52">16.2-7 NrUnorderedTuples</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X86A3CA0F7CC8C320">16.2-8 Tuples</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7BA135297E8DA819">16.2-9 EnumeratorOfTuples</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X86416A31807B0086">16.2-10 IteratorOfTuples</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X85E18A9A87FD4CA2">16.2-11 NrTuples</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7B0143FB83F359B7">16.2-12 PermutationsList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X8629A2908050EB3A">16.2-13 NrPermutationsList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X79C159507B2BF1C9">16.2-14 Derangements</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7C1741B181A9AB9C">16.2-15 NrDerangements</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7A13D8DC8204525F">16.2-16 PartitionsSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7BCD7FC2876386F1">16.2-17 NrPartitionsSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X84A6D15F8107008B">16.2-18 Partitions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X8793AEBD7E529E1D">16.2-19 IteratorOfPartitions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7EBD746A8607D0B8">16.2-20 IteratorOfPartitionsSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X86933C4F795C4EBD">16.2-21 NrPartitions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X820DF201871F2723">16.2-22 OrderedPartitions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X80BB9F4982CA1E8B">16.2-23 NrOrderedPartitions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X8009520C82942461">16.2-24 PartitionsGreatestLE</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7CB8D4FF8592A9BB">16.2-25 PartitionsGreatestEQ</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7A70D4F3809494E7">16.2-26 RestrictedPartitions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X800B43838742FBF4">16.2-27 NrRestrictedPartitions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7F4EDCCA780B469D">16.2-28 SignPartition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7DB9BEB6856EC03D">16.2-29 AssociatedPartition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7A95D8A6820363A8">16.2-30 PowerPartition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X877D997B7F66A119">16.2-31 PartitionTuples</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7F44AD098561DE32">16.2-32 NrPartitionTuples</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X8796C1D783ED9CB4">16.2-33 BetaSet</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap16.html#X83DC50B67D74E674">16.3 <span class="Heading">Fibonacci and Lucas Sequences</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X85AE1D70803A886C">16.3-1 Fibonacci</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7830A03181D67192">16.3-2 Lucas</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap16.html#X821888E77EB43F67">16.4 <span class="Heading">Permanent of a Matrix</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7F0942DD83BBAB7A">16.4-1 Permanent</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap17.html#X87003045878E74DF">17 <span class="Heading">Rational Numbers</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap17.html#X7A76497986DA921F">17.1 <span class="Heading">Rationals: Global Variables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap17.html#X7B6029D18570C08A">17.1-1 Rationals</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap17.html#X826E2AA88679B3DF">17.2 <span class="Heading">Elementary Operations for Rationals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap17.html#X7ED018F5794935F7">17.2-1 IsRat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap17.html#X7BD6E170840F045D">17.2-2 IsPosRat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap17.html#X81179AC87AC951A8">17.2-3 IsNegRat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap17.html#X7D830E7482E7F528">17.2-4 NumeratorRat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap17.html#X81F6B5877A81E727">17.2-5 DenominatorRat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap17.html#X7EB4C646806A2BDE">17.2-6 Rat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap17.html#X7C8F8693825C28A4">17.2-7 Random</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap18.html#X7DFC03C187DE4841">18 <span class="Heading">Cyclotomic Numbers</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap18.html#X79E25C3085AA568F">18.1 <span class="Heading">Operations for Cyclotomics</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X8631458886314588">18.1-1 E</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X863D1E017BC9EB7F">18.1-2 Cyclotomics</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X841C425281A6F775">18.1-3 IsCyclotomic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X869750DA81EA0E67">18.1-4 IsIntegralCyclotomic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X7DD6B95F79321D23">18.1-5 Int</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X7CBA6CB678E2B143">18.1-6 String</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X815D6EC57CBA9827">18.1-7 Conductor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X81DD58BB81FB3426">18.1-8 AbsoluteValue</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X7808ECF37AA9004D">18.1-9 RoundCyc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X7AE2933985BE4C3E">18.1-10 CoeffsCyc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X803478CA7D2D830F">18.1-11 DenominatorCyc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X785F2CAB805DE1BE">18.1-12 ExtRepOfObj</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X7DDD51B983D5BC44">18.1-13 DescriptionOfRootOfUnity</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X8712419182ECD8DD">18.1-14 IsGaussInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X7E6CF4947D0A56F7">18.1-15 IsGaussRat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X7FE3D5637B5485D0">18.1-16 DefaultField</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap18.html#X7EE5FB7181125E02">18.2 <span class="Heading">Infinity and negative Infinity</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X8511B8DF83324C27">18.2-1 IsInfinity</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap18.html#X7F66A62384329705">18.3 <span class="Heading">Comparisons of Cyclotomics</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap18.html#X7B242083873DD74F">18.4 <span class="Heading">ATLAS Irrationalities</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X8414ED887AF36359">18.4-1 <span class="Heading">EB, EC, <span class="SimpleMath">...</span>, EH</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X813CF4327C4B4D29">18.4-2 <span class="Heading">EI and ER</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X8672D7F986CBA116">18.4-3 <span class="Heading">EY, EX, <span class="SimpleMath">...</span>, ES</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X7E5985FC846C5201">18.4-4 <span class="Heading">EM, EL, <span class="SimpleMath">...</span>, EJ</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X844F0EBF849EDEB3">18.4-5 NK</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X812E334E7A869D33">18.4-6 AtlasIrrationality</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap18.html#X79FE34337DF2CD10">18.5 <span class="Heading">Galois Conjugacy of Cyclotomics</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X79EE9097783128C4">18.5-1 GaloisCyc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X7BE001A0811CD599">18.5-2 ComplexConjugate</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X7E361C057E97CA66">18.5-3 StarCyc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X84438F867B0CC299">18.5-4 Quadratic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X7DDDEC3F80543B7D">18.5-5 GaloisMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X7BB9F5957AA8C082">18.5-6 RationalizedMat</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap18.html#X8557FC2D7ACD6105">18.6 <span class="Heading">Internally Represented Cyclotomics</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap18.html#X7D3028777DE39709">18.6-1 SetCyclotomicsLimit</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap19.html#X81AA901181CA568F">19 <span class="Heading">Floats</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap19.html#X7B4092CA7ABB93B0">19.1 <span class="Heading">A sample run</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap19.html#X8606FDCE878850EF">19.2 <span class="Heading">Methods</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X86D5EA93813FB6C4">19.2-1 <span class="Heading">Float creators</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X7BCD34DC7B5A0521">19.2-2 Rat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X7D1EAE11844625F4">19.2-3 Cyc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X7A962B0983FA66E8">19.2-4 SetFloats</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X819050BF8403806E">19.2-5 FLOAT</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X7BD96E0585D5A1EE">19.2-6 EqFloat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X7B3133497DDE839B">19.2-7 PrecisionFloat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X801753137949DD78">19.2-8 SignBit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X7935C65D7B0F47C7">19.2-9 SinCos</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X846E1196844B9E11">19.2-10 Atan2</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X7981510D826EE3E5">19.2-11 Log1p</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X86073E147FB3C0EA">19.2-12 Erf</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X7E03FDEE824D1E8E">19.2-13 <span class="Heading">Infinity testers</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X8151581186F75BA3">19.2-14 <span class="Heading">Standard mathematical operations</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap19.html#X845ACF3A78BD2771">19.3 <span class="Heading">High-precision-specific methods</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap19.html#X7E8F6EFB87A65F78">19.4 <span class="Heading">Complex arithmetic</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X7B0269D983F96677">19.4-1 Argument</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap19.html#X7E57B09C80136484">19.5 <span class="Heading">Interval-specific methods</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X7C34D1D185802F2F">19.5-1 Sup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X78F1E457814FD1FD">19.5-2 Inf</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X829581A485F55996">19.5-3 Mid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X7FE540B387B0012C">19.5-4 AbsoluteDiameter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X7CA771757F441592">19.5-5 RelativeDiameter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X86D22AE57E2D84B2">19.5-6 IsDisjoint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X7A5E0C3E79837EB8">19.5-7 IsSubset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X85191E1679936CE9">19.5-8 IncreaseInterval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X879EE14282DD1539">19.5-9 BlowupInterval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19.html#X7EC15DAE7CBBB42E">19.5-10 BisectInterval</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap20.html#X787B4AB77A2F5E14">20 <span class="Heading">Booleans</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap20.html#X87F9AF65832E7AD2">20.1 <span class="Heading">IsBool (Filter)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap20.html#X7D58580284CF7894">20.1-1 IsBool</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap20.html#X85E648AA8414F303">20.2 <span class="Heading">Fail (Variable)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap20.html#X8294AAC9860E87E5">20.2-1 fail</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap20.html#X862F17B68465B399">20.3 <span class="Heading">Comparisons of Booleans</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap20.html#X79305F9780394190">20.3-1 <span class="Heading">Equality and inequality of Booleans</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap20.html#X7FEF019482AF5923">20.3-2 <span class="Heading">Ordering of Booleans</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap20.html#X79AD41A185FD7213">20.4 <span class="Heading">Operations for Booleans</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap20.html#X7DFE7E518088AA89">20.4-1 <span class="Heading">Logical disjunction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap20.html#X7A64D25F804973CD">20.4-2 <span class="Heading">Logical conjunction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap20.html#X84F5034185D7EC3C">20.4-3 <span class="Heading">Logical negation</span></a>
</span>
</div></div>
</div>
<div class="ContChap"><a href="chap21.html#X7B256AE5780F140A">21 <span class="Heading">Lists</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X86B28F5B781FFD31">21.1 <span class="Heading">List Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7C4CC4EA8299701E">21.1-1 IsList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X870AA9D8798C93DD">21.1-2 IsDenseList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7C71596C82B6EF35">21.1-3 IsHomogeneousList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X80872FAF80EB5DF9">21.1-4 IsTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X79581E0387F7F7A9">21.1-5 IsRectangularTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7C84E16A85C99C8C">21.1-6 IsConstantTimeAccessList</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X7B202D147A5C2884">21.2 <span class="Heading">Basic Operations for Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X8297BBCD79642BE6"><code>21.2-1 <span>\</span>[<span>\</span>]</code></a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X7921047F83F5FA28">21.3 <span class="Heading">List Elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X78791B8B838A8BA0"><code>21.3-1 \{\}</code></a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X8611EF768210625B">21.4 <span class="Heading">List Assignment</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X813FF1637F8D2B7F"><code>21.4-1 \{\}\:\=</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X795EC9D67E34DAB0">21.4-2 Add</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7E98B11B79BA9167">21.4-3 Remove</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X79D7E96F80A2D7C0">21.4-4 CopyListEntries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X79E31DB27C82D6E1">21.4-5 Append</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X7963C8E17EFF86DB">21.5 <span class="Heading">IsBound and Unbind for Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X79EC565A7DCEC938">21.5-1 IsBound</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X866F45D3797FDA00">21.5-2 GetWithDefault</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X78B72FDF7BD63C0B">21.5-3 Unbind</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X7DD65BEA7EDB0CD7">21.6 <span class="Heading">Identical Lists</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X7ED7C0738495556F">21.7 <span class="Heading">Duplication of Lists</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X808A207182B2F84F">21.8 <span class="Heading">Membership Test for Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7B914A287F88ED0A"><code>21.8-1 \in</code></a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X84D6FC7E7E39ED33">21.9 <span class="Heading">Enlarging Internally Represented Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X78BF67A5802E93AD">21.9-1 EmptyPlist</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X8016D50F85147A77">21.10 <span class="Heading">Comparisons of Lists</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X845EEAF083D43CCE">21.11 <span class="Heading">Arithmetic for Lists</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X84D642967B8546B7">21.12 <span class="Heading">Filters Controlling the Arithmetic Behaviour of Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X87ABCEE9809585A0">21.12-1 IsGeneralizedRowVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7FBCA5B58308C158">21.12-2 IsMultiplicativeGeneralizedRowVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7BAD12E67BFC90DE">21.12-3 IsListDefault</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X8428E77B86722D52">21.12-4 NestingDepthA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X84B383B97FD986CD">21.12-5 NestingDepthM</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X7E6A1F66781BE923">21.13 <span class="Heading">Additive Arithmetic for Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X86A85ADC85C451DC">21.13-1 <span class="Heading">Zero for lists</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7B91CE4D814C2D08">21.13-2 <span class="Heading">AdditiveInverse for lists</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X842D123E7EE5E3DB">21.13-3 <span class="Heading">Addition of lists</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7C3DC8BE78DEECDE">21.13-4 <span class="Heading">Subtraction of lists</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X782ED7F27D8C7FC1">21.14 <span class="Heading">Multiplicative Arithmetic for Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X79A8A5627FD42FA5">21.14-1 <span class="Heading">One for lists</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X78C6C1E2849D303A">21.14-2 <span class="Heading">Inverse for lists</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X84FDB95179BFE4CD">21.14-3 <span class="Heading">Multiplication of lists</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X82EA2A5B786181C7">21.14-4 <span class="Heading">Division of lists</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7A0FD70C80B95C00">21.14-5 <span class="Heading">mod for lists</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X84BB2DFB8432A1A4">21.14-6 <span class="Heading">Left quotients of lists</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X8676EFE67972FD06">21.15 <span class="Heading">Mutability Status and List Arithmetic</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X80FDB1457FF582E7">21.15-1 ListWithIdenticalEntries</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X8196FD4779BCCA0C">21.16 <span class="Heading">Finding Positions in Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X79975EC6783B4293">21.16-1 Position</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7FA9648883AE1B88">21.16-2 Positions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7B4B10AE81602D4E">21.16-3 PositionCanonical</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7D2B25B484591506">21.16-4 PositionNthOccurrence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7A122E848464E534">21.16-5 PositionSorted</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X820BA44D85930EBF">21.16-6 PositionSortedBy</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X78BFE9D78347C0DA">21.16-7 PositionSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7FD9C1D37F300206">21.16-8 PositionMaximum</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7E6C763A82C6153B">21.16-9 PositionProperty</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7DA94D278304EC3D">21.16-10 PositionsProperty</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X86C9E5C3863B3C03">21.16-11 PositionBound</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X819F71047AABEA2F">21.16-12 PositionsBound</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X865EF45D87ED1384">21.16-13 PositionNot</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7F42E5AD87EC9D5A">21.16-14 PositionNonZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X87A8C62A867D6DA4">21.16-15 PositionSublist</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X7865747A7CCF5812">21.17 <span class="Heading">Properties and Attributes for Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X83F8EC7C7BF27EFC">21.17-1 IsMatchingSublist</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7FA892828252BB3B">21.17-2 IsDuplicateFree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7BAA9B0E81D4A884">21.17-3 IsSortedList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X80CDAF45782E8DCB">21.17-4 IsSSortedList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X780769238600AFD1">21.17-5 Length</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7B55FB967CDEF468">21.17-6 ConstantTimeAccessList</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X83E558E37D1B44D4">21.18 <span class="Heading">Sorting Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7FE4975F8166884D">21.18-1 Sort</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X791F2B2C7E9B9A46">21.18-2 SortParallel</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X87287FCA81E2B06A">21.18-3 Sortex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X800209E881E7CECB">21.18-4 SortingPerm</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X80ABC25582343910">21.19 <span class="Heading">Sorted Lists and Sets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7B16AD597CB12305"><code>21.19-1 \in</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7B4C0FEE7CDF6F2A">21.19-2 IsEqualSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X79B940567A849216">21.19-3 IsSubsetSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X832C23CC7FCD8892">21.19-4 AddSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7FCA282E789A4F4B">21.19-5 RemoveSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7B3469CD7EFC1A87">21.19-6 UniteSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X8473AA657FEC3D4D">21.19-7 IntersectSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X80B427537EB07D09">21.19-8 SubtractSet</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X7DF510F7848CBBFD">21.20 <span class="Heading">Operations for Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X840C55A77D1BB2E1">21.20-1 Concatenation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7CB0A6AF87C7FAF7">21.20-2 Compacted</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7ECE9056792F28BA">21.20-3 Collected</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X8727F2928467C2F9">21.20-4 DuplicateFreeList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7F5D4DD87E4378AC">21.20-5 AsDuplicateFreeList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7FA272D984EF82ED">21.20-6 Flat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7C4FDB007C3F54A1">21.20-7 Reversed</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X8057372F83374193">21.20-8 Shuffle</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X8075FBDE7B81B4C8">21.20-9 Apply</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7EF6E2BC81DBF6FB">21.20-10 Perform</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X8763882A7D65F979">21.20-11 PermListList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X82CE0DE8828E4303">21.20-12 <span class="Heading">Maximum</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X82F133EC7F89665F">21.20-13 <span class="Heading">Minimum</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X842851EB7E0969F7">21.20-14 <span class="Heading">MaximumList and MinimumList</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7E1593B979BDF2CD">21.20-15 <span class="Heading">Cartesian</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7E76F5A782184823">21.20-16 <span class="Heading">IteratorOfCartesianProduct</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7B5A19098406347A">21.20-17 Permuted</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X86CB7DCE8510F977">21.20-18 List</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7C86D7F7795125F0">21.20-19 Filtered</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X8179B13D80E935FC">21.20-20 Number</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X82801DFA84E11272">21.20-21 First</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7E5B62E780421CE9">21.20-22 Last</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7F06961278166671">21.20-23 ForAll</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7AF82E747A8BDA75">21.20-24 ForAny</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7E5C72F27B657948">21.20-25 Product</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7A04B71C84CFCC2D">21.20-26 Sum</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X834E4DF57F3A20F0">21.20-27 Iterated</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7D150C2881881139">21.20-28 ListN</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X805CA0B68029B47A">21.21 <span class="Heading">Advanced List Manipulations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X8258477D7F72171B">21.21-1 ListX</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7AC321B87A2DCAF5">21.21-2 SetX</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X82B1411E7FBE925F">21.21-3 SumX</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7FB318B47D8783DA">21.21-4 ProductX</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X79596BDE7CAF8491">21.22 <span class="Heading">Ranges</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X86DDC2FF7A50FBEE">21.22-1 IsRange</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X83896BC481536B07">21.22-2 IsRangeRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7D22B2298167A58F">21.22-3 ConvertToRangeRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X7EA3ACE27E43D174">21.23 <span class="Heading">Enumerators</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X7BB462C17962647F">21.23-1 IsQuickPositionList</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap21.html#X81ECC2077D88E112">21.24 <span class="Heading">Plain Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X8438CB908367254C">21.24-1 PlainListCopy</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap21.html#X87BA4EBF80F16B72">21.24-2 IsPlistRep</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap22.html#X7AC531DD79B6938E">22 <span class="Heading">Boolean Lists</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap22.html#X7E7832B0804221AE">22.1 <span class="Heading">IsBlist (Filter)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X7BE078187A08DCEA">22.1-1 IsBlist</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap22.html#X7CC745317FE54C14">22.2 <span class="Heading">Boolean Lists Representing Subsets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X7C597B2D87CA2E6E">22.2-1 BlistList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X874BEF63785AB439">22.2-2 ListBlist</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X85AD5EF77EFD7451">22.2-3 SizeBlist</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X7BA42D03796ED4B3">22.2-4 IsSubsetBlist</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap22.html#X8100080382AECFF9">22.3 <span class="Heading">Set Operations via Boolean Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X7970BD3883C42D91">22.3-1 <span class="Heading">UnionBlist</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X86E1F8DE85E1EE1E">22.3-2 <span class="Heading">IntersectionBlist</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X7D6FC2C58725708C">22.3-3 DifferenceBlist</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap22.html#X8634D25D7B4C6151">22.4 <span class="Heading">Function that Modify Boolean Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X79815EB77CC8A389">22.4-1 UniteBlist</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X7C86C8D3853BE5EB">22.4-2 UniteBlistList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X84EB70D37EB275DF">22.4-3 IntersectBlist</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X7AA138407D5A3BAC">22.4-4 SubtractBlist</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X830645EC846B2E3C">22.4-5 MeetBlist</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X7F14FF35786DAEF3">22.4-6 FlipBlist</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X7E9F6C197A79098F">22.4-7 SetAllBlist</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X87ED45A88688AE8E">22.4-8 ClearAllBlist</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap22.html#X7C71B225841DFC0F">22.5 <span class="Heading">More about Boolean Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap22.html#X8453ADDA810B4C03">22.5-1 IsBlistRep</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap23.html#X82C7E6CF7BA03391">23 <span class="Heading">Row Vectors</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap23.html#X7E383689817D2371">23.1 <span class="Heading">IsRowVector (Filter)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X7DFB22A07836A7A9">23.1-1 IsRowVector</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap23.html#X85516C3179C229DB">23.2 <span class="Heading">Operators for Row Vectors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X785DC60D8482695D">23.2-1 NormedRowVector</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap23.html#X8679F7DD7DFCBD9C">23.3 <span class="Heading">Row Vectors over Finite Fields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X810E46927F9E8F75">23.3-1 <span class="Heading">ConvertToVectorRep</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X83D8F5BB80089279">23.3-2 ImmutableVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X872E17FF829DB50F">23.3-3 NumberFFVector</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap23.html#X85C68AED805E4B9C">23.4 <span class="Heading">Coefficient List Arithmetic</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X800EC03F7E0A5F23">23.4-1 AddVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X7854B2B67E3FE2CA">23.4-2 AddCoeffs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X7BEF28C981C42E16">23.4-3 MultVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X8264B3EE7D56EEDD">23.4-4 CoeffsMod</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap23.html#X7D287281781E16A2">23.5 <span class="Heading">Shifting and Trimming Coefficient Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X80465E9B7A38C176">23.5-1 LeftShiftRowVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X822CCA4781D5C5EC">23.5-2 RightShiftRowVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X78951C0E86D857B5">23.5-3 ShrinkRowVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X85796B6079581023">23.5-4 RemoveOuterCoeffs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap23.html#X7B63F1EB83FA0CF6">23.6 <span class="Heading">Functions for Coding Theory</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X7C9F4D657F9BA5A1">23.6-1 WeightVecFFE</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X85AA5C6587559C1C">23.6-2 DistanceVecFFE</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X7F2F630984A9D3D6">23.6-3 DistancesDistributionVecFFEsVecFFE</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X85135CEB86E61D49">23.6-4 DistancesDistributionMatFFEVecFFE</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X82E5987E81487D18">23.6-5 AClosestVectorCombinationsMatFFEVecFFE</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X7C88671678A2BEB4">23.6-6 CosetLeadersMatFFE</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap23.html#X87FEC1927B3A63C8">23.7 <span class="Heading">Vectors as coefficients of polynomials</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X84DE99D57C29D47F">23.7-1 ValuePol</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X8328088C807AFFAF">23.7-2 ProductCoeffs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X87248AA27F05BDCC">23.7-3 ReduceCoeffs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X7F74B1637CB13B7B">23.7-4 ReduceCoeffsMod</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X825F8F357FB1BF56">23.7-5 PowerModCoeffs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23.html#X833EF7AE80CE8B3C">23.7-6 ShiftedCoeffs</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap24.html#X812CCAB278643A59">24 <span class="Heading">Matrices</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X801E1B5D7EC8DDD3">24.1 <span class="Heading">InfoMatrix (Info Class)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X78EC82D27B4191DA">24.1-1 InfoMatrix</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X866E55A58164FAED">24.2 <span class="Heading">Categories of Matrices</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7E1AE46B862B185F">24.2-1 IsMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7CF42B8A845BC6A9">24.2-2 IsOrdinaryMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X86EC33E17DD12D0E">24.2-3 IsLieMatrix</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X7899335779A39A95">24.3 <span class="Heading">Operators for Matrices</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X7F5AD28E869B66CB">24.4 <span class="Heading">Properties and Attributes of Matrices</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X83A9DC2085D3A972">24.4-1 DimensionsMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X80AE547B8095A5CB">24.4-2 DefaultFieldOfMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X784EC2777C06AFE4">24.4-3 TraceMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X8488D69A7ADDB4E2">24.4-4 DeterminantMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X824B5DC2875118B3">24.4-5 DeterminantMatrixDestructive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X80693FAB7D541804">24.4-6 DeterminantMatrixDivFree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7F8D25897EC1630B">24.4-7 IsEmptyMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X848B80437CE65FF3">24.4-8 IsMonomialMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7EEC8E768178696E">24.4-9 IsDiagonalMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X8740E71C799C0BCC">24.4-10 IsUpperTriangularMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X853A5B988306DBFE">24.4-11 IsLowerTriangularMatrix</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X823FB2398697B957">24.5 <span class="Heading">Matrix Constructions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7DB902CE848D1524">24.5-1 IdentityMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X86D343A77D9B3D4D">24.5-2 NullMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X8508A7EA812BA0CC">24.5-3 EmptyMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X81042E7A7F247ADE">24.5-4 DiagonalMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X87BADF217C19CBE1">24.5-5 DiagonalMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X806C62A67A7D5379">24.5-6 PermutationMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7C52A38C79C36C35">24.5-7 TransposedMatImmutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7DBB40847E2B6252">24.5-8 TransposedMatDestructive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X8634C79E7DB22934">24.5-9 KroneckerProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X845EC4D18054D140">24.5-10 ReflectionMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7DEBC9967DFDFC18">24.5-11 PrintArray</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X79CC5F568252D341">24.6 <span class="Heading">Random Matrices</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7F957F0280A87961">24.6-1 RandomMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7C939B4A7EDF015D">24.6-2 RandomInvertibleMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X84743732846ACB44">24.6-3 RandomUnimodularMat</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X85485DCE809E323A">24.7 <span class="Heading">Matrices Representing Linear Equations and the Gaussian Algorithm</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7A995A74838950E6">24.7-1 RankMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7BA26C3387AB434E">24.7-2 TriangulizedMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X8384CA8E7B3850D3">24.7-3 TriangulizeMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7DA0D5887DB12DC4">24.7-4 NullspaceMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X87684B0F7AB7B7DB">24.7-5 NullspaceMatDestructive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X838A519C7CD2969E">24.7-6 SolutionMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7A7880D27CE7C1FE">24.7-7 SolutionMatDestructive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7AB5AC547809F999">24.7-8 BaseFixedSpace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X871FCAA97C60B2BA">24.8 <span class="Heading">Eigenvectors and eigenvalues</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7A2462CC7B0C9D66">24.8-1 GeneralisedEigenvalues</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X845CA0457D65876D">24.8-2 GeneralisedEigenspaces</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X8413C6FB7CEE9D59">24.8-3 Eigenvalues</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7A6B047281B52FD7">24.8-4 Eigenspaces</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X8506584579D4EA18">24.8-5 Eigenvectors</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X7E5405D085661B29">24.9 <span class="Heading">Elementary Divisors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7AC4D74F81908109">24.9-1 ElementaryDivisorsMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7AA1C9047B102204">24.9-2 ElementaryDivisorsTransformationsMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X85819D3F7A582180">24.9-3 DiagonalizeMat</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X7CA6B51D7AE3172B">24.10 <span class="Heading">Echelonized Matrices</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7D5D6BD07B7E981B">24.10-1 SemiEchelonMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X8251F6F57D346385">24.10-2 SemiEchelonMatDestructive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7EFD1DB5861A54F0">24.10-3 SemiEchelonMatTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X827D7971800DB661">24.10-4 SemiEchelonMats</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X808F493B839BC7A6">24.10-5 SemiEchelonMatsDestructive</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X86B0D4A886BC0C6E">24.11 <span class="Heading">Matrices as Basis of a Row Space</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7AD6B5F5794D9E46">24.11-1 BaseMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X78B094597E382A5F">24.11-2 BaseMatDestructive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X78B94EFF87A455BE">24.11-3 BaseOrthogonalSpaceMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7AFF8BCF80C88B45">24.11-4 SumIntersectionMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X8245D54F7AC532EB">24.11-5 BaseSteinitzVectors</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X79D5E53685F0FBEE">24.12 <span class="Heading">Triangular Matrices</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7A9139D686ACB7D8">24.12-1 DiagonalOfMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X84A78C057F9DAE5E">24.12-2 UpperSubdiagonal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X84D74DEA798A9094">24.12-3 DepthOfUpperTriangularMatrix</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X85B403857F2855F7">24.13 <span class="Heading">Matrices as Linear Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X87FA0A727CDB060B">24.13-1 CharacteristicPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7B52560C792C1A0F">24.13-2 RationalCanonicalFormTransform</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X83F55D4E79BA5D1B">24.13-3 JordanDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X85923C107A4569D0">24.13-4 BlownUpMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X82AC277D84EC5749">24.13-5 BlownUpVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7E06762479A00DF4">24.13-6 CompanionMatrix</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X873822B6830CE367">24.14 <span class="Heading">Matrices over Finite Fields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7DED2522828B6C30">24.14-1 ImmutableMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X8587A62F818AA0D6">24.14-2 ConvertToMatrixRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X84A76F7A7B4166BC">24.14-3 ProjectiveOrder</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X847ADC6779E33A1C">24.14-4 SimultaneousEigenvalues</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X8593A5337D3B2C70">24.15 <span class="Heading">Inverse and Nullspace of an Integer Matrix Modulo an Ideal</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7D8D1E0E83C7F872">24.15-1 InverseMatMod</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7D7DF873826A7C20">24.15-2 BasisNullspaceModN</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X86AE919983B242E2">24.15-3 NullspaceModQ</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X787DF5F07DC7D86E">24.16 <span class="Heading">Special Multiplication Algorithms for Matrices over GF(2)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7C0C26027FAE0C83">24.16-1 PROD_GF2MAT_GF2MAT_SIMPLE</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X81965B7D7F45E088">24.16-2 PROD_GF2MAT_GF2MAT_ADVANCED</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X7F8A71F38201A250">24.17 <span class="Heading">Block Matrices</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7D675B3C79CF8871">24.17-1 AsBlockMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X8633538685551E7A">24.17-2 BlockMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X83FAF4158180041F">24.17-3 MatrixByBlockMatrix</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X782F2EBF80C431D0">24.18 <span class="Heading">Linear Programming</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X845D5F8D7D905CB8">24.18-1 SimplexMethod</a></span>
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</div>
<div class="ContChap"><a href="chap25.html#X8414F20D8412DDA4">25 <span class="Heading">Integral matrices and lattices</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap25.html#X786A64B983339767">25.1 <span class="Heading">Linear equations over the integers and Integral Matrices</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X792315717F5B0294">25.1-1 NullspaceIntMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X7D749F317DBD1E69">25.1-2 SolutionIntMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X82CECB6E7D515CD2">25.1-3 SolutionNullspaceIntMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X7F66E8EA7D1AA2C1">25.1-4 BaseIntMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X8771349D865C9179">25.1-5 BaseIntersectionIntMats</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X7848EF9F83D491C1">25.1-6 ComplementIntMat</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap25.html#X8143C1448069D846">25.2 <span class="Heading">Normal Forms over the Integers</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X783CEC847D81F22A">25.2-1 TriangulizedIntegerMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X7DBE174E8625AFA5">25.2-2 TriangulizedIntegerMatTransform</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X78CD40A687FE2311">25.2-3 TriangulizeIntegerMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X8535AC327932B89F">25.2-4 HermiteNormalFormIntegerMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X7FDA78F979574ACC">25.2-5 HermiteNormalFormIntegerMatTransform</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X87089FEC7FBEEA8F">25.2-6 SmithNormalFormIntegerMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X839C1F9E87273A93">25.2-7 SmithNormalFormIntegerMatTransforms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X80EF38737F6D61DB">25.2-8 DiagonalizeIntMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X81FB746E82BE6CDA">25.2-9 NormalFormIntMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X8221694D7C99197A">25.2-10 AbelianInvariantsOfList</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap25.html#X80F6990983C979FB">25.3 <span class="Heading">Determinant of an integer matrix</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X787599E087F4C0BA">25.3-1 DeterminantIntMat</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap25.html#X79F2EFEC7C3EA80C">25.4 <span class="Heading">Decompositions</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X7911A60384C511AB">25.4-1 Decomposition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X843A976787600F13">25.4-2 LinearIndependentColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X8285776B7DD86925">25.4-3 PadicCoefficients</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X7F5C619B7A9C3EB9">25.4-4 IntegralizedMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X8512FB69824AE353">25.4-5 DecompositionInt</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap25.html#X839C6ABE829355F4">25.5 <span class="Heading">Lattice Reduction</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X7D0FCEF8859E8637">25.5-1 LLLReducedBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X86D23EB885EDE60E">25.5-2 LLLReducedGramMat</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap25.html#X871DB00B803D5177">25.6 <span class="Heading">Orthogonal Embeddings</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X842280C2808FF05D">25.6-1 OrthogonalEmbeddings</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap25.html#X79A692B6819353D4">25.6-2 ShortestVectors</a></span>
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<div class="ContChap"><a href="chap26.html#X856C23B87E50F118">26 <span class="Heading">Vector and Matrix Objects</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X7A7275C27EC61ACE">26.1 <span class="Heading">Concepts and Rules for Vector and Matrix Objects</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X7C6CDBFE7EB083A5">26.2 <span class="Heading">Categories of Vector and Matrix Objects</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7D963FCC7E849BE0">26.2-1 IsVectorObj</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7E7617A0781D1E4B">26.2-2 IsMatrixObj</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7D1ACCBE7E9CF501">26.2-3 IsMatrixOrMatrixObj</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X78CD88A283330E72">26.2-4 IsRowListMatrix</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X877A706186C89ADB">26.3 <span class="Heading">Defining Attributes of Vector and Matrix Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X8662026C7CCDB446">26.3-1 <span class="Heading">BaseDomain</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X85ABF33684865ED5">26.3-2 <span class="Heading">ConstructingFilter</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X818702FD7A2E9D90">26.3-3 <span class="Heading">CompatibleVectorFilter</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X828BA5E1849E3D06">26.3-4 Length</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X820ED34380C10E19">26.3-5 <span class="Heading">NumberRows and NumberColumns</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X7BD7D2837BFDE649">26.4 <span class="Heading">Constructing Vector and Matrix Objects</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X860E84397BD148E9">26.4-1 <span class="Heading">NewVector and NewZeroVector</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X79A6544D86261E82">26.4-2 <span class="Heading">Vector</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7DBA8BF5844F3281">26.4-3 <span class="Heading">ZeroVector</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7AD2210B8047FB01">26.4-4 <span class="Heading">NewMatrix, NewZeroMatrix, NewIdentityMatrix</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X879384D479EB1D82">26.4-5 <span class="Heading">Matrix</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X838F5B6C7C87C8E1">26.4-6 <span class="Heading">ZeroMatrix</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7D807ABC7FCB4E77">26.4-7 <span class="Heading">IdentityMatrix</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X7C7F5250855C4371">26.5 <span class="Heading">Operations for Base Domains of Vector and Matrix Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X85D7A6A782B21E5C">26.5-1 <span class="Heading">OneOfBaseDomain and ZeroOfBaseDomain</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X7954E20987E0B260">26.6 <span class="Heading">Operations for Vector and Matrix Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7FFC60A27FE6FA97">26.6-1 <span class="Heading">Comparison of Vector and Matrix Objects</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7FBBE79478012648">26.6-2 <span class="Heading">Unpack</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X85E896F67CE2F925">26.6-3 <span class="Heading">ChangedBaseDomain</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X83DD8B39864A2C94">26.6-4 <span class="Heading">Randomize</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X7FE662477F36A21F">26.7 <span class="Heading">List Like Operations for Vector Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7D5DF49C7ADB6986">26.7-1 <span class="Heading">Element Access and Assignment for Vector Objects</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7A21731C83EE3BB0">26.7-2 PositionNonZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7ABDE1B685A78326">26.7-3 PositionLastNonZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X790013817E314B2D">26.7-4 ListOp</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X7FDF7655852AEAAE">26.8 <span class="Heading">Arithmetical Operations for Vector Objects</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7F8CE23F7A250072">26.8-1 <span class="Heading">Unary Arithmetical Operations for Vector Objects</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X85A815CA790094CC">26.8-2 <span class="Heading">Binary Arithmetical Operations for Vector Objects</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X876090A684E71C93">26.8-3 AddVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X8039D013817317C3">26.8-4 MultVector</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X7BE9D278852C13BC">26.9 <span class="Heading">Operations for Vector Objects</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7AC470557EC90714">26.9-1 <span class="Heading">ConcatenationOfVectors</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7DBE956E7F9C700E">26.9-2 ExtractSubVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X80EC354D78D7B5A6">26.9-3 CopySubVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X866366E587991171">26.9-4 WeightOfVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X81ACAE017C00F782">26.9-5 DistanceOfVectors</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X81CC13CA7A1FF4AA">26.10 <span class="Heading">Arithmetical Operations for Matrix Objects</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X819F87A07DA7E2DC">26.10-1 <span class="Heading">Unary Arithmetical Operations for Matrix Objects</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7BBB70557A7A9591">26.10-2 <span class="Heading">Binary Arithmetical Operations for Matrix Objects</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X85FAB7E778A71C19">26.11 <span class="Heading">Operations for Matrix Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X870FBE817C884AB5">26.11-1 MatElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7C33059984635480">26.11-2 SetMatElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X838B45F7790E9FDF">26.11-3 ExtractSubMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X793CD4637F237915">26.11-4 MutableCopyMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7ED9E5D4809E3B50">26.11-5 CopySubMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X809A6B3F7EA5E7D8">26.11-6 CompatibleVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7EE70D5A81E9ED72">26.11-7 RowsOfMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7E06762479A00DF4">26.11-8 <span class="Heading">CompanionMatrix</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X7D40EE2084A6C976">26.12 <span class="Heading">Operations for Row List Matrix Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X82C4FCFA808010F8">26.12-1 <span class="Heading">List Access for a Row List Matrix</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7F89BB2482D28AAE">26.12-2 <span class="Heading">List Assignment for a Row List Matrix</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X807518367C96516F">26.12-3 <span class="Heading">Sublist Access for a Row List Matrix</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X8371789181FA136B">26.12-4 <span class="Heading">Sublist Assignment for a Row List Matrix</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X872E63867803ED78"><code>26.12-5 IsBound<span>\</span>[<span>\</span>]</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X79328CB280C71DDB"><code>26.12-6 Unbind<span>\</span>[<span>\</span>]</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7BDD838579E4D2D6">26.12-7 Add</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X86E355D07A41C025">26.12-8 Remove</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X82D0359B81F8D442">26.12-9 Append</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7E234F717BE333EA">26.12-10 ShallowCopy</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7E9F095E85DED480">26.12-11 ListOp</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X7B86A8487B12F9BD">26.13 <span class="Heading">Basic operations for row/column reductions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7B3997D37CC44FCA">26.13-1 MultMatrixRowLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X794636447E8C5553">26.13-2 MultMatrixRowRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X80AF7B267E6B9CE0">26.13-3 MultMatrixColumnRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X843DAFE37F347471">26.13-4 MultMatrixColumnLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X8662EB748629502F">26.13-5 AddMatrixRowsLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7CD05EE984614AB6">26.13-6 AddMatrixRowsRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7B1E1E417CA267A3">26.13-7 AddMatrixColumnsRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X85ECB8C87DFD8F32">26.13-8 AddMatrixColumnsLeft</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X87CCA3117F6B3F0D">26.13-9 SwapMatrixRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X824C8A347EB9D499">26.13-10 SwapMatrixColumns</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X7BEE647484978886">26.14 <span class="Heading">Implementing New Vector and Matrix Objects Types</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X82EEE1D37A94F807">26.15 <span class="Heading">Available Representations of Vector Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7C8050938691A914">26.15-1 IsGF2VectorRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X82A643007EC6D1CA">26.15-2 Is8BitVectorRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X83262B7085FA94E3">26.15-3 IsPlistVectorRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X8730DB7D7E7DA883">26.15-4 IsZmodnZVectorRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap26.html#X7CFD844C7D80D541">26.16 <span class="Heading">Available Representations of Matrix Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X7F6078FF81E912E7">26.16-1 IsGF2MatrixRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X81466B6C7CAC3A7B">26.16-2 Is8BitMatrixRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X80C6031C7DB31A15">26.16-3 IsPlistMatrixRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap26.html#X84D0F3117DA86850">26.16-4 IsZmodnZMatrixRep</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap27.html#X7D28329B7EDB8F47">27 <span class="Heading">Strings and Characters</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap27.html#X7A90690B78260194">27.1 <span class="Heading">IsChar and IsString</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X80CFAE128560E064">27.1-1 IsChar</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X78723B5D795A3B6D">27.1-2 IsString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7B1B45C587A72F96">27.1-3 <span class="Heading">Strings As Lists</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7EA6CA7486D7E9DD">27.1-4 <span class="Heading">Printing Strings</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap27.html#X82E5F5AB818F32DB">27.2 <span class="Heading">Special Characters</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap27.html#X7E70384E7D0B7083">27.3 <span class="Heading">Triple Quoted Strings</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap27.html#X82AEC07487C45ECD">27.4 <span class="Heading">Internally Represented Strings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7A17EDF8785C9F58">27.4-1 IsStringRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7CE2415F7FEC5809">27.4-2 ConvertToStringRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7FFC464683CC8023">27.4-3 CopyToStringRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7D944D507CBB24CD">27.4-4 IsEmptyString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X836078DC829A8221">27.4-5 EmptyString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7DA671FC7F490C16">27.4-6 CharsFamily</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap27.html#X82F980A17FE84AA4">27.5 <span class="Heading">Recognizing Characters</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X78566FD57B95ECBE">27.5-1 IsDigitChar</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X854114A97BAFEAEA">27.5-2 IsLowerAlphaChar</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X87B1A13D81353AD8">27.5-3 IsUpperAlphaChar</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X84634DF67A431D26">27.5-4 IsAlphaChar</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap27.html#X8127954B79B8A0DA">27.6 <span class="Heading">Comparisons of Strings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X79538F138286739A"><code>27.6-1 \=</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X8129E3A785F60093"><code>27.6-2 \<</code></a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap27.html#X7E72717A82A309F5">27.7 <span class="Heading">Operations to Produce or Manipulate Strings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X792FB3A1849FD739">27.7-1 DisplayString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X8482132779EA7A23">27.7-2 DEFAULTDISPLAYSTRING</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7803FBCA79DB5529">27.7-3 ViewString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7BBDF9D383595425">27.7-4 DEFAULTVIEWSTRING</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7B3CC87285DEC23D">27.7-5 PrintString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X81FB5BE27903EC32">27.7-6 String</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X86AACCE987F74FA5">27.7-7 StripLineBreakCharacters</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X865FBB7E788017DD">27.7-8 HexStringInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7BB1059185AB4F84">27.7-9 StringPP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X79C8280A853D8FA9">27.7-10 WordAlp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X798A0F35852ABDAD">27.7-11 LowercaseString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X87A2F2557DE7EE08">27.7-12 LowercaseChar</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7E7E5F5B7FED56A0">27.7-13 UppercaseString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X81E0AEE687200505">27.7-14 UppercaseChar</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X86E897D486DCFEAB">27.7-15 SplitString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X864F0A9078D4DE0E">27.7-16 ReplacedString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X806379367A53D171">27.7-17 NormalizeWhitespace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X8685DE9386E57771">27.7-18 NormalizedWhitespace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X86EBB6EB829723E4">27.7-19 RemoveCharacters</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X84624FEB825EC4B5">27.7-20 JoinStringsWithSeparator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X79F8FFC5876D854A">27.7-21 Chomp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X855820848179CC28">27.7-22 StartsWith</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X8235AD797868E872">27.7-23 StringFormatted</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7848A9D878FD59BB">27.7-24 NumbersString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X787EAB117816578E">27.7-25 StringNumbers</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X82975B6480932683">27.7-26 StringOfMemoryAmount</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap27.html#X844BDC8578A3B508">27.8 <span class="Heading">Character Conversion</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X826D95D680F87D23">27.8-1 IntChar</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X87B6C1AF7E4A6639">27.8-2 CharInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X8159CE81798DDA76">27.8-3 SIntChar</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X78E6611A829DDA3E">27.8-4 CharSInt</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap27.html#X78D9BD857F890C0A">27.9 <span class="Heading">Operations to Evaluate Strings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7B6D118184F692A0">27.9-1 Int</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X87AD395584294FF2">27.9-2 Rat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X796D366B7DDEFF67">27.9-3 IntHexString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7C0C29C87CBA97B7">27.9-4 Ordinal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7DE4CCD285440659">27.9-5 EvalString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7D4B9D7A7995C55D">27.9-6 CrcString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7873D1F28779B490">27.9-7 HexSHA256</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X83FF17E782E6FFF3">27.9-8 Pluralize</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap27.html#X78F20AA1804D524F">27.10 <span class="Heading">Calendar Arithmetic</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X87BA46787FF000E8">27.10-1 DaysInYear</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X8791B0B386D59ADB">27.10-2 DaysInMonth</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7CED84C07CD5E2CF">27.10-3 DMYDay</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7A79DEE07A41B8EF">27.10-4 DayDMY</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X87D03FC0809DB6EC">27.10-5 WeekDay</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7C74C33784CDED6C">27.10-6 StringDate</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X84A6A2637FB35A32">27.10-7 HMSMSec</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X879461D77C81100B">27.10-8 SecHMSM</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X802469C47F886A59">27.10-9 StringTime</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X870A71D47B0E936E">27.10-10 SecondsDMYhms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X78AF8EA887532B5B">27.10-11 DMYhmsSeconds</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap27.html#X78024C8087F3E07F">27.11 <span class="Heading">Obtaining LaTeX Representations of Objects</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap28.html#X867203C5877489A2">28 <span class="Heading">Dictionaries and General Hash Tables</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap28.html#X81560C4083E27955">28.1 <span class="Heading">Using Dictionaries</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap28.html#X7B571EA282AF70D7">28.2 <span class="Heading">Dictionaries</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap28.html#X7E78E3E983A5C895">28.2-1 NewDictionary</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap28.html#X86BD015B7B889329">28.3 <span class="Heading">Dictionaries via Binary Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap28.html#X865D5BE1830A448D">28.3-1 DictionaryByPosition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap28.html#X87F7247E784021C2">28.3-2 IsDictionary</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap28.html#X7D776BC67ABDDCCE">28.3-3 IsLookupDictionary</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap28.html#X86C4F0507AD98B8A">28.3-4 AddDictionary</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap28.html#X808C885D7E267285">28.3-5 KnowsDictionary</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap28.html#X863706BF847A47EB">28.3-6 LookupDictionary</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap28.html#X8444087381BBA88A">28.4 <span class="Heading">General Hash Tables</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap28.html#X85CD6C9B85DE7C54">28.5 <span class="Heading">Hash keys</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap28.html#X79DEEB5783513838">28.5-1 DenseIntKey</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap28.html#X87FC10AC81E5F6BA">28.5-2 SparseIntKey</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap28.html#X84D1A83C8247E7FB">28.6 <span class="Heading">Dense hash tables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap28.html#X874FAA447930C7DA">28.6-1 DenseHashTable</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap28.html#X7FDB74417A19E674">28.7 <span class="Heading">Sparse hash tables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap28.html#X8757D3A785290640">28.7-1 SparseHashTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap28.html#X80FDDF957887B4FC">28.7-2 DoubleHashArraySize</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap29.html#X7AA1073C7E943DD7">29 <span class="Heading">Records</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap29.html#X864F92347B5A3FF0">29.1 <span class="Heading">IsRecord and RecNames</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap29.html#X782A998E7D9EC406">29.1-1 IsRecord</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap29.html#X837F1E1F866FB1A0">29.1-2 RecNames</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap29.html#X7EAAE25D7A17F778">29.2 <span class="Heading">Accessing Record Elements</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap29.html#X806DE3BD78742CA4">29.3 <span class="Heading">Record Assignment</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap29.html#X86BC2672803863FB">29.4 <span class="Heading">Identical Records</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap29.html#X83A7E6607B1D63BC">29.5 <span class="Heading">Comparisons of Records</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap29.html#X79BE8D0E829E7ACE">29.6 <span class="Heading">IsBound and Unbind for Records</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap29.html#X7A13E8F87CAAA0AF">29.6-1 IsBound</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap29.html#X7CA9AEFE7DB71604">29.6-2 Unbind</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap29.html#X784897E180815EDA">29.7 <span class="Heading">Record Access Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap29.html#X87BF90FA7F7A3B1B">29.7-1 NameRNam</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap29.html#X78199B6B84A017B9">29.7-2 RNamObj</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap29.html#X7821AC097821AC09"><code>29.7-3 \.</code></a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap30.html#X8050A8037984E5B6">30 <span class="Heading">Collections</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X8084F03A78ABD4F8">30.1 <span class="Heading">IsCollection (Filter)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X79C9FC7F86E2738C">30.1-1 IsCollection</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X85D8D8F684B02DDF">30.2 <span class="Heading">Collection Families</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X84E5A67E87D8DD66">30.2-1 CollectionsFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X856AC2DF7F7CBAAF">30.2-2 IsCollectionFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X864BB3748546F63F">30.2-3 ElementsFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X78C38017804B2EA7">30.2-4 CategoryCollections</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X85ABC1B4829778C7">30.2-5 DeclareCategoryCollections</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X7C3722DF8736FFDB">30.3 <span class="Heading">Lists and Collections</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X877128A77826DD69">30.3-1 IsListOrCollection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X7EF8910F82B45EC7">30.3-2 Enumerator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X80CD7DDC7D0C60D5">30.3-3 EnumeratorSorted</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X85E149177AC547C3">30.3-4 EnumeratorByFunctions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X7F12F40E87F3C3A7">30.3-5 List</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X82CE157A7FAD8036">30.3-6 SortedList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X7E399AC97FD98217">30.3-7 SSortedList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X8289FCCC8274C89D">30.3-8 AsList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X7BCA5C6181391007">30.3-9 AsSortedList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X856D927378C33548">30.3-10 AsSSortedList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X79B130FC7906FB4C">30.3-11 Elements</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X79AD18737E70B414">30.4 <span class="Heading">Attributes and Properties for Collections</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X7969C48780C5C1BC">30.4-1 IsEmpty</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X808A4061809A6E67">30.4-2 IsFinite</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X7E3402D6799D3C24">30.4-3 IsTrivial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X7F192373850B85B9">30.4-4 IsNonTrivial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X78EF6A137E8F66B0">30.4-5 IsWholeFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X858ADA3B7A684421">30.4-6 Size</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X865507568182424E">30.4-7 Representative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X8026085680270D37">30.4-8 RepresentativeSmallest</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X7F8FEA3278239ADE">30.5 <span class="Heading">Operations for Collections</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X79CA175481F8105F">30.5-1 IsSubset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X851069107CACF98E">30.5-2 <span class="Heading">Intersection</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X799F0E2F7A502DBA">30.5-3 <span class="Heading">Union</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X825AC0F07E010B07">30.5-4 Difference</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X82D39CF980FDBFFA">30.6 <span class="Heading">Membership Test for Collections</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X84B7FA8C7C94400F"><code>30.6-1 \in</code></a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X8151A51884B7EE2C">30.7 <span class="Heading">Random Elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X7FF906E57D6936F8">30.7-1 Random</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X811B5BD47DC5356B">30.7-2 PseudoRandom</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X7EBA01EB83BC65A9">30.7-3 RandomList</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X85A3F00985453F95">30.8 <span class="Heading">Iterators</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X83ADF8287ED0668E">30.8-1 Iterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X8688C20B828FC129">30.8-2 IteratorSorted</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X87168A827E5B28E4">30.8-3 IsIterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X8055FC557B5D899E">30.8-4 IsDoneIterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X879F62F77D1D1179">30.8-5 NextIterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X858A28667D137C4B">30.8-6 IteratorList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X7DB80BE68271247E">30.8-7 TrivialIterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X82677D8F817D6701">30.8-8 IteratorByFunctions</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap31.html#X7E651AC287AFDCC1">31 <span class="Heading">Domains and their Elements</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap31.html#X859C7AB97B34F55F">31.1 <span class="Heading">Operational Structure of Domains</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap31.html#X84FA03F87A17B059">31.2 <span class="Heading">Equality and Comparison of Domains</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap31.html#X82039A218274826F">31.3 <span class="Heading">Constructing Domains</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap31.html#X7EA77DE17DD8A231">31.4 <span class="Heading">Changing the Structure</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap31.html#X860FCCBE7A41412F">31.5 <span class="Heading">Changing the Representation</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap31.html#X7D72F11B82F4A036">31.6 <span class="Heading">Domain Categories</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap31.html#X7CBDD36E7B7BE286">31.7 <span class="Heading">Parents</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7BC856CC7F116BB0">31.7-1 Parent</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap31.html#X7B58FDEF80338DD6">31.8 <span class="Heading">Constructing Subdomains</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap31.html#X86D579707B112970">31.9 <span class="Heading">Operations for Domains</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X86B4AC017FAF4D12">31.9-1 IsGeneralizedDomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7E353DD1838AB223">31.9-2 GeneratorsOfDomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X826A21287FD3ACC0">31.9-3 Domain</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap31.html#X7C2B0C1280237CB0">31.10 <span class="Heading">Attributes and Properties of Elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X81278E53800BF64D">31.10-1 Characteristic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X8046262384895B2A">31.10-2 OneImmutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X8040AC7A79FFC442">31.10-3 ZeroImmutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X86DEB543824C40EB">31.10-4 MultiplicativeZeroOp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X814D78347858EC13">31.10-5 IsOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X82BDA47282F9BBA7">31.10-6 IsZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7CB5896082D29173">31.10-7 IsIdempotent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X78EE524E83624057">31.10-8 InverseImmutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X84BB723C81D55D63">31.10-9 AdditiveInverseImmutable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X84F59A2687C62763">31.10-10 Order</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap31.html#X7B3BC7BA7BB2646D">31.11 <span class="Heading">Comparison Operations for Elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7EF67D047F03CA6F">31.11-1 <span class="Heading">\= and \<</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7EFE013B8634D214">31.11-2 CanEasilyCompareElements</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap31.html#X7A2914307963E370">31.12 <span class="Heading">Arithmetic Operations for Elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X8481C9B97B214C23">31.12-1 <span class="Heading">\+, \*, \/, \^, \mod</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7A37082878DB3930">31.12-2 LeftQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X80761843831B468E">31.12-3 Comm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X86A62A937A42B82E">31.12-4 LieBracket</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7E8F1FB87C229BB0">31.12-5 Sqrt</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap31.html#X80A2D8A7874B268B">31.13 <span class="Heading">Relations Between Domains</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7C03098C838ADE40">31.13-1 UseSubsetRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X78039B628262BFA8">31.13-2 UseFactorRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X839BE6467E8474D9">31.13-3 UseIsomorphismRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X863C35007C7AA914">31.13-4 InstallSubsetMaintenance</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7BB7EE5078EF6F47">31.13-5 InstallFactorMaintenance</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X79F97F0F78D89186">31.13-6 InstallIsomorphismMaintenance</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap31.html#X7B97A0307EA161E5">31.14 <span class="Heading">Useful Categories of Elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7FBD4F65861C2DF2">31.14-1 IsExtAElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7F346AA47AEC39AB">31.14-2 IsNearAdditiveElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X78D042B486E1D7F7">31.14-3 IsAdditiveElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7CE2353F836F6E0A">31.14-4 IsNearAdditiveElementWithZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X87F3552A789C572D">31.14-5 IsAdditiveElementWithZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X84B0929982B51CB4">31.14-6 IsNearAdditiveElementWithInverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7C0E4AE883947778">31.14-7 IsAdditiveElementWithInverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X860D1E387DD5CCCF">31.14-8 IsExtLElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X809E0C097E480AF1">31.14-9 IsExtRElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X797D3B2A7A2B2F53">31.14-10 IsMultiplicativeElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X82BC294F7D388AE8">31.14-11 IsMultiplicativeElementWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X8703BFC2841BBD63">31.14-12 IsMultiplicativeElementWithZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7FDB14E57814FA3B">31.14-13 IsMultiplicativeElementWithInverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X802F34F280B29DF4">31.14-14 IsVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X799AEDE180C31276">31.14-15 IsNearRingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X84BF40CA86C07361">31.14-16 IsRingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7C724689784EEF3D">31.14-17 IsNearRingElementWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X875B67208017608E">31.14-18 IsRingElementWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X80CD04ED85B6B2F9">31.14-19 IsNearRingElementWithInverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X8113834E84FD0435">31.14-20 IsRingElementWithInverse</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap31.html#X7ABEF00C870789D2">31.15 <span class="Heading">Useful Categories for all Elements of a Family</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7979AFAA80FF795A">31.15-1 IsAssociativeElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X78A286418205CE44">31.15-2 IsAdditivelyCommutativeElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X8137FA8D86714AC0">31.15-3 IsCommutativeElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X810D2E5E832594AA">31.15-4 IsFiniteOrderElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X796957D0805A0221">31.15-5 IsJacobianElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap31.html#X7844399D7847AB24">31.15-6 IsZeroSquaredElement</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap32.html#X7C9734B880042C73">32 <span class="Heading">Mappings</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X859A13548515A5D7">32.1 <span class="Heading">Direct Products and their Elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X87FD9FE787023FF0">32.1-1 IsDirectProductElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X78F8A1168280E06D">32.1-2 DirectProductFamily</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X7CF6FEFB8290D5CB">32.2 <span class="Heading">Creating Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X79D0D2F07A14D039">32.2-1 GeneralMappingByElements</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7D55E1977ED70E01">32.2-2 <span class="Heading">MappingByFunction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X865FC25A87D36F3D">32.2-3 InverseGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7BD2D5A87CD6B213">32.2-4 RestrictedInverseGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7ED1E4E27CCE2DCA">32.2-5 CompositionMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X86486B687B7077AC">32.2-6 CompositionMapping2</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7A926D167C3155F6">32.2-7 IsCompositionMappingRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X87775B438008DCA5">32.2-8 ConstituentsCompositionMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X795FF8DC785F110A">32.2-9 ZeroMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7EBAE0368470A603">32.2-10 IdentityMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X86452F8587CBAEA0">32.2-11 <span class="Heading">Embedding</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X8769E8DA80BC96C1">32.2-12 <span class="Heading">Projection</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X800014D683A81009">32.2-13 RestrictedMapping</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X7E5A430D7F838F1C">32.3 <span class="Heading">Properties and Attributes of (General) Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X83C7494E828CC9C8">32.3-1 IsTotal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X86D44C8A78BF1981">32.3-2 IsSingleValued</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7CC95EB282854385">32.3-3 IsMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7F065FD7822C0A12">32.3-4 IsInjective</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X784ECE847E005B8F">32.3-5 IsSurjective</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X878F56AB7B342767">32.3-6 IsBijective</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7B6FD7277CDE9FCB">32.3-7 Range</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7DE8173F80E07AB1">32.3-8 Source</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X784F871383FB599B">32.3-9 UnderlyingRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X786581DE871A47D0">32.3-10 UnderlyingGeneralMapping</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X83B4FF15847F06FC">32.4 <span class="Heading">Images under Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7D23C1CE863DACD8">32.4-1 ImagesSource</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X85ADB89B7C8DD7D0">32.4-2 ImagesRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7D51184B7EE5B2CF">32.4-3 ImagesElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X8781348F7F5796A0">32.4-4 ImagesSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7CFAB0157BFB1806">32.4-5 ImageElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X87F4D35A826599C6">32.4-6 <span class="Heading">Image</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X86114B2E7E77488C">32.4-7 <span class="Heading">Images</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X79BB1EC07C828667">32.5 <span class="Heading">Preimages under Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X78EF1FE77B0973C0">32.5-1 PreImagesRange</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7FBB830C8729E995">32.5-2 PreImagesElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7D212F727CAE971A">32.5-3 PreImageElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7AE24A1586B7DE79">32.5-4 PreImagesRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X856BAFC87B2D2811">32.5-5 PreImagesSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X836FAEAC78B55BF4">32.5-6 <span class="Heading">PreImage</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X85C8590E832002EF">32.5-7 <span class="Heading">PreImages</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X7E2E16277940FA0B">32.6 <span class="Heading">Arithmetic Operations for General Mappings</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X834E02BB7D4B4AE5">32.7 <span class="Heading">Mappings which are Compatible with Algebraic Structures</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X8008FCCC7F4C731F">32.8 <span class="Heading">Magma Homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7DC72CF28539A251">32.8-1 IsMagmaHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X8181676787E760A2">32.8-2 MagmaHomomorphismByFunctionNC</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X79D0216E871B7051">32.8-3 NaturalHomomorphismByGenerators</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X806F892C862F29F9">32.9 <span class="Heading">Mappings that Respect Multiplication</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7BEFF95883EAEC78">32.9-1 RespectsMultiplication</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7EE4DA097AE9CBC1">32.9-2 RespectsOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7F27AE9C84A4DF90">32.9-3 RespectsInverses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X819DD174829BF3AE">32.9-4 IsGroupGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X81A5A5CF846E5FBF">32.9-5 KernelOfMultiplicativeGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7F09B6E28080DCB4">32.9-6 CoKernelOfMultiplicativeGeneralMapping</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X8455A5A67C35178B">32.10 <span class="Heading">Mappings that Respect Addition</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7A3321E878925C3A">32.10-1 RespectsAddition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X8130D8907B92F746">32.10-2 RespectsAdditiveInverses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7D342736781EB280">32.10-3 RespectsZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7B99EF287A8A0BD9">32.10-4 IsAdditiveGroupGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7EC0E9907D6631D6">32.10-5 KernelOfAdditiveGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X813C6D7980213F41">32.10-6 CoKernelOfAdditiveGeneralMapping</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X7C24431C81532575">32.11 <span class="Heading">Linear Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X87842ED97FA19973">32.11-1 RespectsScalarMultiplication</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X780BE6307A3271A9">32.11-2 IsLeftModuleGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7F6841107E59107F">32.11-3 IsLinearMapping</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X7E88C32A82E942DA">32.12 <span class="Heading">Ring Homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7C8DA031799B79D5">32.12-1 IsRingGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7988102883675606">32.12-2 IsRingWithOneGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X86B14F908601DEA9">32.12-3 IsAlgebraGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X842AD44679C5BDC2">32.12-4 IsAlgebraWithOneGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X8324DA78879DF4D7">32.12-5 IsFieldHomomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X7E4A55567BED0F88">32.13 <span class="Heading">General Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X8656AB8A7D672CAE">32.13-1 IsGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X791690817E23D90C">32.13-2 IsConstantTimeAccessGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X81CFF5F87BBEA8AD">32.13-3 IsEndoGeneralMapping</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap32.html#X7D6F78587C00CDD0">32.14 <span class="Heading">Technical Matters Concerning General Mappings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7D28581F82481163">32.14-1 IsSPGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X80D02AD183E01F16">32.14-2 IsGeneralMappingFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X86CFADBA7F2FE446">32.14-3 FamilyRange</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7C3736E281A9E505">32.14-4 FamilySource</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7AE54FB67E2E6374">32.14-5 FamiliesOfGeneralMappingsAndRanges</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7E1E26E37C413F6F">32.14-6 GeneralMappingsFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap32.html#X7CF92CC37A6BBDA5">32.14-7 TypeOfDefaultGeneralMapping</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap33.html#X838651287FCCEFD8">33 <span class="Heading">Relations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap33.html#X7DED7F1F78D31785">33.1 <span class="Heading">General Binary Relations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X788D722F82165551">33.1-1 IsBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X7A1D8EEF8034B0B5">33.1-2 BinaryRelationByElements</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X81878EEF873B34D5">33.1-3 <span class="Heading">IdentityBinaryRelation</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X80DDCDD387BA23F2">33.1-4 EmptyBinaryRelation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap33.html#X7899E59181C46EBB">33.2 <span class="Heading">Properties and Attributes of Binary Relations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X79D69B667F5FE8FE">33.2-1 IsReflexiveBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X785916A181555368">33.2-2 IsSymmetricBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X7823942478124563">33.2-3 IsTransitiveBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X870F72C38550A0A4">33.2-4 IsAntisymmetricBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X782B7C8A8136532F">33.2-5 IsPreOrderBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X7A1228207AB4FBA3">33.2-6 IsPartialOrderBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X80D3735C84D1CDC2">33.2-7 IsHasseDiagram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X82D6CB4B7CCE9E25">33.2-8 IsEquivalenceRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X85E2FD8B82652876">33.2-9 Successors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X7B4D22A17E752A91">33.2-10 DegreeOfBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X8278E4457C3C3A0D">33.2-11 PartialOrderOfHasseDiagram</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap33.html#X78032F927F078E19">33.3 <span class="Heading">Binary Relations on Points</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X79E40E9385274F89">33.3-1 BinaryRelationOnPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X7D9323C283867515">33.3-2 RandomBinaryRelationOnPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X8315C7A47CEB6BB3">33.3-3 <span class="Heading">AsBinaryRelationOnPoints</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap33.html#X7D9A14AE799142EF">33.4 <span class="Heading">Closure Operations and Other Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X8252B17C864A4904">33.4-1 ReflexiveClosureBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X820811E9785A7274">33.4-2 SymmetricClosureBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X853BFAD9858DCDF7">33.4-3 TransitiveClosureBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X79672B3A7BCB6991">33.4-4 HasseDiagramBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X85C22B3D812957C0">33.4-5 StronglyConnectedComponents</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X86AAE6027A3AEF72">33.4-6 PartialOrderByOrderingFunction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap33.html#X7DAA67338458BB64">33.5 <span class="Heading">Equivalence Relations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X7A44D73D8150266A">33.5-1 EquivalenceRelationByPartition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X82CD1C00810F6042">33.5-2 EquivalenceRelationByRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X7B70215E7E3F9CA4">33.5-3 EquivalenceRelationByPairs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X7C5AA9B97EE42DA5">33.5-4 EquivalenceRelationByProperty</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap33.html#X85A2A8E27AF52769">33.6 <span class="Heading">Attributes of and Operations on Equivalence Relations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X877389B683DD8F1A">33.6-1 EquivalenceRelationPartition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X79DC914C82D7903B">33.6-2 GeneratorsOfEquivalenceRelationPartition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X82BE360381476D92">33.6-3 JoinEquivalenceRelations</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap33.html#X79EE13287DEB11B1">33.7 <span class="Heading">Equivalence Classes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X8424996186DB14EA">33.7-1 IsEquivalenceClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X78F967E77EB16386">33.7-2 EquivalenceClassRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X879439897EF4D728">33.7-3 EquivalenceClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap33.html#X7BB985BA7FD7A82E">33.7-4 EquivalenceClassOfElement</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap34.html#X7E4AAA7382D42361">34 <span class="Heading">Orderings</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap34.html#X79B1262585CE5427">34.1 <span class="Heading">IsOrdering (Filter)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X7EFDF115780934AF">34.1-1 IsOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X85E6445C87283BEC">34.1-2 OrderingsFamily</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap34.html#X85C4CAA784BD7F01">34.2 <span class="Heading">Building new orderings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X78B5D91278EFAFC9">34.2-1 OrderingByLessThanFunctionNC</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X813D5BEB80506CE4">34.2-2 OrderingByLessThanOrEqualFunctionNC</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap34.html#X7F62235B87C20A54">34.3 <span class="Heading">Properties and basic functionality</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X84FA448B7B4DDFDC">34.3-1 IsWellFoundedOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X867AC932843AD921">34.3-2 IsTotalOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X814E5E7D85EDCAC7">34.3-3 IsIncomparableUnder</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X872497B9782B97B4">34.3-4 FamilyForOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X7D08ED6882015BFB">34.3-5 LessThanFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X857E800583E9026D">34.3-6 LessThanOrEqualFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X87F51D737C695D41">34.3-7 IsLessThanUnder</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X8308B7DF7AAF6D9C">34.3-8 IsLessThanOrEqualUnder</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap34.html#X834CD021878745BC">34.4 <span class="Heading">Orderings on families of associative words</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X7C1808AE84B989AE">34.4-1 IsOrderingOnFamilyOfAssocWords</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X8175B8887868F29A">34.4-2 IsTranslationInvariantOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X816CD4BD82D41ED0">34.4-3 IsReductionOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X7B6051C282EA88D5">34.4-4 OrderingOnGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X79B2DEB786680F51">34.4-5 LexicographicOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X802EB44B7E7B1F57">34.4-6 ShortLexOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X7B6ED9327E0A2099">34.4-7 IsShortLexOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X849DD7C6782333D5">34.4-8 WeightLexOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X7C7D7954784F5C73">34.4-9 IsWeightLexOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X7E7FAEA484148947">34.4-10 WeightOfGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X79D1019E7C3E575E">34.4-11 BasicWreathProductOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X7CB765477FBC3383">34.4-12 IsBasicWreathProductOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X7E6DF1B17F53642E">34.4-13 WreathProductOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X7F0EE6E987148C96">34.4-14 IsWreathProductOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap34.html#X7901AA4479EDBE72">34.4-15 LevelsOfGenerators</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap35.html#X873E502F7D21C39C">35 <span class="Heading">Magmas</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap35.html#X7E1248B186E7BB44">35.1 <span class="Heading">Magma Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X87D3F38B7EAB13FA">35.1-1 IsMagma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X86071DE7835F1C7C">35.1-2 IsMagmaWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X83E4903D7FBB2E24">35.1-3 IsMagmaWithInversesIfNonzero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X82CBFF648574B830">35.1-4 IsMagmaWithInverses</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap35.html#X808F1A148398733D">35.2 <span class="Heading">Magma Generation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X839147CF813312D6">35.2-1 Magma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X7854B23286B17321">35.2-2 MagmaWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X7A2B51F67EF4DA28">35.2-3 MagmaWithInverses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X7F629A498383A0AD">35.2-4 MagmaByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X84DABBEB803107EB">35.2-5 MagmaWithOneByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X82C08CFB854E3F1A">35.2-6 MagmaWithInversesByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X8268EAA47E4A3A64">35.2-7 Submagma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X7F295EBC7A9CE87E">35.2-8 SubmagmaWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X79441F1F7A277E28">35.2-9 SubmagmaWithInverses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X84ED076D7E46AB79">35.2-10 AsMagma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X87EEEC018129F0F4">35.2-11 AsSubmagma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X8553F44D8123B2C6">35.2-12 IsMagmaWithZeroAdjoined</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X8620878D7FD98823">35.2-13 InjectionZeroMagma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X7B353674859BF659">35.2-14 UnderlyingInjectionZeroMagma</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap35.html#X782215B982F2F01C">35.3 <span class="Heading">Magmas Defined by Multiplication Tables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X85CD1E7678295CA6">35.3-1 MagmaByMultiplicationTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X865526C881645D65">35.3-2 MagmaWithOneByMultiplicationTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X7EDAFB987EE8A770">35.3-3 MagmaWithInversesByMultiplicationTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X828BED4580D28FB8">35.3-4 MagmaElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X849BDCC27C4C3191">35.3-5 <span class="Heading">MultiplicationTable</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap35.html#X87036FCE868FFEE9">35.4 <span class="Heading">Attributes and Properties for Magmas</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X872E05B478EC20CA">35.4-1 GeneratorsOfMagma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X87DD93EC8061DD81">35.4-2 GeneratorsOfMagmaWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X83A901B1857C8489">35.4-3 GeneratorsOfMagmaWithInverses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X7A2BF4527E08803C">35.4-4 <span class="Heading">Centralizer</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X847ABE6F781C7FE8">35.4-5 Centre</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X7C651C9C78398FFF">35.4-6 Idempotents</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X7C83B5A47FD18FB7">35.4-7 IsAssociative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X857B0E507D745ADB">35.4-8 IsCentral</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X830A4A4C795FBC2D">35.4-9 IsCommutative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X7EE2EA5F7EB7FEC2">35.4-10 MultiplicativeNeutralElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X7B39F93C8136D642">35.4-11 MultiplicativeZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X867DB05A8218FB1E">35.4-12 SquareRoots</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap35.html#X837DA95883CFB985">35.4-13 TrivialSubmagmaWithOne</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap36.html#X7CB0D2F780D15136">36 <span class="Heading">Words</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap36.html#X79AEC832815B9317">36.1 <span class="Heading">Categories of Words and Nonassociative Words</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap36.html#X843F5C3A82239398">36.1-1 IsWord</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap36.html#X804B616579F223D8">36.1-2 IsWordCollection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap36.html#X808FA6F97E16502F">36.1-3 IsNonassocWord</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap36.html#X7F81276C80F690DC">36.1-4 IsNonassocWordCollection</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap36.html#X852C815F85DBE4BD">36.2 <span class="Heading">Comparison of Words</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap36.html#X7CA51DD7874115DF"><code>36.2-1 \=</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap36.html#X82D4C7BE803166D6"><code>36.2-2 \<</code></a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap36.html#X7A60A8E57AF13901">36.3 <span class="Heading">Operations for Words</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap36.html#X7EC17930781D104A">36.3-1 MappedWord</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap36.html#X7F51B17983019D3E">36.4 <span class="Heading">Free Magmas</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap36.html#X7CFFD9027DDD1555">36.4-1 <span class="Heading">FreeMagma</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap36.html#X86DB748080B4A9B9">36.4-2 <span class="Heading">FreeMagmaWithOne</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap36.html#X84C2F9037EEE9CED">36.5 <span class="Heading">External Representation for Nonassociative Words</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap37.html#X78C56A0A87CE380E">37 <span class="Heading">Associative Words</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap37.html#X7AB546CB7B929253">37.1 <span class="Heading">Categories of Associative Words</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7FA8DA728773BA89">37.1-1 IsAssocWord</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap37.html#X82E7EA7F7FD31EC3">37.2 <span class="Heading">Free Groups, Monoids and Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X8215999E835290F0">37.2-1 <span class="Heading">FreeGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X8601654A7C4AF1E7">37.2-2 IsFreeGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X814203E281F3272E">37.2-3 AssignGeneratorVariables</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap37.html#X8405BECB7AC4EB61">37.3 <span class="Heading">Comparison of Associative Words</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X8206153078E97B90"><code>37.3-1 \=</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7BB12B9D7F990899"><code>37.3-2 \<</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X805C519682B0A7ED">37.3-3 IsShortLexLessThanOrEqual</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X84875E08847B39E1">37.3-4 IsBasicWreathLessThanOrEqual</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap37.html#X79AF6C757A3547BD">37.4 <span class="Heading">Operations for Associative Words</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X87CD4C6978A7936A">37.4-1 Length</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7F5ED4357A9C12E6">37.4-2 ExponentSumWord</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X82CC92C17AF6FFA0">37.4-3 Subword</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X8509A0A4851981BB">37.4-4 PositionWord</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X79186218787C224A">37.4-5 <span class="Heading">SubstitutedWord</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X8486BFE1844CFE59">37.4-6 EliminatedWord</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap37.html#X7D357E047ABD2C6B">37.5 <span class="Heading">Operations for Associative Words by their Syllables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X842D0B547CE93CF2">37.5-1 NumberSyllables</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7E91575F848F4526">37.5-2 ExponentSyllable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7F2A8CFD811C73B1">37.5-3 GeneratorSyllable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7B4F7A167E844FA5">37.5-4 SubSyllables</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap37.html#X80A9F39582ED296E">37.6 <span class="Heading">Representations for Associative Words</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7E3612247B3E241B">37.6-1 IsLetterAssocWordRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7E36F7897D82417F">37.6-2 IsLetterWordsFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7C84789D7BB161E9">37.6-3 IsBLetterAssocWordRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X8719E7F27CDA1995">37.6-4 IsBLetterWordsFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7886F8BD83CD8081">37.6-5 IsSyllableAssocWordRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7869716C84EA9D81">37.6-6 IsSyllableWordsFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X83F669828481FC32">37.6-7 Is16BitsFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7BD7647C7B088389">37.6-8 LetterRepAssocWord</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7AC8EC757CFB9A51">37.6-9 AssocWordByLetterRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap37.html#X7934D3D5797102EC">37.7 <span class="Heading">The External Representation for Associative Words</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap37.html#X7DC99E4284093FBB">37.8 <span class="Heading">Straight Line Programs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7F69FF3F7C6694CB">37.8-1 IsStraightLineProgram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7AECA57280DA3195">37.8-2 StraightLineProgram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X81A8AFC47F8E4B91">37.8-3 LinesOfStraightLineProgram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X820A592881D57802">37.8-4 NrInputsOfStraightLineProgram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7847D32B863E822F">37.8-5 ResultOfStraightLineProgram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X8098EAAF7D344466">37.8-6 StringOfResultOfStraightLineProgram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X8274C7948248C053">37.8-7 CompositionOfStraightLinePrograms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7A582FA97C786640">37.8-8 IntegratedStraightLineProgram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7C9CABD17BE4850F">37.8-9 RestrictOutputsOfSLP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7EF202F17DCA5D1C">37.8-10 IntermediateResultOfSLP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X8085CF79856B2889">37.8-11 IntermediateResultOfSLPWithoutOverwrite</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X873244F37FAA717A">37.8-12 IntermediateResultsOfSLPWithoutOverwrite</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X837101F982C35035">37.8-13 ProductOfStraightLinePrograms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X84C83CE98194FD03">37.8-14 SlotUsagePattern</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap37.html#X8188799182D82A92">37.9 <span class="Heading">Straight Line Program Elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X85A5838482944FA5">37.9-1 IsStraightLineProgElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X78889E5B7E1B3BFF">37.9-2 StraightLineProgElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X81BC263A7E45E775">37.9-3 StraightLineProgGens</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7BEAE8AC809B27DC">37.9-4 EvalStraightLineProgElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap37.html#X7D85D1DF84DC68E3">37.9-5 StretchImportantSLPElement</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap38.html#X7CA8FCFD81AA1890">38 <span class="Heading">Rewriting Systems</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap38.html#X8287CBE183EBE5D7">38.1 <span class="Heading">Operations on rewriting systems</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap38.html#X842C0ED87986F7AA">38.1-1 IsRewritingSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap38.html#X833EAA8C86356F42">38.1-2 Rules</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap38.html#X7C38C2EF817F9E0A">38.1-3 OrderOfRewritingSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap38.html#X8340EB2280DE6CCC">38.1-4 ReducedForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap38.html#X8006790B86328CE8">38.1-5 <span class="Heading">IsConfluent</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap38.html#X870A1E1C7FB45A55">38.1-6 ConfluentRws</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap38.html#X8134689C7B576946">38.1-7 IsReduced</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap38.html#X864C82FD7FBA31A6">38.1-8 ReduceRules</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap38.html#X81E6B5CB789A7C3A">38.1-9 AddRule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap38.html#X7FA0B54D7C533DDC">38.1-10 AddRuleReduced</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap38.html#X7BD6299E85561DC3">38.1-11 MakeConfluent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap38.html#X795DC25886007DFE">38.1-12 GeneratorsOfRws</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap38.html#X81B812C778CB1E4E">38.2 <span class="Heading">Operations on elements of the algebra</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap38.html#X81BB38CC793F7CE2">38.2-1 ReducedProduct</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap38.html#X8318649681DF783B">38.3 <span class="Heading">Properties of rewriting systems</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap38.html#X7B647DB77D138A49">38.3-1 IsBuiltFromAdditiveMagmaWithInverses</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap38.html#X7F8B7848851784DF">38.4 <span class="Heading">Rewriting in Groups and Monoids</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap38.html#X8751F8FA7DC989A2">38.5 <span class="Heading">Developing rewriting systems</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap39.html#X8716635F7951801B">39 <span class="Heading">Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X822370B47DEA37B1">39.1 <span class="Heading">Group Elements</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X86A022F9800121F8">39.2 <span class="Heading">Creating Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D7B075385435151">39.2-1 <span class="Heading">Group</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7F81960287F3E32A">39.2-2 GroupByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8589EF9C7B658B94">39.2-3 GroupWithGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X79C44528864044C5">39.2-4 GeneratorsOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A0747F17B50D967">39.2-5 AsGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E4143A08040BB47">39.2-6 ConjugateGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7939B3177BBD61E4">39.2-7 IsGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X845874BA82E1A11F">39.2-8 InfoGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7BA181CA81D785BB">39.3 <span class="Heading">Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7C82AA387A42DCA0">39.3-1 Subgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X842AD37E79CE953E">39.3-2 <span class="Heading">Index (<strong class="pkg">GAP</strong> operation)</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8014135884DCC53E">39.3-3 IndexInWholeGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7904AC9D7E9A3BB7">39.3-4 AsSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7839D8927E778334">39.3-5 IsSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X838186F9836F678C">39.3-6 IsNormal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8390B5117A10CC52">39.3-7 IsCharacteristicSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X84F5464983655590">39.3-8 ConjugateSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D9990EB837075A4">39.3-9 ConjugateSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82ABF80780CC27AF">39.3-10 IsSubnormal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X829766158665FB54">39.3-11 SubgroupByProperty</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E95101F80583E77">39.3-12 SubgroupShell</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7B855B0485C3C6C5">39.4 <span class="Heading">Closures of (Sub)groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D13FC1F8576FFD8">39.4-1 ClosureGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81A20A397C308483">39.4-2 ClosureGroupAddElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82F59F6680D1B0D5">39.4-3 ClosureGroupDefault</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A7AF14A8052F055">39.4-4 ClosureSubgroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7E19F92284F6684E">39.5 <span class="Heading">Expressing Group Elements as Words in Generators</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7FE8A3B08458A1BF">39.5-1 EpimorphismFromFreeGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8357294D7B164106">39.5-2 Factorization</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X871508DD808EB487">39.5-3 GrowthFunctionOfGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X87BF1B887C91CA2E">39.6 <span class="Heading">Structure Descriptions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8199B74B84446971">39.6-1 StructureDescription</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X81002AA87DDBC02F">39.7 <span class="Heading">Cosets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8412ABD57986B9FC">39.7-1 RightCoset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X835F48248571364F">39.7-2 RightCosets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X85884F177B5D98AE">39.7-3 CanonicalRightCosetElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D7625A1861D9DAB">39.7-4 IsRightCoset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X78F4F0D8838F5ABF">39.7-5 IsBiCoset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82F6ABE378B928D1">39.7-6 CosetDecomposition</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X83C723878230D616">39.8 <span class="Heading">Transversals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X85C65D06822E716F">39.8-1 RightTransversal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X78B98B257E981046">39.9 <span class="Heading">Double Cosets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E51ED757D17254B">39.9-1 DoubleCoset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7F53DABD79BA4F72">39.9-2 RepresentativesContainedRightCosets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A5EFABB86E6D4D5">39.9-3 DoubleCosets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X85ED464F878EF24C">39.9-4 IsDoubleCoset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A25B1C886CF8C6A">39.9-5 DoubleCosetRepsAndSizes</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X84AE7EE77E5FB30E">39.9-6 InfoCoset</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7D474F8F87E4E5D9">39.10 <span class="Heading">Conjugacy Classes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7B2F207F7F85F5B8">39.10-1 ConjugacyClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X871B570284BBA685">39.10-2 ConjugacyClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D6ED84C86C2979B">39.10-3 ConjugacyClassesByRandomSearch</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X852B3634789D770E">39.10-4 ConjugacyClassesByOrbits</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8733F87B7E4C9903">39.10-5 NrConjugacyClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7BD2A4427B7FE248">39.10-6 RationalClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81E9EF0A811072E8">39.10-7 RationalClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X877691247DE23386">39.10-8 GaloisGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X83DD148D7DA2ABA9">39.10-9 <span class="Heading">IsConjugate</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81A92F828400FC8A">39.10-10 NthRootsInGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X804F0F037F06E25E">39.11 <span class="Heading">Normal Structure</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X87B5370C7DFD401D">39.11-1 <span class="Heading">Normalizer</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7C4E00297E37AA44">39.11-2 Core</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7CF497C77B1E8938">39.11-3 PCore</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7BDEA0A98720D1BB">39.11-4 NormalClosure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D25E7DC7834A703">39.11-5 NormalIntersection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X811B8A4683DDE1F9">39.11-6 ComplementClassesRepresentatives</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8581F4E77B11C610">39.11-7 InfoComplement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7C39EE3E836D6BC6">39.12 <span class="Heading">Specific and Parametrized Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X829759F67D4247CA">39.12-1 TrivialSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A9A3D5578CE33A0">39.12-2 CommutatorSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7CC17CF179ED7EF2">39.12-3 DerivedSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7B10B58F83DDE56E">39.12-4 CommutatorLength</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X780552B57C30DD8F">39.12-5 FittingSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X788C856C82243274">39.12-6 FrattiniSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81D86CCE84193E4F">39.12-7 PrefrattiniSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X83D5C8B8865C85F1">39.12-8 PerfectResiduum</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8250D99A830DA832">39.12-9 SolvableRadical</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81F647FA83D8854F">39.12-10 Socle</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8440C61080CDAA14">39.12-11 SupersolvableResiduum</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X796DA805853FAC90">39.12-12 PRump</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7FF0BBDD80E8F6BF">39.13 <span class="Heading">Sylow Subgroups and Hall Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7AA351308787544C">39.13-1 SylowSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8605F3FE7A3B8E12">39.13-2 SylowComplement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7EDBA19E828CD584">39.13-3 HallSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X832E8E6B8347B13F">39.13-4 SylowSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X87A245E180D27147">39.13-5 ComplementSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82FE5DFD84F8A3C6">39.13-6 HallSystem</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X87AF37E980382499">39.14 <span class="Heading">Subgroups characterized by prime powers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7F069ACC83DB3374">39.14-1 Omega</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X83DB33747F069ACC">39.14-2 Agemo</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7B75879B8085120A">39.15 <span class="Heading">Group Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7DA27D338374FD28">39.15-1 IsCyclic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X813C952F80E775D4">39.15-2 IsElementaryAbelian</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X87D062608719F2CD">39.15-3 IsNilpotentGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E3056237C6A5D43">39.15-4 NilpotencyClassOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8755147280C84DBB">39.15-5 IsPerfectGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X809C78D5877D31DF">39.15-6 IsSolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D7456077D3D1B86">39.15-7 IsPolycyclicGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7AADF2E88501B9FF">39.15-8 IsSupersolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X83977EB97A8E2290">39.15-9 IsMonomialGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A6685D7819AEC32">39.15-10 IsSimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X78CC9764803601E7">39.15-11 IsAlmostSimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7C1709A986B00F97">39.15-12 IsQuasisimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7C6AA6897C4409AC">39.15-13 <span class="Heading">IsomorphismTypeInfoFiniteSimpleGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8492B05B822AC58C">39.15-14 SimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X839CDD8C7AE39FD6">39.15-15 SimpleGroupsIterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X872E93F586F54FCE">39.15-16 SmallSimpleGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7EB47BF27D8CBF72">39.15-17 AllSmallNonabelianSimpleGroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81E22D07871DF37E">39.15-18 IsFinitelyGeneratedGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8648EDA287829755">39.15-19 IsSubsetLocallyFiniteGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8089F18C810B7E3E">39.15-20 IsPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7F232B3F8261CE25">39.15-21 IsPowerfulPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7ED4A14F7A235617">39.15-22 IsRegularPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X87356BAA7E9E2142">39.15-23 PrimePGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X863434AD7DDE514B">39.15-24 PClassPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X840A4F937ABF15E1">39.15-25 RankPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81130F9A7CFCF6BF">39.15-26 IsPSolvable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X87415A8485FCF510">39.15-27 IsPNilpotent</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7F8264FA796B2B7D">39.16 <span class="Heading">Numerical Group Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X812827937F403300">39.16-1 AbelianInvariants</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D44470C7DA59C1C">39.16-2 Exponent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X843E0CCA8351FDF4">39.16-3 EulerianFunction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7AEDEDF67CFED672">39.17 <span class="Heading">Subgroup Series</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7BDD116F7833800F">39.17-1 ChiefSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7AC93E977AC9ED58">39.17-2 ChiefSeriesThrough</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8724E15F81B51173">39.17-3 ChiefSeriesUnderAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A0E7A8B8495B79D">39.17-4 SubnormalSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81CDCBD67BC98A5A">39.17-5 CompositionSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82C0D0217ACB2042">39.17-6 DisplayCompositionSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A879948834BD889">39.17-7 DerivedSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A9AA1577CEC891F">39.17-8 DerivedLength</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X83F057E5791944D6">39.17-9 <span class="Heading">ElementaryAbelianSeries</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X782BD7A47D6B6503">39.17-10 InvariantElementaryAbelianSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X879D55A67DB42676">39.17-11 LowerCentralSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8428592E8773CD7B">39.17-12 UpperCentralSeriesOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7809B7ED792669F3">39.17-13 PCentralSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82A34BD681F24A94">39.17-14 JenningsSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7C08A8B77EC09CFF">39.17-15 DimensionsLoewyFactors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X84112774812180DD">39.17-16 AscendingChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7C5029EE86D7FC96">39.17-17 IntermediateGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X781661FB78DC83B5">39.17-18 IntermediateSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X783CDAA67BDD8195">39.17-19 StructuralSeriesOfGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X84091B0A7E401E2B">39.18 <span class="Heading">Factor Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X80FC390C7F38A13F">39.18-1 NaturalHomomorphismByNormalSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E6EED0185B27C48">39.18-2 FactorGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7816FA867BF1B8ED">39.18-3 CommutatorFactorGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7BB93B9778C5A0B2">39.18-4 MaximalAbelianQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7FC83E4C783572E7">39.18-5 HasAbelianFactorGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7FAC018680B766B7">39.18-6 HasElementaryAbelianFactorGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X822A3AB27919BC1E">39.18-7 CentralizerModulo</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7D8EFB2F85AA24EE">39.19 <span class="Heading">Sets of Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7DDE67C67E871336">39.19-1 ConjugacyClassSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7C5BBF487977B8CD">39.19-2 IsConjugacyClassSubgroupsRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E986BF48393113A">39.19-3 ConjugacyClassesSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8486C25380853F9B">39.19-4 ConjugacyClassesMaximalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X798BF55C837DB188">39.19-5 MaximalSubgroupClassReps</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X85DAFB7582A88463">39.19-6 LowIndexSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X80399CD4870FFC4B">39.19-7 AllSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X861CD8DA790D81C2">39.19-8 MaximalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X80237A847E24E6CF">39.19-9 NormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82ECAA427C987318">39.19-10 MaximalNormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X86FDD9BA819F5644">39.19-11 MinimalNormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A823C5A810910C3">39.19-12 CharacteristicSubgroups</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7FA267497CFC0550">39.20 <span class="Heading">Subgroup Lattice</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7B104E2C86166188">39.20-1 LatticeSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X78928A3582882BFD">39.20-2 ClassElementLattice</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E5DF287825EE7BA">39.20-3 DotFileLatticeSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X815CDA447C5DB285">39.20-4 MaximalSubgroupsLattice</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8138997C871EDF96">39.20-5 MinimalSupergroupsLattice</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X87BE970D7B18E2C5">39.20-6 LowLayerSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X87FABD5F87AD2568">39.20-7 ContainedConjugates</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X79C3619C849F97B8">39.20-8 ContainingConjugates</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8111F50C798B0D76">39.20-9 MinimalFaithfulPermutationDegree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7BA3484E7AE0A0E1">39.20-10 RepresentativesPerfectSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7B2233D180DF77A1">39.20-11 ConjugacyClassesPerfectSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7BFE573187B4BEF8">39.20-12 Zuppos</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82C12E2C81963B23">39.20-13 InfoLattice</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X85E613D57F28AEFF">39.21 <span class="Heading">Specific Methods for Subgroup Lattice Computations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X86462A567DDBA6BC">39.21-1 LatticeByCyclicExtension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X78918D83835A0EDF">39.21-2 InvariantSubgroupsElementaryAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7AD7804A803910AC">39.21-3 SubgroupsSolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7F60BBB8874DFE40">39.21-4 SizeConsiderFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X833C51BD7E7812C4">39.21-5 ExactSizeConsiderFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A2C774B7CFF3E07">39.21-6 InfoPcSubgroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X79F894537D526B61">39.22 <span class="Heading">Special Generating Sets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82FD78AF7F80A0E2">39.22-1 GeneratorsSmallest</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A258CCF79552198">39.22-2 LargestElementGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X81D15723804771E2">39.22-3 MinimalGeneratingSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X814DBABC878D5232">39.22-4 SmallGeneratingSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7D1574457B152333">39.22-5 IndependentGeneratorsOfAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X86F835DA8264A0CE">39.22-6 IndependentGeneratorExponents</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7CA0B6A27E0BE6B8">39.23 <span class="Heading">1-Cohomology</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X847BEC137A49BAF4">39.23-1 <span class="Heading">OneCocycles</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E6438D5834ACCDA">39.23-2 OneCoboundaries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X80400ABD7F40FAA0">39.23-3 OCOneCocycles</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X811E1CF07DABE924">39.23-4 ComplementClassesRepresentativesEA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8199B1D27D487897">39.23-5 InfoCoh</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X80A4B0F282977074">39.24 <span class="Heading">Schur Covers and Multipliers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7F619DDA7DD6C43B">39.24-1 EpimorphismSchurCover</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7DD1E37987612042">39.24-2 SchurCover</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X792BC39D7CEB1D27">39.24-3 AbelianInvariantsMultiplier</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X819E8AEC835F8CD1">39.24-4 Epicentre</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8739CD4686301A0E">39.24-5 NonabelianExteriorSquare</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E1C8CD77CDB9F71">39.24-6 EpimorphismNonabelianExteriorSquare</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7BF8DB3D8300BB3F">39.24-7 IsCentralFactor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7F4240CD782B6032">39.24-8 <span class="Heading">Covering groups of symmetric groups</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7DDA6BC1824F78FD">39.24-9 BasicSpinRepresentationOfSymmetricGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X844CFFDE80F6AD15">39.24-10 SchurCoverOfSymmetricGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7E0F4896795E34FC">39.24-11 DoubleCoverOfAlternatingGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X7BD95B8D879B73A3">39.25 <span class="Heading">2-Cohomology</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A1EBC3A7AB0D614">39.25-1 TwoCohomologyGeneric</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7A65366879BB3977">39.25-2 FpGroupCocycle</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X865722987E0E19B6">39.26 <span class="Heading">Tests for the Availability of Methods</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X798F13EA810FB215">39.26-1 CanEasilyTestMembership</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7C2A89607BDFD920">39.26-2 CanEasilyComputeWithIndependentGensAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X83245C82835D496C">39.26-3 CanComputeSize</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X8268965487364912">39.26-4 CanComputeSizeAnySubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X82DDE00D82A32083">39.26-5 CanComputeIndex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X7BE7C36B84C23511">39.26-6 CanComputeIsSubset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X87D62C2C7C375E2D">39.26-7 KnowsHowToDecompose</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap39.html#X83A9997586694DC0">39.27 <span class="Heading">Specific functions for Normalizer calculation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap39.html#X84ABCA997D294B36">39.27-1 NormalizerViaRadical</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap40.html#X83702FC27B3C3098">40 <span class="Heading">Group Homomorphisms</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X81A7BB0F7D81B247">40.1 <span class="Heading">Creating Group Homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7F348F497C813BE0">40.1-1 GroupHomomorphismByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7AB15AF5830F2A6B">40.1-2 GroupHomomorphismByImagesNC</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7A59F2C47BD41DC8">40.1-3 GroupGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7BC6C20E7CEDBFC5">40.1-4 <span class="Heading">GroupHomomorphismByFunction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X785AB6057F736344">40.1-5 AsGroupGeneralMappingByImages</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X794043AC7E4FAF9E">40.2 <span class="Heading">Operations for Group Homomorphisms</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X7A121B9E7F78138A">40.3 <span class="Heading">Efficiency of Homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X84CFBB577BAFFD4D">40.3-1 <span class="Heading">Mappings given on generators</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X86C2BE2481FDC8EE">40.3-2 <span class="Heading">Action homomorphisms</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X802C5A887D8A7CC4">40.3-3 <span class="Heading">Mappings given by functions</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X87497C207B7D7511">40.3-4 <span class="Heading">Other operations</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X80B8ABEC7CC20DFB">40.3-5 ImagesSmallestGenerators</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X7BA90DA481A1C6D6">40.4 <span class="Heading">Homomorphism for very large groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X7FFD731684606BC6">40.5 <span class="Heading">Nice Monomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X78849F81804C44B3">40.5-1 IsHandledByNiceMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7965086E82ABCF41">40.5-2 NiceMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7B47BE0983E93A83">40.5-3 NiceObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X8652149F7F291EE3">40.5-4 IsCanonicalNiceMonomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X783030917CB43959">40.6 <span class="Heading">Group Automorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7E52E99487562F3A">40.6-1 ConjugatorIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X79ED68CF8139F08A">40.6-2 ConjugatorAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7E937A947856D9DA">40.6-3 InnerAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7F31FECC7A3D4A8A">40.6-4 IsConjugatorIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X78FE7E307E86525A">40.6-5 ConjugatorOfConjugatorIsomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X79640F3682BDBFC1">40.7 <span class="Heading">Groups of Automorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X87677B0787B4461A">40.7-1 AutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7FC631B786C1DC8B">40.7-2 IsGroupOfAutomorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7B767B9D827EB0FC">40.7-3 AutomorphismDomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7F87D5957D9B991E">40.7-4 IsAutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X8476738A7BF9BADA">40.7-5 InnerAutomorphismsAutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7957AC21782B6C8C">40.7-6 InnerAutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7FC9B6EA7CAADC0A">40.7-7 InducedAutomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X7A8E961C7F1A57B3">40.8 <span class="Heading">Calculating with Group Automorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X85691E8386107403">40.8-1 AssignNiceMonomorphismAutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7C9FB0A57BFF6CC0">40.8-2 NiceMonomorphismAutomGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X81B79CC27F47D429">40.9 <span class="Heading">Searching for Homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7B536A32827788C6">40.9-1 IsomorphismGroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7D0C3D5E864CE954">40.9-2 AllHomomorphismClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X791D12B7845610CE">40.9-3 AllHomomorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X790C261184EEAB94">40.9-4 GQuotients</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X83B417BE7C508DC4">40.9-5 IsomorphicSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7AABA9A27E30BF2B">40.9-6 MorClassLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap40.html#X81FC3CEF85CED3DC">40.10 <span class="Heading">Representations for Group Homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X82B77A5F7F9EDBC7">40.10-1 IsGroupGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X863805187A24B5E3">40.10-2 MappingGeneratorsImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7DFBBAB18126B4D9">40.10-3 IsGroupGeneralMappingByAsGroupGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X78707DF57C3927EB">40.10-4 IsPreimagesByAsGroupGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X83E10338798F552B">40.10-5 IsPermGroupGeneralMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X83DADD9F7CAD829B">40.10-6 IsToPermGroupGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X798E72E77EC85D4A">40.10-7 IsGroupGeneralMappingByPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X86FF63B784FB8F85">40.10-8 IsPcGroupGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X79A853B579B250C0">40.10-9 IsToPcGroupGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X7BE2A2EB80DC5CFF">40.10-10 IsFromFpGroupGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap40.html#X81090C207F4F6423">40.10-11 IsFromFpGroupStdGensGeneralMappingByImages</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap41.html#X87115591851FB7F4">41 <span class="Heading">Group Actions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X83661AFD7B7BD1D9">41.1 <span class="Heading">About Group Actions</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X81B8F9CD868CD953">41.2 <span class="Heading">Basic Actions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7FE417DD837987B4">41.2-1 OnPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7960924D84B5B18F">41.2-2 OnRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X832DF5327ECA0E44">41.2-3 OnLeftInverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X85AA04347CD117F9">41.2-4 OnSets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X832CC5F87EEA4A7E">41.2-5 OnTuples</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X80DAA1D2855B1456">41.2-6 OnPairs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7C10492081D72376">41.2-7 OnSetsSets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7E23686E7A9D3A20">41.2-8 OnSetsDisjointSets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7ADE244E819035FF">41.2-9 OnSetsTuples</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7FF556CD7E6739A9">41.2-10 OnTuplesSets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X844E902382EB4151">41.2-11 OnTuplesTuples</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X86DC2DD5829CAD9A">41.2-12 OnLines</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7FA394D27E721E2B">41.2-13 OnIndeterminates</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7BA8D76586F1F06E">41.2-14 Permuted</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X85124D197F0F9C4D">41.2-15 OnSubspacesByCanonicalBasis</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X82181CA07A5B2056">41.3 <span class="Heading">Action on canonical representatives</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X81E0FF0587C54543">41.4 <span class="Heading">Orbits</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X80E0234E7BD79409">41.4-1 Orbit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X86BCAE17869BBEAA">41.4-2 Orbits</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X86BC8B958123F953">41.4-3 <span class="Heading">OrbitsDomain</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X799910CF832EDC45">41.4-4 OrbitLength</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X8032F73078DF2DDB">41.4-5 <span class="Heading">OrbitLengths</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X8520E2487F7E98AF">41.4-6 <span class="Heading">OrbitLengthsDomain</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X797BD60E7ACEF1B1">41.5 <span class="Heading">Stabilizers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7C34EC437EF598BF">41.5-1 OrbitStabilizer</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X86FB962786397E02">41.5-2 Stabilizer</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X78C3A8568414BC44">41.5-3 OrbitStabilizerAlgorithm</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X7A9389097BAF670D">41.6 <span class="Heading">Elements with Prescribed Images</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X857DC7B085EB0539">41.6-1 RepresentativeAction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X87F73CCA7921DE65">41.7 <span class="Heading">The Permutation Image of an Action</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X78E6A002835288A4">41.7-1 <span class="Heading">ActionHomomorphism</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X85A8E93D786C3C9C">41.7-2 Action</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X86FF54A383B73967">41.7-3 SparseActionHomomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X7FED50ED7ACA5FB2">41.8 <span class="Heading">Action of a group on itself</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X784D417D87F4E58D">41.8-1 FactorCosetAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X8561DEBA79E01ABD">41.8-2 RegularActionHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X835317A7847477D4">41.8-3 AbelianSubfactorAction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X807AA91E841D132B">41.9 <span class="Heading">Permutations Induced by Elements and Cycles</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7807A33381DCAB26">41.9-1 <span class="Heading">Permutation</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X81D4EA42810974A0">41.9-2 PermutationCycle</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X80AF6E0683CA7F14">41.9-3 Cycle</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7F559E897B333758">41.9-4 CycleLength</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7F3B387A7FD8AE5E">41.9-5 Cycles</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X83040A6080C2C6C6">41.9-6 CycleLengths</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X87FDA6838065CDCB">41.9-7 <span class="Heading">CycleIndex</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X850A84618421392A">41.10 <span class="Heading">Tests for Actions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X79B15750851828CB">41.10-1 <span class="Heading">IsTransitive</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X8295D733796B7A37">41.10-2 <span class="Heading">Transitivity</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X8166A6A17C8D6E73">41.10-3 <span class="Heading">RankAction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7B77040F8543CD6E">41.10-4 <span class="Heading">IsSemiRegular</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7CF02C4785F0EAB5">41.10-5 <span class="Heading">IsRegular</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7CB1D74280F92AFC">41.10-6 <span class="Heading">Earns</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X84C19AD68247B760">41.10-7 <span class="Heading">IsPrimitive</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X7E9D3D0B7A9A8572">41.11 <span class="Heading">Block Systems</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X84FE699F85371643">41.11-1 <span class="Heading">Blocks</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X79936EB97AAD1144">41.11-2 <span class="Heading">MaximalBlocks</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7941DB6380B74510">41.11-3 <span class="Heading">RepresentativesMinimalBlocks</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X835658B07B28EF3B">41.11-4 AllBlocks</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap41.html#X7FD3D2D2788709B7">41.12 <span class="Heading">External Sets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X8264C3C479FF0A8B">41.12-1 IsExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7C90F648793E47DD">41.12-2 ExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7B9DB15D80CE28B4">41.12-3 ActingDomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X86153CB087394DC1">41.12-4 FunctionAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X86A0CC1479A5932A">41.12-5 HomeEnumerator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X879DE63C7858453C">41.12-6 IsExternalSubset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X87D1EA1486D86233">41.12-7 ExternalSubset</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7E081F568407317F">41.12-8 IsExternalOrbit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7FB656AE7A066C35">41.12-9 ExternalOrbit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7BAFF02B7D6DF9F2">41.12-10 StabilizerOfExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X867262FA82FDD592">41.12-11 <span class="Heading">ExternalOrbits</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7A64EF807CE8893E">41.12-12 <span class="Heading">ExternalOrbitsStabilizers</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X8048AE727A7F1A2F">41.12-13 CanonicalRepresentativeOfExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X8071A8D784DC8325">41.12-14 CanonicalRepresentativeDeterminatorOfExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X85E9A6A77B8D00B8">41.12-15 ActorOfExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X8190A8247F29A5C7">41.12-16 UnderlyingExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap41.html#X7A3D87DE809FBFD4">41.12-17 SurjectiveActionHomomorphismAttr</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap42.html#X80F808307A2D5AB8">42 <span class="Heading">Permutations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap42.html#X80F07BE2811D4BAC">42.1 <span class="Heading">IsPerm (Filter)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X7AA69C6686FC49EA">42.1-1 IsPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X82069E437D2DF9AA">42.1-2 IsPermCollection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X819628B083B3939B">42.1-3 PermutationsFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X83C711557DEB4B36">42.1-4 PERM_INVERSE_THRESHOLD</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap42.html#X7A21DE5779D21A6D">42.2 <span class="Heading">Comparison of Permutations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X7CEC03A9808E2E7C"><code>42.2-1 \=</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X7BC944F57A04AFF2">42.2-2 DistancePerms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X83A917F67D45C7EA">42.2-3 SmallestGeneratorPerm</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap42.html#X82C255E2821C0721">42.3 <span class="Heading">Moved Points of Permutations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X84EF0A697F7A87DC">42.3-1 SmallestMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X84AA603987C94AC0">42.3-2 LargestMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X85E61B9C7A6B0CCA">42.3-3 MovedPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X85E7B1E28430F49E">42.3-4 NrMovedPoints</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap42.html#X79BE80267F4AA2B0">42.4 <span class="Heading">Sign and Cycle Structure</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X7BE5011B7C0DB704">42.4-1 SignPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X7944D1447804A69A">42.4-2 CycleStructurePerm</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap42.html#X7B3194EC869D936D">42.5 <span class="Heading">Creating Permutations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X7A9DCFD986958C1E">42.5-1 ListPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X78D611D17EA6E3BC">42.5-2 PermList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X8087DCC780B9656A">42.5-3 MappingPermListList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X7EF8388E7DA8E600">42.5-4 RestrictedPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X80665A5D800CAFE1">42.5-5 CycleFromList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap42.html#X8353AB8987E35DF3">42.5-6 AsPermutation</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap43.html#X85ED46007CED6191">43 <span class="Heading">Permutation Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap43.html#X7F38777E7BBE12AE">43.1 <span class="Heading">IsPermGroup (Filter)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7879877482F59676">43.1-1 IsPermGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap43.html#X85D769FF85545AAB">43.2 <span class="Heading">The Natural Action</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X84CFA16D858B00B8">43.2-1 OrbitPerms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X81F98222818DA35B">43.2-2 OrbitsPerms</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap43.html#X7E468B64860D5604">43.3 <span class="Heading">Computing a Permutation Representation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X80B7B1C783AA1567">43.3-1 IsomorphismPermGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X8086628878AFD3EA">43.3-2 SmallerDegreePermutationRepresentation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap43.html#X834208CD7C2956A3">43.4 <span class="Heading">Symmetric and Alternating Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X8129BE59781478E1">43.4-1 IsNaturalSymmetricGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X85CA6AD17BE90C95">43.4-2 IsSymmetricGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X8514BE9E79C608E0">43.4-3 IsAlternatingGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7ED60F7E81F1B614">43.4-4 SymmetricParentGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap43.html#X83F8D3B578A7BEEB">43.5 <span class="Heading">Primitive Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7E50211A7B92455F">43.5-1 ONanScottType</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7E89A46A86A3F4A2">43.5-2 SocleTypePrimitiveGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap43.html#X7FA58C3A8283F3BD">43.6 <span class="Heading">Stabilizer Chains</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap43.html#X7C2406B97E057196">43.7 <span class="Heading">Randomized Methods for Permutation Groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap43.html#X7C7EA55C80E457FA">43.8 <span class="Heading">Construction of Stabilizer Chains</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X80B5CF78829495C2">43.8-1 StabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X790C27B8783EDE68">43.8-2 StabChainOptions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X87E1292E85A5D31C">43.8-3 DefaultStabChainOptions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X86D64D2B81D58431">43.8-4 StabChainBaseStrongGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7BEC5F5A7851CAAB">43.8-5 MinimalStabChain</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap43.html#X81D7FCE47AC7F942">43.9 <span class="Heading">Stabilizer Chain Records</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap43.html#X7ECF8A4586346FD4">43.10 <span class="Heading">Operations for Stabilizer Chains</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7FBE6EB57EBE8B7D">43.10-1 BaseStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7D2A190D8308ED39">43.10-2 BaseOfGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7EF36DC78465026A">43.10-3 SizeStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X8384170881B9B531">43.10-4 StrongGeneratorsStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X87F473777EFDE867">43.10-5 GroupStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X87FB6DED80692D3F">43.10-6 OrbitStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7AC8F165875906DE">43.10-7 IndicesStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7CF607BC82C2C202">43.10-8 ListStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7F40E52D7B0438BF">43.10-9 ElementsStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X780875477CD2A57D">43.10-10 IteratorStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X861062AE87ACF340">43.10-11 InverseRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X79D2248C8787EAF2">43.10-12 SiftedPermutation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7B870C217D0B9997">43.10-13 MinimalElementCosetStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X87435B7884D9B353">43.10-14 LargestElementStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X809B2C3B7C5F77AB">43.10-15 ApproximateSuborbitsStabilizerPermGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap43.html#X8188051F79E72A95">43.11 <span class="Heading">Low Level Routines to Modify and Create Stabilizer Chains</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X86B31E6A81AE5FCB">43.11-1 CopyStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7E167E557B567C6A">43.11-2 CopyOptionsDefaults</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X87FF64AB87BFC779">43.11-3 ChangeStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X8778B4657D3FD97B">43.11-4 ExtendStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7E5E9F727D0B19D9">43.11-5 ReduceStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X85BF290D848C4091">43.11-6 RemoveStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X84E4906B86E5C089">43.11-7 EmptyStabChain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X80C7D2E87E6EE357">43.11-8 InsertTrivialStabilizer</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7B47B379824F6150">43.11-9 IsFixedStabilizer</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X8373007880EBF736">43.11-10 AddGeneratorsExtendSchreierTree</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap43.html#X86C78160854C7F30">43.12 <span class="Heading">Backtrack</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7BE3F03C80BF8B08">43.12-1 SubgroupProperty</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7EE7DDCC87C4BC31">43.12-2 ElementProperty</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X7A2D046B83DD5F5F">43.12-3 TwoClosure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap43.html#X861461AB7964DC64">43.12-4 InfoBckt</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap43.html#X78A68F5A80ADD1B6">43.13 <span class="Heading">Working with large degree permutation groups</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap44.html#X7CF51CB48610A07D">44 <span class="Heading">Matrix Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap44.html#X86CEA60E7C04744C">44.1 <span class="Heading">IsMatrixGroup (Filter)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7E6093FF85F1C3A1">44.1-1 IsMatrixGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap44.html#X7FD808E386FAF9B0">44.2 <span class="Heading">Attributes and Properties for Matrix Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7E55258C783C50CA">44.2-1 DimensionOfMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7D540083793CD496">44.2-2 DefaultFieldOfMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X78A9F0E580DA613A">44.2-3 FieldOfMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X832D18C77ED608DE">44.2-4 TransposedMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X84B36A827E5EFC35">44.2-5 IsFFEMatrixGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap44.html#X7F4B0B397AAC7659">44.3 <span class="Heading">Actions of Matrix Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7BD4F38E8624735D">44.3-1 ProjectiveActionOnFullSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7F8EA8D583C1E9B2">44.3-2 ProjectiveActionHomomorphismMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X849C451A80B4A210">44.3-3 BlowUpIsomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap44.html#X7934EED77891BE6B">44.4 <span class="Heading">GL and SL</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X781387AF7999EA99">44.4-1 IsGeneralLinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X86F9A27D7AFAEB5A">44.4-2 IsNaturalGL</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X816677CD821261FA">44.4-3 IsSpecialLinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X84134F08781EB943">44.4-4 IsNaturalSL</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7ED43D4F7E993A31">44.4-5 IsSubgroupSL</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap44.html#X7CA4097C79F5BD90">44.5 <span class="Heading">Invariant Forms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7C08385A81AB05E1">44.5-1 InvariantBilinearForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X8652FBF781940AC3">44.5-2 IsFullSubgroupGLorSLRespectingBilinearForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X82F22079852130C9">44.5-3 InvariantSesquilinearForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7B35A8AF7D8F0313">44.5-4 IsFullSubgroupGLorSLRespectingSesquilinearForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7BCACC007EB9B613">44.5-5 InvariantQuadraticForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X84AB04A67DFC0274">44.5-6 IsFullSubgroupGLorSLRespectingQuadraticForm</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap44.html#X7FB0138F79E8C5E7">44.6 <span class="Heading">Matrix Groups in Characteristic 0</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X850821F78558C829">44.6-1 IsCyclotomicMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7FEDB2E17EE02674">44.6-2 IsRationalMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7F737FC4795F3E48">44.6-3 IsIntegerMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X86F9CC1E7DB97CB6">44.6-4 IsNaturalGLnZ</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7B0E70127F5D2EAF">44.6-5 IsNaturalSLnZ</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7DE412A37A6975B3">44.6-6 InvariantLattice</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7CC4D6DC81739698">44.6-7 NormalizerInGLnZ</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7DAFB71F86525DE7">44.6-8 CentralizerInGLnZ</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X8217762A863F1382">44.6-9 ZClassRepsQClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X84FD9FC97FB90795">44.6-10 IsBravaisGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7AAE301C83116451">44.6-11 BravaisGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X788C7D9C7C2301C5">44.6-12 BravaisSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7F5FF1A481E08AD5">44.6-13 BravaisSupergroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X79B7CD797A420720">44.6-14 NormalizerInGLnZBravaisGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap44.html#X868288377CFA8D1B">44.7 <span class="Heading">Acting OnRight and OnLeft</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X7D1318A6780CD88B">44.7-1 CrystGroupDefaultAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap44.html#X792D237385977BE6">44.7-2 SetCrystGroupDefaultAction</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap45.html#X86007B0083F60470">45 <span class="Heading">Polycyclic Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X7F18A01785DBAC4E">45.1 <span class="Heading">Polycyclic Generating Systems</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X87F7E31879AFA06C">45.2 <span class="Heading">Computing a Pcgs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X84C3750C7A4EEC34">45.2-1 Pcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X8635E61A7DB73BA6">45.2-2 IsPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7B561B1685CEC2AB">45.2-3 CanEasilyComputePcgs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X7CAAD6D2838354D9">45.3 <span class="Heading">Defining a Pcgs Yourself</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7E139C3D80847D76">45.3-1 PcgsByPcSequence</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X816C5E8E7F71C9D8">45.4 <span class="Heading">Elementary Operations for a Pcgs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7DD0DF677AC1CF10">45.4-1 RelativeOrders</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X80D526848427A5C6">45.4-2 IsFiniteOrdersPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X866C3A5382FF231A">45.4-3 IsPrimeOrdersPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X827A7B097A959579">45.4-4 PcSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7903702E8194EF29">45.4-5 GroupOfPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X878FB11887524E2C">45.4-6 OneOfPcgs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X84243AA07DA5A827">45.5 <span class="Heading">Elementary Operations for a Pcgs and an Element</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7B941D4A7CAFCD73">45.5-1 RelativeOrderOfPcElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X78134914842E2F5F">45.5-2 ExponentOfPcElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X848DAEBF7DC448A5">45.5-3 ExponentsOfPcElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X829BCB267CDBC5C0">45.5-4 DepthOfPcElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7D47966479EA2890">45.5-5 LeadingExponentOfPcElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X87AF746B8328F5D0">45.5-6 PcElementByExponents</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7F8BD7A87DB3933A">45.5-7 LinearCombinationPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X8066B66D8069BAB4">45.5-8 SiftedPcElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7B52ADE7878A749A">45.5-9 CanonicalPcElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7A94AA357DB2F86C">45.5-10 ReducedPcElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X8702D76D8284CF3E">45.5-11 CleanedTailPcElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X830A0D037DBEAA97">45.5-12 HeadPcElementByNumber</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X7EF61EA4822870E7">45.6 <span class="Heading">Exponents of Special Products</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X868D6DB07D349460">45.6-1 ExponentsConjugateLayer</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X874F70697FE7B6DF">45.6-2 ExponentsOfRelativePower</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X78CAF32F864EF656">45.6-3 ExponentsOfConjugate</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X875689897DD0CAFC">45.6-4 ExponentsOfCommutator</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X8676397383093D1E">45.7 <span class="Heading">Subgroups of Polycyclic Groups – Induced Pcgs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X81FA878C854D63F8">45.7-1 IsInducedPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X83F6759184937F1B">45.7-2 InducedPcgsByPcSequence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X86308E80843BF9E5">45.7-3 ParentPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7F0EB20080590B23">45.7-4 InducedPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X8332F1197DF6FEDE">45.7-5 InducedPcgsByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7AF82BD079D811E5">45.7-6 InducedPcgsByPcSequenceAndGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X845FF8CA783D6CB3">45.7-7 LeadCoeffsIGS</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X800287C680C5DEC3">45.7-8 ExtendedPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X817E16D67B31389B">45.7-9 SubgroupByPcgs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X84068D2478C134C1">45.8 <span class="Heading">Subgroups of Polycyclic Groups – Canonical Pcgs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X80D122B986B42F80">45.8-1 IsCanonicalPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X816F6B4187032A10">45.8-2 CanonicalPcgs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X8294F5EF81B7ABA0">45.9 <span class="Heading">Factor Groups of Polycyclic Groups – Modulo Pcgs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7FE689A37E559F66">45.9-1 ModuloPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X868207D77D09D915">45.9-2 IsModuloPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X8027CC9878031D74">45.9-3 NumeratorOfModuloPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X87DBE2797D51B2F1">45.9-4 DenominatorOfModuloPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7E923519832F2D08"><code>45.9-5 \mod</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X876A41F97FBA7754">45.9-6 CorrespondingGeneratorsByModuloPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X8480852A7D49BC3F">45.9-7 CanonicalPcgsByGeneratorsWithImages</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X8254C0F485F945BD">45.10 <span class="Heading">Factor Groups of Polycyclic Groups in their Own Representation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X806C2D827E04ACF3">45.10-1 ProjectedPcElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X82F39CCE7C928D3A">45.10-2 ProjectedInducedPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X816813A078B93A6B">45.10-3 LiftedPcElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X83C60F1587577D65">45.10-4 LiftedInducedPcgs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X83FE235E7B208EC0">45.11 <span class="Heading">Pcgs and Normal Series</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7E7E89C278DDE20D">45.11-1 IsPcgsElementaryAbelianSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X863A20B57EA37BAC">45.11-2 PcgsElementaryAbelianSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7BCC1E2A80544CC7">45.11-3 IndicesEANormalSteps</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7FCE308887F621FC">45.11-4 EANormalSeriesByPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X79675266796D7254">45.11-5 IsPcgsCentralSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X8187FCF483659E69">45.11-6 PcgsCentralSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7FB73FEB7BED5BFA">45.11-7 IndicesCentralNormalSteps</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X82266ADA86B2A689">45.11-8 CentralNormalSeriesByPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X786E60AF7B61BF9E">45.11-9 IsPcgsPCentralSeriesPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X86F19DBD7D346E7F">45.11-10 PcgsPCentralSeriesPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X863968F08509E7D4">45.11-11 IndicesPCentralNormalStepsPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7A92C9EA7BAF60CA">45.11-12 PCentralNormalSeriesByPcgsPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7EA5BC3B7FE9D98D">45.11-13 IsPcgsChiefSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7E7326947EAE4BC9">45.11-14 PcgsChiefSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7C05E84A78CA405E">45.11-15 IndicesChiefNormalSteps</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X83C5ABC587074B14">45.11-16 ChiefNormalSeriesByPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7A954E3887189842">45.11-17 IndicesNormalSteps</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7947B0FB87F44042">45.11-18 NormalSeriesByPcgs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X7E624B4E8224DE2D">45.12 <span class="Heading">Sum and Intersection of Pcgs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7833DAAA7C07CFD7">45.12-1 SumFactorizationFunctionPcgs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X83039CF97D27D819">45.13 <span class="Heading">Special Pcgs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7C8A82FA786AC021">45.13-1 IsSpecialPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X827EB7767BACD023">45.13-2 <span class="Heading">SpecialPcgs</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X82DC7CE682140588">45.13-3 LGWeights</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X824645C97E347EEE">45.13-4 LGLayers</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7A655F4C7D9AE130">45.13-5 LGFirst</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7C3912F77B12C8B6">45.13-6 LGLength</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X814C35BF7C9A8DEF">45.13-7 IsInducedPcgsWrtSpecialPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7C14AE5C82FB0771">45.13-8 InducedPcgsWrtSpecialPcgs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X7E86EB517DC08809">45.14 <span class="Heading">Action on Subfactors Defined by a Pcgs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7A9BB9D0817CA949">45.14-1 VectorSpaceByPcgsOfElementaryAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X81FC09DD7FC06C6E">45.14-2 LinearAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7C2135B98732BBC3">45.14-3 LinearActionLayer</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X79C2D6BF7DD69ED6">45.14-4 AffineAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7E4CB1358524497B">45.14-5 AffineActionLayer</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X7EEA8D638492F432">45.15 <span class="Heading">Orbit Stabilizer Methods for Polycyclic Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7CFCCF607A30B5EE">45.15-1 StabilizerPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7A87E72F86813132">45.15-2 Pcgs_OrbitStabilizer</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X7A19DF1E7E841074">45.16 <span class="Heading">Operations which have Special Methods for Groups with Pcgs</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap45.html#X79DCCF6D80351859">45.17 <span class="Heading">Conjugacy Classes in Solvable Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X79593F667A68A21D">45.17-1 ClassesSolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap45.html#X7B358D3B7E236973">45.17-2 CentralizerSizeLimitConsiderFunction</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap46.html#X7EAD57C97EBF7E67">46 <span class="Heading">Pc Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap46.html#X78E9E4D778A57A96">46.1 <span class="Heading">The Family Pcgs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X79EDB35E82C99304">46.1-1 FamilyPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X80893D2A7FFC791B">46.1-2 IsFamilyPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X85C1596A867BE93D">46.1-3 InducedPcgsWrtFamilyPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X8333ACCB7F530406">46.1-4 IsParentPcgsFamilyPcgs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap46.html#X842526BE7FEFE8BD">46.2 <span class="Heading">Elements of Pc Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X869DCE7D86E32337">46.2-1 <span class="Heading">Comparison of elements of pc groups</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X7D1B700882FC6C78">46.2-2 <span class="Heading">Arithmetic operations for elements of pc groups</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap46.html#X87B866C386B386E4">46.3 <span class="Heading">Pc Groups versus Fp Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X7D1F506D7830B1D9">46.3-1 IsPcGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X7D2735A18111FE39">46.3-2 IsomorphismFpGroupByPcgs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap46.html#X8581887880556E0C">46.4 <span class="Heading">Constructing Pc Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X84C10D1F7CB5274F">46.4-1 PcGroupFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X7E958DB281E070FD">46.4-2 SingleCollector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X86A08D887E049347">46.4-3 SetConjugate</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X7B25997C7DF92B6D">46.4-4 SetCommutator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X7BC319BA8698420C">46.4-5 SetPower</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X84F0521486672C3C">46.4-6 GroupByRws</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X7DF4835F79667099">46.4-7 IsConfluent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X7E6226597DFE5F8F">46.4-8 IsomorphismRefinedPcGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X821560A387762DD1">46.4-9 RefinedPcGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap46.html#X83F69FE27B024E24">46.5 <span class="Heading">Computing Pc Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X81C55D4F825C36D4">46.5-1 PcGroupWithPcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X873CEB137BA1CD6E">46.5-2 IsomorphismPcGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X82BE14A986FA6882">46.5-3 IsomorphismSpecialPcGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap46.html#X85696AB9791DF047">46.6 <span class="Heading">Saving a Pc Group</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X8593253380D84508">46.6-1 GapInputPcGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap46.html#X8391EE8D782D0C9E">46.7 <span class="Heading">Operations for Pc Groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap46.html#X877AAB887D4507E3">46.8 <span class="Heading"><span class="SimpleMath">2</span>-Cohomology and Extensions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X78E6E11E8285E288">46.8-1 TwoCoboundaries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X784FCA207B8694A6">46.8-2 TwoCocycles</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X838065F97F60468F">46.8-3 TwoCohomology</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X8236AD927A5A0E5A">46.8-4 Extensions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X7B3BE908867CE4F9">46.8-5 Extension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X83DCB5AB7B6EE785">46.8-6 SplitExtension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X7EAC6B8B7ABEEB86">46.8-7 ModuleOfExtension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X824F2B2E7C11ABAF">46.8-8 CompatiblePairs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X854FFEF187C4AAB9">46.8-9 ExtensionRepresentatives</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X84E2DA897FAAF6D8">46.8-10 SplitExtension</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap46.html#X874E4B107BD78F5A">46.9 <span class="Heading">Coding a Pc Presentation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X79948F1D7D4FF8D9">46.9-1 CodePcgs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X8041C2D88721EEA9">46.9-2 CodePcGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X826BFDA07A707C54">46.9-3 PcGroupCode</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap46.html#X81D211D8838B875C">46.10 <span class="Heading">Random Isomorphism Testing</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap46.html#X84F6F9787CB2CF16">46.10-1 RandomIsomorphismTest</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap47.html#X7AA982637E90B35A">47 <span class="Heading">Finitely Presented Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap47.html#X7824C8167B3CFAB1">47.1 <span class="Heading">IsSubgroupFpGroup and IsFpGroup</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7AF7E2B48199452C">47.1-1 IsSubgroupFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X850B9DF17D90C3A2">47.1-2 IsFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X8370BF3B78D0B14D">47.1-3 InfoFpGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap47.html#X7D55E56E790F85FD">47.2 <span class="Heading">Creating Finitely Presented Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7EF4179E78BC7313"><code>47.2-1 \/</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7CE0FA5F8695241E">47.2-2 FactorGroupFpGroupByRels</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7B3D290B87B6EFE4">47.2-3 ParseRelators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X85EAA789848B528E">47.2-4 StringFactorizationWord</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap47.html#X84D693EC872DAA55">47.3 <span class="Heading">Comparison of Elements of Finitely Presented Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X797D29628203CBD6"><code>47.3-1 \=</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7B350C718573B8DF"><code>47.3-2 \<</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X87512CF485CC4128">47.3-3 FpElmComparisonMethod</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X82CB9EC982CDAEAC">47.3-4 SetReducedMultiplication</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap47.html#X7B0B2781796800AD">47.4 <span class="Heading">Preimages in the Free Group</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X85CF3931849FB441">47.4-1 FreeGroupOfFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X79C77C5184CA02B6">47.4-2 FreeGeneratorsOfFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X87BA180287CD1F71">47.4-3 RelatorsOfFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X8447A2397A1E524B">47.4-4 UnderlyingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7F34C8017DC03FDB">47.4-5 ElementOfFpGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap47.html#X869143D284F3379D">47.5 <span class="Heading">Operations for Finitely Presented Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7AB7187779EDC9BA">47.5-1 PseudoRandom</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap47.html#X7BD0CEBA7B225416">47.6 <span class="Heading">Coset Tables and Coset Enumeration</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7F7F31E47D7F6EF8">47.6-1 CosetTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X87D175757C581E62">47.6-2 TracedCosetFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7EC1B0EE876E478A">47.6-3 FactorCosetAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X82926A7F8365A341">47.6-4 CosetTableBySubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7DE601F179E6FD09">47.6-5 CosetTableFromGensAndRels</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X822B188F87E9E642">47.6-6 CosetTableDefaultMaxLimit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7A80A00E7E088E44">47.6-7 CosetTableDefaultLimit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X829D31A981CB2AF4">47.6-8 MostFrequentGeneratorFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7912E6577B577A5C">47.6-9 IndicesInvolutaryGenerators</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap47.html#X85B882F782D7AFD0">47.7 <span class="Heading">Standardization of coset tables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X85FD1D637EF1EBE7">47.7-1 CosetTableStandard</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X85FCD8DF81BA94D5">47.7-2 StandardizeTable</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap47.html#X87C3FA0784A85309">47.8 <span class="Heading">Coset tables for subgroups in the whole group</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X846EC8AB7803114D">47.8-1 CosetTableInWholeGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X857F239583AFE0B7">47.8-2 SubgroupOfWholeGroupByCosetTable</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap47.html#X7E17A14E823F953D">47.9 <span class="Heading">Augmented Coset Tables and Rewriting</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X80F8BF1D867DA7C1">47.9-1 AugmentedCosetTableInWholeGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7AF67CFD846C1159">47.9-2 AugmentedCosetTableMtc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7F3F09C778552811">47.9-3 AugmentedCosetTableRrs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X86B65EA186140244">47.9-4 RewriteWord</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap47.html#X87FBDA2B815A8776">47.10 <span class="Heading">Low Index Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X85C5151380E19122">47.10-1 LowIndexSubgroupsFpGroupIterator</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap47.html#X81003D217D92E342">47.11 <span class="Heading">Converting Groups to Finitely Presented Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7F28268F850F454E">47.11-1 IsomorphismFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X81B2B3B6812FD62D">47.11-2 IsomorphismFpGroupByGenerators</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap47.html#X826604AA7F18BFA3">47.12 <span class="Heading">New Presentations and Presentations for Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X78D87FA68233C401">47.12-1 IsomorphismSimplifiedFpGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap47.html#X86E7CE077D82133D">47.13 <span class="Heading">Preimages under Homomorphisms from an FpGroup</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7ABC3C917D41A74B">47.13-1 SubgroupOfWholeGroupByQuotientSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X8047D7A37B27FEEA">47.13-2 IsSubgroupOfWholeGroupByQuotientRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X84E6CEA28611C112">47.13-3 AsSubgroupOfWholeGroupByQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7DA1151D84289FC9">47.13-4 DefiningQuotientHomomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap47.html#X846072F779B51087">47.14 <span class="Heading">Quotient Methods</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7B5DDADC80F5796B">47.14-1 PQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X86EB30A7867EEF16">47.14-2 EpimorphismQuotientSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7CA738DB80B20D67">47.14-3 EpimorphismPGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X7CA20E2582DC45FD">47.14-4 EpimorphismNilpotentQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X869F70CC818C946D">47.14-5 SolvableQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X79A4D3B68110F48A">47.14-6 EpimorphismSolvableQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X81167847832DD3B1">47.14-7 LargerQuotientBySubgroupAbelianization</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap47.html#X81451C4B8463B848">47.15 <span class="Heading">Abelian Invariants for Subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X83B63ED8826F4268">47.15-1 AbelianInvariantsSubgroupFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X804F664180BA2134">47.15-2 AbelianInvariantsSubgroupFpGroupMtc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X8586137B7AAA6C10">47.15-3 <span class="Heading">AbelianInvariantsSubgroupFpGroupRrs</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X850E4CD784F6EAA8">47.15-4 AbelianInvariantsNormalClosureFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X801635B28079E56A">47.15-5 AbelianInvariantsNormalClosureFpGroupRrs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap47.html#X86C43E3B81ED25DC">47.16 <span class="Heading">Testing Finiteness of Finitely Presented Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X82F444F67BE0E4FE">47.16-1 IsInfiniteAbelianizationGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap47.html#X85C9FD548394C1E2">47.16-2 NewmanInfinityCriterion</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap48.html#X782985197BE809BF">48 <span class="Heading">Presentations and Tietze Transformations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X867D00387957450F">48.1 <span class="Heading">Creating Presentations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X797867B287AD92F8">48.1-1 PresentationFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X8637837A79422445">48.1-2 TzSort</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X849429BC7D435F77">48.1-3 GeneratorsOfPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7D6F40A87F24D3D6">48.1-4 FpGroupPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X84E056C57AFEDEA8">48.1-5 PresentationViaCosetTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7E1F2658827FC228">48.1-6 SimplifiedFpGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X8118FECE7AD1879B">48.2 <span class="Heading">Subgroup Presentations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7DB32FA97DAC5AC8">48.2-1 PresentationSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X857365CD87ADC29E">48.2-2 <span class="Heading">PresentationSubgroupRrs</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7FCE7ED581CF7897">48.2-3 PrimaryGeneratorWords</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X80BA10F780EAE68E">48.2-4 PresentationSubgroupMtc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7D6A52837BEE5C3D">48.2-5 PresentationNormalClosureRrs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7A7E5D0084DB0B4F">48.2-6 PresentationNormalClosure</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X7BC960AB7E8DE419">48.3 <span class="Heading">Relators in a Presentation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X8365BAFA785FCD8D">48.3-1 TietzeWordAbstractWord</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X8573E91C838B1D13">48.3-2 AbstractWordTietzeWord</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X867F64FA840B3F81">48.4 <span class="Heading">Printing Presentations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X847EA6737C21171C">48.4-1 TzPrintGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X821B63DD82894443">48.4-2 TzPrintRelators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X852C52C37FAAB7DD">48.4-3 TzPrintLengths</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7D7B3F46865443E4">48.4-4 TzPrintStatus</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X85F8DAE27F06C32B">48.4-5 TzPrintPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7CA8BA51802655FC">48.4-6 TzPrint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X82F6B0EE7C7C7901">48.4-7 TzPrintPairs</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X82455E5885D73FFF">48.5 <span class="Heading">Changing Presentations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7F632A6D8685855D">48.5-1 AddGenerator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X83A5667086FD538A">48.5-2 TzNewGenerator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X78D1BCE67FA852D8">48.5-3 AddRelator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7B11E89E78A22EBF">48.5-4 RemoveRelator</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X829B634286471AB7">48.6 <span class="Heading">Tietze Transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7C4A30328224C466">48.6-1 TzGo</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X78C3D23387DAC35A">48.6-2 SimplifyPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X801D3D8984E1CA55">48.6-3 TzGoGo</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X7D19E30080290FC7">48.7 <span class="Heading">Elementary Tietze Transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X85989AF886EC2BF6">48.7-1 <span class="Heading">TzEliminate</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7DF4BBDF839643DD">48.7-2 TzSearch</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X87F7A87A7ACF2445">48.7-3 TzSearchEqual</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X80D31A0F7C2A51BD">48.7-4 TzFindCyclicJoins</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X7D2FACCF79F57040">48.8 <span class="Heading">Tietze Transformations that introduce new Generators</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X846DB23E8236FF8A">48.8-1 <span class="Heading">TzSubstitute</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7ADE3B437C19B94D">48.8-2 TzSubstituteCyclicJoins</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X85E703997A0212EE">48.9 <span class="Heading">Tracing generator images through Tietze transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7D855FA08242898A">48.9-1 TzInitGeneratorImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7AB9A06F80FB3659">48.9-2 OldGeneratorsOfPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X798B38F87C082C45">48.9-3 TzImagesOldGens</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7AC41B117DBB87D6">48.9-4 TzPreImagesNewGens</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7F086D0E7AD6173B">48.9-5 TzPrintGeneratorImages</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X7D9E283D81CCCF1A">48.10 <span class="Heading">The Decoding Tree Procedure</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7ACBFE2F78D72A31">48.10-1 DecodeTree</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap48.html#X856F37537E9927EE">48.11 <span class="Heading">Tietze Options</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X8178683283214D88">48.11-1 TzOptions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap48.html#X7BC90B6882DE6D10">48.11-2 TzPrintOptions</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap49.html#X7D5C75647DB168F1">49 <span class="Heading">Group Products</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap49.html#X7D39232A84CD8DBD">49.1 <span class="Heading">Direct Products</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap49.html#X861BA02C7902A4F4">49.1-1 DirectProduct</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap49.html#X87FE512E7DB7346C">49.2 <span class="Heading">Semidirect Products</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap49.html#X7D905A5778D7ACDE">49.2-1 <span class="Heading">SemidirectProduct</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap49.html#X815AFC537B215D7B">49.3 <span class="Heading">Subdirect Products</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap49.html#X82112D768085AD98">49.3-1 SubdirectProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap49.html#X814204E97812894C">49.3-2 SubdirectProducts</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap49.html#X7DF2AEBC8518FFA4">49.4 <span class="Heading">Wreath Products</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap49.html#X8786EFBC78D7D6ED">49.4-1 WreathProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap49.html#X8589DCFA7C2E5FAA">49.4-2 WreathProductImprimitiveAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap49.html#X82B8DD1C868A3726">49.4-3 WreathProductProductAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap49.html#X80634C3180E0C593">49.4-4 KuKGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap49.html#X801A358E879A0FF0">49.4-5 ListWreathProductElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap49.html#X7ECB076E81D8D402">49.4-6 WreathProductElementList</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap49.html#X7AC1AD17833117DF">49.5 <span class="Heading">Free Products</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap49.html#X837AC5A081EECF50">49.5-1 <span class="Heading">FreeProduct</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap49.html#X798FDA1386A0EAC6">49.6 <span class="Heading">Embeddings and Projections for Group Products</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap49.html#X784149B8847B20FF">49.6-1 Embedding</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap49.html#X86F275AC7C625626">49.6-2 Projection</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap50.html#X81B00B667D2BD022">50 <span class="Heading">Group Libraries</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X839981CC7D9B671B">50.1 <span class="Heading">Basic Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X8489BECB78664847">50.1-1 TrivialGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7A7C473D87B31F3B">50.1-2 CyclicGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X81CCC3BF8005A2D7">50.1-3 AbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X8778256286E50743">50.1-4 ElementaryAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7F43050D8587E767">50.1-5 FreeAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X838DE1AB7B3D70FF">50.1-6 DihedralGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X8233A853818CAF33">50.1-7 IsDihedralGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7E9844EF7C47EEB0">50.1-8 DicyclicGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7F260D177FD4BE4C">50.1-9 IsDicyclicGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X86E76B3A796BEFA8">50.1-10 ExtraspecialGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7E54D3E778E6A53E">50.1-11 <span class="Heading">AlternatingGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X858666F97BD85ABB">50.1-12 <span class="Heading">SymmetricGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X788FA7DE84E0FE6A">50.1-13 MathieuGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X8469DBBF82F8E5C3">50.1-14 SuzukiGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X87E5B0F679CA7FE4">50.1-15 ReeGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7D0FFDA4793995FC">50.1-16 <span class="Heading">Generator Names</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X8674AAA578FE4AEE">50.2 <span class="Heading">Classical Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X85D607DD82AF3E27">50.2-1 <span class="Heading">GeneralLinearGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7CA3F7BF83992C6B">50.2-2 <span class="Heading">SpecialLinearGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X866D4E2B816BDFA5">50.2-3 GeneralUnitaryGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X82A2AADE805DCDE9">50.2-4 SpecialUnitaryGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X8142A8B07811CA90">50.2-5 <span class="Heading">SymplecticGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7C2051CB7B94CEB1">50.2-6 GeneralOrthogonalGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X78D4EEF27AA2DCFD">50.2-7 SpecialOrthogonalGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X8365E0AB8338DA3F">50.2-8 Omega</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X79C3C61A7D83A6D0">50.2-9 GeneralSemilinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7D3779237CB5B49C">50.2-10 SpecialSemilinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7F0DBEB880D2D574">50.2-11 ProjectiveGeneralLinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X86784EDA80224B74">50.2-12 ProjectiveSpecialLinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7E471ADE7E095604">50.2-13 ProjectiveGeneralUnitaryGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7A88FE2B7EF9C804">50.2-14 ProjectiveSpecialUnitaryGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7DEDE2537B8FFFF5">50.2-15 ProjectiveSymplecticGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X87AAB20A8434356B">50.2-16 ProjectiveGeneralOrthogonalGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X835E0D3384C4AB6B">50.2-17 ProjectiveSpecialOrthogonalGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7F546F907A37DF15">50.2-18 ProjectiveOmega</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X824925DB7C3A2FA6">50.2-19 ProjectiveGeneralSemilinearGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X86BD9AE27CCAB1A6">50.2-20 ProjectiveSpecialSemilinearGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X85B9F2D379616C35">50.3 <span class="Heading">Conjugacy Classes in Classical Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X831789117E93171E">50.3-1 NrConjugacyClassesGL</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X817EBD6E841285CD">50.4 <span class="Heading">Constructors for Basic Groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X82676ED5826E9E2E">50.5 <span class="Heading">Selection Functions</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X7A884ECF813C2026">50.6 <span class="Heading">Finite Perfect Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X866A25F882A4E97B">50.6-1 SizesPerfectGroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7906BBA7818E9415">50.6-2 <span class="Heading">PerfectGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7E1CB2D18085FF9D">50.6-3 PerfectIdentification</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7D68BE547FE5C0F5">50.6-4 NumberPerfectGroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X866356A684F6B15E">50.6-5 SizeNumbersPerfectGroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X845419F07BB92867">50.6-6 <span class="Heading">DisplayInformationPerfectGroups</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X875C5BE67BAB7F71">50.6-7 <span class="Heading">More about the Perfect Groups Library</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap50.html#X7873506D873EDB95">50.7 <span class="Heading">Irreducible Maximal Finite Integral Matrix Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X8693FD647EF3C53B">50.7-1 ImfNumberQQClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X8705F64B7E19DDC7">50.7-2 DisplayImfInvariants</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X8604A2167B2E8434">50.7-3 ImfInvariants</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X78935B307B909101">50.7-4 ImfMatrixGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X84BF34B27CD5E85C">50.7-5 IsomorphismPermGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap50.html#X7CEDB6CE7BAC4518">50.7-6 IsomorphismPermGroupImfGroup</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap51.html#X8665D8737FDD5B10">51 <span class="Heading">Semigroups and Monoids</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X80AF5F307DBDC2B4">51.1 <span class="Heading">Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7B412E5B8543E9B7">51.1-1 IsSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7F55D28F819B2817">51.1-2 <span class="Heading">Semigroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X8678D40878CC09A1">51.1-3 Subsemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X782B7BDD8252581C">51.1-4 IsSubsemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X79FBBEC9841544F3">51.1-5 SemigroupByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X80ED104F85AE5134">51.1-6 AsSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7B1EEA3E82BFE09F">51.1-7 AsSubsemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X78147A247963F23B">51.1-8 GeneratorsOfSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X79776D7C8399F2CF">51.1-9 IsGeneratorsOfSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7C72E4747BF642BB">51.1-10 <span class="Heading">FreeSemigroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7E67E13F7A01F8D3">51.1-11 SemigroupByMultiplicationTable</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X872FE34A7814C0DC">51.2 <span class="Heading">Monoids</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X861C523483C6248C">51.2-1 IsMonoid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7F95328B7C7E49EA">51.2-2 <span class="Heading">Monoid</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X8322D01E84912FD7">51.2-3 Submonoid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X85129EE387CC4D28">51.2-4 MonoidByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7B22038F832B9C0F">51.2-5 AsMonoid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7C9A12DE8287B2D3">51.2-6 AsSubmonoid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X83CA2E7279C44718">51.2-7 GeneratorsOfMonoid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7EC77C0184587181">51.2-8 TrivialSubmonoid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X79FA3FA978CA2E43">51.2-9 <span class="Heading">FreeMonoid</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7BFE938E857CA27D">51.2-10 MonoidByMultiplicationTable</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X840847B6810BD0E1">51.3 <span class="Heading">Inverse semigroups and monoids</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X78B13FED7AFB4326">51.3-1 InverseSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X80D9B9A98736051B">51.3-2 InverseMonoid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X87C373597F787250">51.3-3 GeneratorsOfInverseSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7A3B262C85B6D475">51.3-4 GeneratorsOfInverseMonoid</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7C4C6EE681E7A57E">51.3-5 IsInverseSubsemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X78274024827F306D">51.4 <span class="Heading">Properties of Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7C4663827C5ACEF1">51.4-1 IsRegularSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X87532A76854347E0">51.4-2 IsRegularSemigroupElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7AFDE0F17AE516C5">51.4-3 InversesOfSemigroupElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X836F4692839F4874">51.4-4 IsSimpleSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X8193A60F839C064E">51.4-5 IsZeroSimpleSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X85F7E5CD86F0643B">51.4-6 IsZeroGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7FFEC81F7F2C4EAA">51.4-7 IsReesCongruenceSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X83F1529479D56665">51.4-8 IsInverseSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X7BB32D508183C0F1">51.5 <span class="Heading">Ideals of semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7D5CEE4D7D4318ED">51.5-1 SemigroupIdealByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7F01FFB18125DED5">51.5-2 ReesCongruenceOfSemigroupIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7A3FF85984345540">51.5-3 IsLeftSemigroupIdeal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X7914691E7DFFE27A">51.6 <span class="Heading">Congruences on semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X78E34B737F0E009F">51.6-1 IsSemigroupCongruence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X822DB78579BCB7B5">51.6-2 IsReesCongruence</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X87CE9EAB7EE3A128">51.7 <span class="Heading">Quotients</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X80EF3E6F842BE64E">51.7-1 IsQuotientSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7CAD3D1687956F7F">51.7-2 HomomorphismQuotientSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X87120C46808F7289">51.7-3 QuotientSemigroupPreimage</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X80C6C718801855E9">51.8 <span class="Heading">Green's Relations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X786CEDD4814A9079">51.8-1 GreensRRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X8364D69987D49DE1">51.8-2 IsGreensRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X82A11A087AFB3EB0">51.8-3 IsGreensClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7AA204C8850F9070">51.8-4 IsGreensLessThanOrEqual</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X86FE5F5585EBCF13">51.8-5 RClassOfHClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X78C56F4A78E0088A">51.8-6 EggBoxOfDClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X803237F17ACD44E3">51.8-7 DisplayEggBoxOfDClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X87C75A9D86122D93">51.8-8 GreensRClassOfElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X844D20467A644811">51.8-9 GreensRClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7CB4A18685B850E2">51.8-10 GroupHClassOfGreensDClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X79D740EF7F0E53BD">51.8-11 IsGroupHClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7F5860927CAD920F">51.8-12 IsRegularDClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X81AF2EAB7CEF8C19">51.8-13 DisplaySemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap51.html#X8225A9EC87A255E6">51.9 <span class="Heading">Rees Matrix Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X8526AA557CDF6C49">51.9-1 ReesMatrixSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X78D2A48C87FC8E38">51.9-2 ReesMatrixSubsemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7964B5C97FB9C07D">51.9-3 IsomorphismReesMatrixSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7F6B852B81488C86">51.9-4 IsReesMatrixSemigroupElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7A0DE1F28470295E">51.9-5 ReesMatrixSemigroupElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7F03BE707AC7F8A0">51.9-6 IsReesMatrixSubsemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X780BB78A79275244">51.9-7 IsReesMatrixSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7CACF4D686AF1D19">51.9-8 Matrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X82FC5D6980C66AC4">51.9-9 <span class="Heading">Rows and columns</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7D9719F887AFCF8F">51.9-10 UnderlyingSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap51.html#X7D1D9A0382064B8F">51.9-11 AssociatedReesMatrixSemigroupOfDClass</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap52.html#X7DE7C52A7C4BDADE">52 <span class="Heading">Finitely Presented Semigroups and Monoids</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap52.html#X78C80F1A84C58E1E">52.1 <span class="Heading">IsSubsemigroupFpSemigroup (Filter)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X8496E23C80453C33">52.1-1 IsSubsemigroupFpSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X8239EF2B853411E9">52.1-2 IsFpSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X81ABBE997A4C19B7">52.1-3 IsElementOfFpSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X7DC8A5D380AFE5DB">52.1-4 FpGrpMonSmgOfFpGrpMonSmgElement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap52.html#X865E230B83982E66">52.2 <span class="Heading">Creating Finitely Presented Semigroups and Monoids</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X84745EC6789FEB4C"><code>52.2-1 \/</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X822F04B2833BE254">52.2-2 FactorFreeSemigroupByRelations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X869F966B8196F28C">52.2-3 IsomorphismFpSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap52.html#X85E7C8407C9D5FBE">52.3 <span class="Heading">Comparison of Elements of Finitely Presented Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X7DD9D81F863EBE31"><code>52.3-1 \=</code></a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap52.html#X7CD806CA7E0A1438">52.4 <span class="Heading">Preimages in the Free Semigroup or Monoid</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X784B3DB686E7080C">52.4-1 UnderlyingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X847012347856C55E">52.4-2 ElementOfFpSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X8726523779601873">52.4-3 FreeSemigroupOfFpSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X79A39402806B5EB7">52.4-4 FreeGeneratorsOfFpSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X862BE9FA7C987CAB">52.4-5 RelationsOfFpSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap52.html#X87693BDC79DC6EBF">52.5 <span class="Heading">Rewriting Systems and the Knuth-Bendix Procedure</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X7D8F804E814D894D">52.5-1 ReducedConfluentRewritingSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X7A3F8AE285C41D80">52.5-2 KB_REW</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X87A3823483E4FF86">52.5-3 <span class="Heading">KnuthBendixRewritingSystem</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X7966343587A04AFF">52.5-4 SemigroupOfRewritingSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X80B8115C8147F605">52.5-5 FreeSemigroupOfRewritingSystem</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap52.html#X812C28217F3E6720">52.6 <span class="Heading">Todd-Coxeter Procedure</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap52.html#X7C24508A7F677520">52.6-1 CosetTableOfFpSemigroup</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap53.html#X860026B880BCB2A5">53 <span class="Heading">Transformations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap53.html#X7CF9291C7CC42340">53.1 <span class="Heading">The family and categories of transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7B6259467974FB70">53.1-1 IsTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7A6747CE85F2E6EA">53.1-2 IsTransformationCollection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7E58AFA1832FF064">53.1-3 TransformationFamily</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap53.html#X80F3086F87E93DF8">53.2 <span class="Heading">Creating transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X86ADBDE57A20E323">53.2-1 Transformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X8040642687531E7F">53.2-2 Transformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7E82EBD68455EE4A">53.2-3 TransformationByImageAndKernel</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X85D1071484CE004C">53.2-4 Idempotent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7C2A3FC9782F2099">53.2-5 TransformationOp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7D6FCC417DE86CD1">53.2-6 TransformationNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X8475448F87E8CB8A">53.2-7 <span class="Heading">RandomTransformation</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X8268A58685BEFD6F">53.2-8 IdentityTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7F1E4B5184210D2B">53.2-9 ConstantTransformation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap53.html#X7F81A18B813C9DF0">53.3 <span class="Heading">Changing the representation of a transformation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7C5360B2799943F3">53.3-1 AsTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X846A6F6B7B715188">53.3-2 RestrictedTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X8708AE247F5B129B">53.3-3 PermutationOfImage</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap53.html#X812CEC008609A8A2">53.4 <span class="Heading">Operators for transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X853031E37F214D10"><code>53.4-1 \^</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X824A8E247DE7E53E"><code>53.4-2 \^</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7F6573E67D27D822"><code>53.4-3 \*</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X828CA7137F97C124"><code>53.4-4 \/</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7856D91E8709EF5B">53.4-5 LeftQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X84B8E294826A9377"><code>53.4-6 \<</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7D454AAD851AE07E"><code>53.4-7 \=</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X83DBA2A18719EFA8">53.4-8 PermLeftQuoTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X8275DFAA8270BB59">53.4-9 IsInjectiveListTrans</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X834A313B7DAF06D5">53.4-10 ComponentTransformationInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X82F5DEEC837B60A3">53.4-11 PreImagesOfTransformation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap53.html#X86DE4F7A7C535820">53.5 <span class="Heading">Attributes for transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X78A209C87CF0E32B">53.5-1 DegreeOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7AEC9E6687B3505A">53.5-2 ImageListOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X839A6D6082A21D1F">53.5-3 ImageSetOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X818EBB167C7EA37B">53.5-4 RankOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X844F00F982D5BD3C">53.5-5 MovedPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7FA6A4B57FDA003D">53.5-6 NrMovedPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X86C0DDDC7881273A">53.5-7 SmallestMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X8383A7727AC97724">53.5-8 LargestMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7CCFE27E83676572">53.5-9 SmallestImageOfMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7E7172567C3A3E63">53.5-10 LargestImageOfMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X8083794579274E87">53.5-11 FlatKernelOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X80FCB5048789CF75">53.5-12 KernelOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X860306EB7FAAD2D4">53.5-13 InverseOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7BB9DB6E8558356D">53.5-14 Inverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X863216CB7AF88BED">53.5-15 IndexPeriodOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X85FE9F20810BCC70">53.5-16 SmallestIdempotentPower</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X858E944481F6B591">53.5-17 ComponentsOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X8640AE1C79201470">53.5-18 NrComponentsOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X784650B583CEAF7D">53.5-19 ComponentRepsOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7EAA15557D55D93B">53.5-20 CyclesOfTransformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X786EB02A829260DB">53.5-21 CycleTransformationInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X845869E0815A6AA6">53.5-22 LeftOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7F19C9C77F9F8981">53.5-23 TrimTransformation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap53.html#X810D23017A5527B7">53.6 <span class="Heading">Displaying transformations</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap53.html#X7B51CE257B814B09">53.7 <span class="Heading">Semigroups of transformations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7EAF835D7FE4026F">53.7-1 IsTransformationSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7EA699C687952544">53.7-2 DegreeOfTransformationSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X7D2B0685815B4053">53.7-3 FullTransformationSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X85C58E1E818C838C">53.7-4 IsFullTransformationSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X78F29C817CF6827F">53.7-5 IsomorphismTransformationSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap53.html#X820ECE00846E480F">53.7-6 AntiIsomorphismTransformationSemigroup</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap54.html#X7D6495F77B8A77BD">54 <span class="Heading">Partial permutations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap54.html#X87B0D6657A3F2B0E">54.1 <span class="Heading">The family and categories of partial permutations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7EECE133792B30FC">54.1-1 IsPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X8262A827790DD1CC">54.1-2 IsPartialPermCollection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7E63D17780F64FBA">54.1-3 PartialPermFamily</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap54.html#X7B9D451D7FDA1DD8">54.2 <span class="Heading">Creating partial permutations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X8538BAE77F2FB2F8">54.2-1 PartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X81188D9F83F64222">54.2-2 PartialPermOp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X80ABBF4883C79060">54.2-3 RestrictedPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X849668DD7B0B9E3B">54.2-4 JoinOfPartialPerms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X81E2B6977E28CD00">54.2-5 MeetOfPartialPerms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X80EFB142817A0A9F">54.2-6 EmptyPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7E6ADC8583C31530">54.2-7 <span class="Heading">RandomPartialPerm</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap54.html#X8779F0997D0FDA78">54.3 <span class="Heading">Attributes for partial permutations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X8612A4DC864E7959">54.3-1 DegreeOfPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X8413D0EF7DEE1FFF">54.3-2 CodegreeOfPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7C1ABD8A80E95B39">54.3-3 RankOfPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X784A14F787E041D7">54.3-4 DomainOfPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7CD84B107831E0FC">54.3-5 ImageOfPartialPermCollection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X8333293F87F654FA">54.3-6 ImageListOfPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7F0724A07A14DCF7">54.3-7 ImageSetOfPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X82AAFF938623422E">54.3-8 FixedPointsOfPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X82FE981A87FAA2DC">54.3-9 MovedPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7FAF969C84CDC742">54.3-10 NrFixedPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X81F5C64E7DAD27A7">54.3-11 NrMovedPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X84A49C977E1E29AA">54.3-12 SmallestMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7D4290A785ABC86D">54.3-13 LargestMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X85280F1A7B1014BA">54.3-14 SmallestImageOfMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7A95CD437BC1CB1A">54.3-15 LargestImageOfMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X873A9F717DA75CBC">54.3-16 IndexPeriodOfPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7C04AA377F080722">54.3-17 SmallestIdempotentPower</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X8185065E788BDD0D">54.3-18 ComponentsOfPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7CB51EB67FFA95E9">54.3-19 NrComponentsOfPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7AAAAE4082B30E18">54.3-20 ComponentRepsOfPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7A8FB86C78C49F85">54.3-21 LeftOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X857FC10C81507E8B">54.3-22 One</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7D90CF497D58D759">54.3-23 MultiplicativeZero</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap54.html#X8585AA8B78E9CDFB">54.4 <span class="Heading">Changing the representation of a partial permutation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X81B32CB182489ACA">54.4-1 AsPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X87EC67747B260E98">54.4-2 AsPartialPerm</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap54.html#X848CD855802C6CE1">54.5 <span class="Heading">Operators and operations for partial permutations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7B8630027B7F0BCC">54.5-1 Inverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X792D3BA278DAB869"><code>54.5-2 \^</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7DC25BC47AAA9C73"><code>54.5-3 \/</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X8213CD6E7C461169"><code>54.5-4 \^</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X86469D597F8BC7CE"><code>54.5-5 \*</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X869DBDF67FA3817B"><code>54.5-6 \/</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X82E3A3E186A4F2D2">54.5-7 LeftQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X8659E9E57AC8D9CE"><code>54.5-8 \<</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7828338C7DB8AAC7"><code>54.5-9 \=</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X8382A0F8875CEB08">54.5-10 PermLeftQuoPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7C7F5EAB7E9A381D">54.5-11 PreImagePartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X797A6CC084068219">54.5-12 ComponentPartialPermInt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X87B1ED93785257C1">54.5-13 NaturalLeqPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X81BD69307D294A1C">54.5-14 ShortLexLeqPartialPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X83560BE678ACF855">54.5-15 TrimPartialPerm</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap54.html#X7849595B81D063EE">54.6 <span class="Heading">Displaying partial permutations</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap54.html#X7CCC82E07A73EB55">54.7 <span class="Heading">Semigroups and inverse semigroups of partial permutations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7D161674800B50E0">54.7-1 IsPartialPermSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7D7F0BAB82F0D820">54.7-2 DegreeOfPartialPermSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X81D271B380995F8A">54.7-3 SymmetricInverseSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7C8AEA50834060DD">54.7-4 IsSymmetricInverseSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7EA51F087CF7621F">54.7-5 NaturalPartialOrder</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap54.html#X7FE18EBE79B9C17C">54.7-6 IsomorphismPartialPermSemigroup</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap55.html#X7D0D096B81365B02">55 <span class="Heading">Additive Magmas</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap55.html#X82A4AB7B812B063B">55.1 <span class="Heading">(Near-)Additive Magma Categories</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X8129E95D83227658">55.1-1 IsNearAdditiveMagma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X7DADE4577D0A7208">55.1-2 IsNearAdditiveMagmaWithZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X7FC3A9C178185942">55.1-3 IsNearAdditiveGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X8565FD0C847BAA3A">55.1-4 IsAdditiveMagma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X785B41A67D791783">55.1-5 IsAdditiveMagmaWithZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X7B8FBD9082CE271B">55.1-6 IsAdditiveGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap55.html#X7C39F9DE7CA22688">55.2 <span class="Heading">(Near-)Additive Magma Generation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X79C947CF8060335A">55.2-1 NearAdditiveMagma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X80F57FB47E1DB380">55.2-2 NearAdditiveMagmaWithZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X872307537ECC5755">55.2-3 NearAdditiveGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X85122CFD7BDAD668">55.2-4 NearAdditiveMagmaByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X81880460851DEFBC">55.2-5 NearAdditiveMagmaWithZeroByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X85F120B68576B267">55.2-6 NearAdditiveGroupByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X7AA6092683FC0F9C">55.2-7 SubnearAdditiveMagma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X784859197D89A548">55.2-8 SubnearAdditiveMagmaWithZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X844C49BA807AB99F">55.2-9 SubnearAdditiveGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap55.html#X799E6CC28737BF1B">55.3 <span class="Heading">Attributes and Properties for (Near-)Additive Magmas</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X82D471327A9CA960">55.3-1 IsAdditivelyCommutative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X804B178884002A40">55.3-2 GeneratorsOfNearAdditiveMagma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X7EB9ABF880DCAE01">55.3-3 GeneratorsOfNearAdditiveMagmaWithZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X7EA15714795D71CF">55.3-4 GeneratorsOfNearAdditiveGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X851EA2E67F0C9A75">55.3-5 AdditiveNeutralElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X78FB0A5C86DC86F9">55.3-6 TrivialSubnearAdditiveMagmaWithZero</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap55.html#X7BB03781863BE4EB">55.4 <span class="Heading">Operations for (Near-)Additive Magmas</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X845E915B87D2AC16">55.4-1 <span class="Heading">ClosureNearAdditiveGroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap55.html#X8142D994794B700A">55.4-2 ShowAdditionTable</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap56.html#X81897F6082CACB59">56 <span class="Heading">Rings</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap56.html#X839FC48687C25FCD">56.1 <span class="Heading">Generating Rings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X80FD843C8221DAC9">56.1-1 IsRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X820B172A860A5B1A">56.1-2 <span class="Heading">Ring</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X83AFFCC77DE6ABDA">56.1-3 <span class="Heading">DefaultRing</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7D736E027DFD8961">56.1-4 RingByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X839E609480495E27">56.1-5 DefaultRingByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7D0428D87E63288C">56.1-6 GeneratorsOfRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X860E4AC78520D27E">56.1-7 Subring</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X819B0AFE79C78C34">56.1-8 <span class="Heading">ClosureRing</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X8350500B8576F833">56.1-9 Quotient</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap56.html#X8776C3F97A731E70">56.2 <span class="Heading">Ideals of Rings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7C486A7C821D79F0">56.2-1 TwoSidedIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7C8E196478C7431A">56.2-2 TwoSidedIdealNC</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7DF623847B338850">56.2-3 IsTwoSidedIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X86C998178690DAE0">56.2-4 TwoSidedIdealByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X82D8B07281EB0AC7">56.2-5 LeftIdealByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X858EAEAF87751428">56.2-6 RightIdealByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X86AAF5F9800E97EE">56.2-7 GeneratorsOfTwoSidedIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7B20BD2B7FAFBD64">56.2-8 GeneratorsOfLeftIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X80F2239F8653FF74">56.2-9 GeneratorsOfRightIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X81D81D027C2F8D06">56.2-10 LeftActingRingOfIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X83D9D7408706B69A">56.2-11 AsLeftIdeal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap56.html#X790DD00586F9B8B8">56.3 <span class="Heading">Rings With One</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7E601FBD8020A0F3">56.3-1 IsRingWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X80942A318417366E">56.3-2 <span class="Heading">RingWithOne</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X851115EC79B8C393">56.3-3 RingWithOneByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7F9F122C831BCDD1">56.3-4 GeneratorsOfRingWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7D0BADF178D4DDF8">56.3-5 SubringWithOne</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap56.html#X797F5869874BDBFB">56.4 <span class="Heading">Properties of Rings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X87A7D5B584713B52">56.4-1 IsIntegralRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X789A917085DB7527">56.4-2 IsUniqueFactorizationRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7D4BB44187C55BF2">56.4-3 IsLDistributive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X79A5AEE786AED315">56.4-4 IsRDistributive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X86716D4F7B968604">56.4-5 IsDistributive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X82DECD237D49D937">56.4-6 IsAnticommutative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7EC0FEC88535E8CC">56.4-7 IsZeroSquaredRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X799BEF8581971A13">56.4-8 IsJacobianRing</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap56.html#X8130085978A9B3C4">56.5 <span class="Heading">Units and Factorizations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X85CBFBAE78DE72E8">56.5-1 IsUnit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X853C045B7BA6A580">56.5-2 Units</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7B307F217DDC7E20">56.5-3 IsAssociated</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7A69C9097E17D161">56.5-4 Associates</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7B1A9A4C7C59FB36">56.5-5 StandardAssociate</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7EB6803C789E027D">56.5-6 StandardAssociateUnit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7CD7C64A7D961A18">56.5-7 IsIrreducibleRingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7AA107AE7F79C6D8">56.5-8 IsPrime</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X82D6EDC685D12AE2">56.5-9 Factors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X8559CC7B80C479F1">56.5-10 PadicValuation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap56.html#X7F12BB99865EB7BF">56.6 <span class="Heading">Euclidean Rings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X808B8E8E80D48E4A">56.6-1 IsEuclideanRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X784234088350D4E4">56.6-2 EuclideanDegree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7A93FA788318B147">56.6-3 EuclideanQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7B5E9639865E91BA">56.6-4 EuclideanRemainder</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X876B7532801A1B35">56.6-5 QuotientRemainder</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap56.html#X7E9CF2C07C4A6CEE">56.7 <span class="Heading">Gcd and Lcm</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7DE207718456F98F">56.7-1 <span class="Heading">Gcd</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7836D50F8341D6E1">56.7-2 GcdOp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7ABB91EF838075EF">56.7-3 <span class="Heading">GcdRepresentation</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X81392E7F84956341">56.7-4 GcdRepresentationOp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X836DB8B47A0219FB">56.7-5 ShowGcd</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7ABA92057DD6C7AF">56.7-6 <span class="Heading">Lcm</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7FB6C5A67AC1E8C1">56.7-7 LcmOp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X8555913A83D716A4">56.7-8 QuotientMod</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X805A35D684B7A952">56.7-9 PowerMod</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X87711E6F8024A358">56.7-10 InterpolatedPolynomial</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap56.html#X7B13484581169439">56.8 <span class="Heading">Homomorphisms of Rings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7DE9CC5B877C91DA">56.8-1 RingGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X78C1016284F08026">56.8-2 RingHomomorphismByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7D01646A7CCBEDBB">56.8-3 RingHomomorphismByImagesNC</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X83D53D98809EC461">56.8-4 NaturalHomomorphismByIdeal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap56.html#X81D526A57B375AAD">56.9 <span class="Heading">Small Rings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7E86DCB7812DF04C">56.9-1 SmallRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7F2EE9AF83DCE641">56.9-2 NumberSmallRings</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X8070D20B86148929">56.9-3 Subrings</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X83629803819C4A6F">56.9-4 Ideals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X82AD6F187B550060">56.9-5 DirectSum</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap56.html#X7E7B1B727EA434CF">56.9-6 RingByStructureConstants</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap57.html#X8183A6857B0C3633">57 <span class="Heading">Modules</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap57.html#X87A33EFD7CC179C1">57.1 <span class="Heading">Generating modules</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X7C62FE5282E9C505">57.1-1 IsLeftOperatorAdditiveGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X7ED323027B291BDF">57.1-2 IsLeftModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X7F76B1FD84775025">57.1-3 GeneratorsOfLeftOperatorAdditiveGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X7C7684EF867323C2">57.1-4 GeneratorsOfLeftModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X7EB3E46D7BC4A35C">57.1-5 AsLeftModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X7F19AD3D799D0469">57.1-6 IsRightOperatorAdditiveGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X8479A5AA7DF25F50">57.1-7 IsRightModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X7DBC4BCB876EEE1C">57.1-8 GeneratorsOfRightOperatorAdditiveGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X8586A83B85F176F6">57.1-9 GeneratorsOfRightModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X79ED1D7D7F0AE59A">57.1-10 LeftModuleByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X86F070E0807DC34E">57.1-11 LeftActingDomain</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap57.html#X7934FAE97B6D2AD8">57.2 <span class="Heading">Submodules</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X8465103F874BC07B">57.2-1 Submodule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X83CF3AD18050C982">57.2-2 SubmoduleNC</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X7C68C4E287481EC0">57.2-3 ClosureLeftModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X7980BC20856B2B7D">57.2-4 TrivialSubmodule</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap57.html#X85BD57F27F513D3E">57.3 <span class="Heading">Free Modules</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X7C4832187F3D9228">57.3-1 IsFreeLeftModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X7C043E307E344AEE">57.3-2 FreeLeftModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X7E6926C6850E7C4E">57.3-3 Dimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X802DB9FB824B0167">57.3-4 IsFiniteDimensional</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X7909E8E785420F0E">57.3-5 UseBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X7C8F844783F4FA09">57.3-6 IsRowModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X81FCC1D780435CF1">57.3-7 IsMatrixModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X853E085C868196EF">57.3-8 IsFullRowModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X848041A47BC4B038">57.3-9 FullRowModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X814CEA62842CF5BB">57.3-10 IsFullMatrixModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap57.html#X7A0C871B7C446F1F">57.3-11 FullMatrixModule</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap58.html#X80A8E676814A19FD">58 <span class="Heading">Fields and Division Rings</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap58.html#X82B74B458705B3CE">58.1 <span class="Heading">Generating Fields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X7F2CAA9E7A16913D">58.1-1 IsDivisionRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X7A5AE30E7C0F457C">58.1-2 IsField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X871AA7D58263E9AC">58.1-3 Field</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X7D9F7FD4786691EE">58.1-4 DefaultField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X7C298A40852C2AFF">58.1-5 DefaultFieldByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X7EF624958648D0FA">58.1-6 GeneratorsOfDivisionRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X7AA715317A81261B">58.1-7 GeneratorsOfField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X8641861A8550F8BE">58.1-8 DivisionRingByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X7C193B7D7AFB29BE">58.1-9 AsDivisionRing</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap58.html#X7C53566A839B57F6">58.2 <span class="Heading">Subfields of Fields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X7FE1FA217A08DCE5">58.2-1 Subfield</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X82A0E79A7B9799E0">58.2-2 FieldOverItselfByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X86DB31B57FB4F570">58.2-3 PrimitiveElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X7DD27F927BD57FDE">58.2-4 PrimeField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X84B6F1E67AD0E33D">58.2-5 IsPrimeField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X7845CECE86A83219">58.2-6 DegreeOverPrimeField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X7ADDCBF47E2ED3D4">58.2-7 DefiningPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X8173DA4982DB1E8A">58.2-8 RootOfDefiningPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X82718B3B818DC699">58.2-9 FieldExtension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X83490C65819D85FE">58.2-10 Subfields</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap58.html#X7D9A02B07D08FA40">58.3 <span class="Heading">Galois Action</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X80CAA5BA82F09ED2">58.3-1 GaloisGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X8738C6687D784BB5">58.3-2 MinimalPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X80FE7E017C2D255C">58.3-3 TracePolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X838515278587FF01">58.3-4 Norm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X7DD17EB581200AD6">58.3-5 <span class="Heading">Traces of field elements and matrices</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X837A4A5781F8EE92">58.3-6 Conjugates</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap58.html#X8236A8B47E6AAD93">58.3-7 NormalBase</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap59.html#X7893ABF67A028802">59 <span class="Heading">Finite Fields</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap59.html#X7B9DCCCC83400B47">59.1 <span class="Heading">Finite Field Elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X7D3DF32C84FEBD25">59.1-1 IsFFE</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X7AA52FAF7EDEDD56">59.1-2 Z</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X8612BCEA816CF1B9">59.1-3 IsLexOrderedFFE</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap59.html#X7A79399283EF78D0">59.2 <span class="Heading">Operations for Finite Field Elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X828E846E7C1EA3DD">59.2-1 DegreeFFE</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X7B049A3478B369E4">59.2-2 LogFFE</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X79F48E337FC2746A">59.2-3 IntFFE</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X7DABD827848BCC2A">59.2-4 IntFFESymm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X8009968782F18888">59.2-5 IntVecFFE</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X807959EE82CED148">59.2-6 AsInternalFFE</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X794AEB148410825B">59.2-7 RootFFE</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap59.html#X81B54A8378734C33">59.3 <span class="Heading">Creating Finite Fields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X7979F51D7C43AB05">59.3-1 DefaultField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X8592DBB086A8A9BE">59.3-2 GaloisField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X788B1ECD83C70516">59.3-3 PrimitiveRoot</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap59.html#X7A5F075185CE5B06">59.4 <span class="Heading">Frobenius Automorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X8758E4AB7D0A1955">59.4-1 FrobeniusAutomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap59.html#X869919BB7EBE5741">59.5 <span class="Heading">Conway Polynomials</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X7C2425A786F09054">59.5-1 ConwayPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X78A7C1247E129AD9">59.5-2 IsCheapConwayPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X7ECC593583E68A6C">59.5-3 RandomPrimitivePolynomial</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap59.html#X78EE3656879C3B88">59.6 <span class="Heading">Printing, Viewing and Displaying Finite Field Elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap59.html#X80DAAA5E7C79C94C">59.6-1 ViewObj</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap60.html#X80510B5880521FDC">60 <span class="Heading">Abelian Number Fields</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap60.html#X7D4E43E5799753B5">60.1 <span class="Heading">Construction of Abelian Number Fields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap60.html#X80D21D80850EFA4B">60.1-1 CyclotomicField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap60.html#X80E5AD028143E11E">60.1-2 AbelianNumberField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap60.html#X82F53C65802FF551">60.1-3 GaussianRationals</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap60.html#X81B5FE06781DB824">60.2 <span class="Heading">Operations for Abelian Number Fields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap60.html#X7B0AB0FB7A4136C4">60.2-1 Factors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap60.html#X87D78F5E875F2E8A">60.2-2 IsNumberField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap60.html#X7D202D707D5708FA">60.2-3 IsAbelianNumberField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap60.html#X84CAE4627F0CD639">60.2-4 IsCyclotomicField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap60.html#X87E7313D8070B9CC">60.2-5 GaloisStabilizer</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap60.html#X7D2421AC8491D2BE">60.3 <span class="Heading">Integral Bases of Abelian Number Fields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap60.html#X7F52BEA0862E06F2">60.3-1 ZumbroichBase</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap60.html#X87DB9C2C858B722A">60.3-2 LenstraBase</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap60.html#X7E4AB4B17C7BA10C">60.4 <span class="Heading">Galois Groups of Abelian Number Fields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap60.html#X7B55A90582E818F3">60.4-1 GaloisGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap60.html#X8643D4B47A827D9D">60.4-2 ANFAutomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap60.html#X85E9E90D7FE877CC">60.5 <span class="Heading">Gaussians</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap60.html#X80BD5EAB879F096E">60.5-1 GaussianIntegers</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap60.html#X7BFD33D27BFB7C5A">60.5-2 IsGaussianIntegers</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap61.html#X7DAD6700787EC845">61 <span class="Heading">Vector Spaces</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X8754F7207CFDA38B">61.1 <span class="Heading">IsLeftVectorSpace (Filter)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X80290A908241706B">61.1-1 IsLeftVectorSpace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X87AD06FE873619EA">61.2 <span class="Heading">Constructing Vector Spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X805413157CE9BECF">61.2-1 VectorSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X78C9826780BC9AE6">61.2-2 Subspace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7B001BAF7D5FD5D0">61.2-3 AsVectorSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7D4F84C27EDAC89B">61.2-4 AsSubspace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X789FB2D883E53662">61.3 <span class="Heading">Operations and Attributes for Vector Spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X849651C6830C94A1">61.3-1 GeneratorsOfLeftVectorSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X86DC71A9835430FD">61.3-2 TrivialSubspace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X8125675583357131">61.4 <span class="Heading">Domains of Subspaces of Vector Spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7975E41A7B29C3FD">61.4-1 Subspaces</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7A8F5C367FAE3D1B">61.4-2 IsSubspacesVectorSpace</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X828AA09B87F14FAD">61.5 <span class="Heading">Bases of Vector Spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X8739510881F5D862">61.5-1 IsBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X837BE54C80DE368E">61.5-2 Basis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7C8EBFF5805F8C51">61.5-3 CanonicalBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X8786D40B84120F38">61.5-4 RelativeBasis</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X839B9C4880EBFB5F">61.6 <span class="Heading">Operations for Vector Space Bases</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7B1F17AE8027A590">61.6-1 BasisVectors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X81E8AE88843B70FF">61.6-2 UnderlyingLeftModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X80B32F667BF6AFD8">61.6-3 Coefficients</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7D305AB3834889BF">61.6-4 LinearCombination</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7EB0D16A7EC2DEE3">61.6-5 EnumeratorByBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X855625D47979005D">61.6-6 IteratorByBasis</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X82809D6C82DE4EC2">61.7 <span class="Heading">Operations for Special Kinds of Bases</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7CC2B3DD81628CE9">61.7-1 IsCanonicalBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X86DE147F8606B739">61.7-2 IsIntegralBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7FC051C579D61223">61.7-3 IsNormalBasis</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X7C11B9C3819F3EA2">61.8 <span class="Heading">Mutable Bases</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7F466FB47F7E9F00">61.8-1 IsMutableBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X8115C061819E5172">61.8-2 MutableBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7EC90F4F7BCAF8D4">61.8-3 NrBasisVectors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7BA87512823A8CFD">61.8-4 ImmutableBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X85B50AC77A22108B">61.8-5 IsContainedInSpan</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7B52C99B84316F61">61.8-6 CloseMutableBasis</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X7D937EBC7DE2819B">61.9 <span class="Heading">Row and Matrix Spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X79B305CE87511C4B">61.9-1 IsRowSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7A2BBBA07B2BE8F8">61.9-2 IsMatrixSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X83724C157F4FDFB4">61.9-3 IsGaussianSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X80209A8785126AAB">61.9-4 FullRowSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X876B66C37A7B749F">61.9-5 FullMatrixSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X8534A750878478D0">61.9-6 DimensionOfVectors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X865A540F85FAE2DF">61.9-7 IsSemiEchelonized</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X87DCA09579589106">61.9-8 SemiEchelonBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7C3CC5F97FA048A4">61.9-9 IsCanonicalBasisFullRowModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X83D282697C1A3148">61.9-10 IsCanonicalBasisFullMatrixModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7D6537F87E940344">61.9-11 NormedRowVectors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X815C69A57C042C34">61.9-12 SiftedVector</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X7F61CECA84CEF39D">61.10 <span class="Heading">Vector Space Homomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X82013D328645E370">61.10-1 LeftModuleGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X85F5293983E47B5A">61.10-2 LeftModuleHomomorphismByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X8477E6C3872A6DBB">61.10-3 LeftModuleHomomorphismByMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X8494AA5D7C3B88AD">61.10-4 NaturalHomomorphismBySubspace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X80015C78876B4F1E">61.10-5 Hom</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X8680ADD381ECF879">61.10-6 End</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7A9A08EA79259659">61.10-7 IsFullHomModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7C4737687E76A24A">61.10-8 IsPseudoCanonicalBasisFullHomModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X84F87C327A1856F2">61.10-9 IsLinearMappingsModule</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X81503EB77FCE648D">61.11 <span class="Heading">Vector Spaces Handled By Nice Bases</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X826FD4BC7BA0559D">61.11-1 NiceFreeLeftModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X807B8032780C59A4">61.11-2 NiceVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X79350786800C2DD8">61.11-3 NiceFreeLeftModuleInfo</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X8388E0248690C214">61.11-4 NiceBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X82BC30A487967F5B">61.11-5 IsBasisByNiceBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X79D1DEA679AEDA40">61.11-6 IsHandledByNiceBasis</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X8238195B851D3C44">61.12 <span class="Heading">How to Implement New Kinds of Vector Spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7DE34C3E837FCBC3">61.12-1 DeclareHandlingByNiceBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7E6077F0830A28DA">61.12-2 NiceBasisFiltersInfo</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X7A374553786DF5E7">61.12-3 CheckForHandlingByNiceBasis</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap61.html#X78515F448644204E">61.13 <span class="Heading">Tensor Products and Exterior and Symmetric Powers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X81B2276A7EBA8ED1">61.13-1 TensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X787BB7FF85F0AD68">61.13-2 ExteriorPower</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap61.html#X79E2C2AF842E8419">61.13-3 SymmetricPower</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap62.html#X7DDBF6F47A2E021C">62 <span class="Heading">Algebras</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X830EDB5F85645FFB">62.1 <span class="Heading">InfoAlgebra (Info Class)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X8665F459841AAD53">62.1-1 InfoAlgebra</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X8686DEBA85D3F3B6">62.2 <span class="Heading">Constructing Algebras by Generators</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7B213851791A594B">62.2-1 Algebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X80FE16EA84EE56CD">62.2-2 AlgebraWithOne</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X7A7B00127DC9DD40">62.3 <span class="Heading">Constructing Algebras as Free Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X83484C917D8F7A1A">62.3-1 FreeAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7FBD04B07B85623D">62.3-2 FreeAlgebraWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X87835FFE79D2E068">62.3-3 FreeAssociativeAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X845A777584A7D711">62.3-4 FreeAssociativeAlgebraWithOne</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X7E8F45547CC07CE5">62.4 <span class="Heading">Constructing Algebras by Structure Constants</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7CC58DFD816E6B65">62.4-1 AlgebraByStructureConstants</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X80D528A57FD64873">62.4-2 AlgebraWithOneByStructureConstants</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X804ADF0280F67CDC">62.4-3 StructureConstantsTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7F1203A1793411DF">62.4-4 EmptySCTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X817BD086876EC1C4">62.4-5 SetEntrySCTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7F333822780B6731">62.4-6 GapInputSCTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7C23ED85814C0371">62.4-7 TestJacobi</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X78B633CE7A5B9F9A">62.4-8 IdentityFromSCTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7F2A71608602635D">62.4-9 QuotientFromSCTable</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X79B7C3078112E7E1">62.5 <span class="Heading">Some Special Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X83DF4BCC7CE494FC">62.5-1 QuaternionAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7B807702782F56FF">62.5-2 ComplexificationQuat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X78C88A38853A8443">62.5-3 OctaveAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7D88E42B7DE087B0">62.5-4 FullMatrixAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X78B8BA77869DAA13">62.5-5 NullAlgebra</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X7DF5989886BE611E">62.6 <span class="Heading">Subalgebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X8396643D7A49EEAD">62.6-1 Subalgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7C6B08657BD836C3">62.6-2 SubalgebraNC</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X83ECF489846F00B0">62.6-3 SubalgebraWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7A11B177868E76AA">62.6-4 SubalgebraWithOneNC</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7FDD953A84CFC3D2">62.6-5 TrivialSubalgebra</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X81EE8C1F7D7A7CF8">62.7 <span class="Heading">Ideals of Algebras</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X7DC95D6982C9D7B0">62.8 <span class="Heading">Categories and Properties of Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7FEDFAA383AB20D2">62.8-1 IsFLMLOR</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X85C1E13A877DF2C8">62.8-2 IsFLMLORWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X801ED693808F6C84">62.8-3 IsAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X80B21AC27DE6D068">62.8-4 IsAlgebraWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X839BAC687B4E1A1D">62.8-5 IsLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X877DF13387831A6A">62.8-6 IsSimpleAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7C5AECE87D79D075">62.8-7 IsFiniteDimensional</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X82B3A9077D0CB453">62.8-8 IsQuaternion</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X7E9273E47CF38CF1">62.9 <span class="Heading">Attributes and Operations for Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X83B055F37EBF2438">62.9-1 GeneratorsOfAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7FA408307A5A420E">62.9-2 GeneratorsOfAlgebraWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7D309FD37D94B196">62.9-3 ProductSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X875CD2B37EE9A8A2">62.9-4 PowerSubalgebraSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X788F4E6184E5C863">62.9-5 AdjointBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X800A410B8536E6DD">62.9-6 IndicesOfAdjointBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7BA35CB28062D407">62.9-7 AsAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X878323367D0B68EB">62.9-8 AsAlgebraWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7A922D26805AFF99">62.9-9 AsSubalgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7B964BC37A975E48">62.9-10 AsSubalgebraWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7C280DAC7F840B60">62.9-11 MutableBasisOfClosureUnderAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7BA1739D7F8B3A2B">62.9-12 MutableBasisOfNonassociativeAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X8467B687823371F9">62.9-13 MutableBasisOfIdealInNonassociativeAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7C591B7C7DEA7EEB">62.9-14 DirectSumOfAlgebras</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7D0EB1437D3D9495">62.9-15 FullMatrixAlgebraCentralizer</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X850C29907A509533">62.9-16 RadicalOfAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X82571785846CF05C">62.9-17 CentralIdempotentsOfAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7CFB230582C26DAA">62.9-18 DirectSumDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X85C58364833E014C">62.9-19 LeviMalcevDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7DCA2568870A2D34">62.9-20 Grading</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X7E94B857847F95C1">62.10 <span class="Heading">Homomorphisms of Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X83CE798C7D39E368">62.10-1 AlgebraGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7A7F97ED8608C882">62.10-2 AlgebraHomomorphismByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X8326D1BD79725462">62.10-3 AlgebraHomomorphismByImagesNC</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X8057E55B864567AD">62.10-4 AlgebraWithOneGeneralMappingByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X866F32B5846E5857">62.10-5 AlgebraWithOneHomomorphismByImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X80BF4D6A7FDC959A">62.10-6 AlgebraWithOneHomomorphismByImagesNC</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X825149467C57DEFC">62.10-7 AlgebraHomomorphismByFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X8712E5C1861CC32C">62.10-8 NaturalHomomorphismByIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X8705A9C68102FEA3">62.10-9 OperationAlgebraHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7B249E8E86D895F0">62.10-10 NiceAlgebraMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X79D770777D873F80">62.10-11 IsomorphismFpAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7FB760F9813B0789">62.10-12 IsomorphismMatrixAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7F8D3DF2863EC50D">62.10-13 IsomorphismSCAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7F34244B81979696">62.10-14 RepresentativeLinearOperation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap62.html#X818DE6C57D1A4B33">62.11 <span class="Heading">Representations of Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X8055B87F7ADBD66B">62.11-1 LeftAlgebraModuleByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X8026B99B7955A355">62.11-2 RightAlgebraModuleByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7F28A47E876427E0">62.11-3 BiAlgebraModuleByGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X852524F581613359">62.11-4 LeftAlgebraModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X8222F2B67D753036">62.11-5 RightAlgebraModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X84517770868DDA02">62.11-6 BiAlgebraModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X79AAB50D83A14A43">62.11-7 GeneratorsOfAlgebraModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X82B708BD84F3DAB1">62.11-8 IsAlgebraModuleElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X80E786467F9163F9">62.11-9 IsLeftAlgebraModuleElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X863756787E2B6E75">62.11-10 IsRightAlgebraModuleElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X85654EF07F708AC3">62.11-11 LeftActingAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X826298B37E1B1520">62.11-12 RightActingAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X8308408D86CFC3C9">62.11-13 ActingAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7C325A507EC9BA18">62.11-14 IsBasisOfAlgebraModuleElementSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X789863037B0E35D2">62.11-15 MatrixOfAction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X8742A7D27F26AFAB">62.11-16 SubAlgebraModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X86E0515987192F0E">62.11-17 LeftModuleByHomomorphismToMatAlg</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7EE41297867E41A8">62.11-18 RightModuleByHomomorphismToMatAlg</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X8729F0A678A4A09C">62.11-19 AdjointModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X84813BCD80BDF3C4">62.11-20 FaithfulModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7E16630185CE2C10">62.11-21 ModuleByRestriction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7885AAC87FDCF649">62.11-22 NaturalHomomorphismBySubAlgebraModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X85D0F3758551DADC">62.11-23 DirectSumOfAlgebraModules</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap62.html#X7D7A6486803B15CE">62.11-24 TranslatorSubalgebra</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap63.html#X85A22A8286596D02">63 <span class="Heading">Finitely Presented Algebras</span></a>
</div>
<div class="ContChap"><a href="chap64.html#X78559D4C800AF58A">64 <span class="Heading">Lie Algebras</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap64.html#X80A607C47B7A2E69">64.1 <span class="Heading">Lie Objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X87F121978775AF48">64.1-1 LieObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X83E5DD4381D9A65D">64.1-2 IsLieObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X8725993C7BF386EE">64.1-3 LieFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X81D9F5C6876FE93B">64.1-4 UnderlyingFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X874B2B2A7F5A9A78">64.1-5 UnderlyingRingElement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap64.html#X789A44F283C16B2B">64.2 <span class="Heading">Constructing Lie algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7D362350824FA115">64.2-1 LieAlgebraByStructureConstants</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7EEB79EE855E124C">64.2-2 RestrictedLieAlgebraByStructureConstants</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7C840A9F85D28C81">64.2-3 LieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7F7B34BD80F0F1C8">64.2-4 FreeLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X8735EE937A0081F0">64.2-5 FullMatrixLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X821B6C197C08878B">64.2-6 RightDerivations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7933F05F7DE342AB">64.2-7 SimpleLieAlgebra</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap64.html#X798391F47E835F85">64.3 <span class="Heading">Distinguished Subalgebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X8111F58E7DE3E25C">64.3-1 LieCentre</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X811444717EEDCC34">64.3-2 LieCentralizer</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7E62B6B37A75E09D">64.3-3 LieNormalizer</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7C95C0057C977747">64.3-4 LieDerivedSubalgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7D072F6D7A3D0BAF">64.3-5 LieNilRadical</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X8445C9F17F7CBEA1">64.3-6 LieSolvableRadical</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X86114F157DFF6523">64.3-7 CartanSubalgebra</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap64.html#X7A72840882F7A9B6">64.4 <span class="Heading">Series of Ideals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7DEF89A8869809F5">64.4-1 LieDerivedSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7900D17E7BA26A48">64.4-2 LieLowerCentralSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X86A8701C868828C7">64.4-3 LieUpperCentralSeries</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap64.html#X8208CE5F8286155F">64.5 <span class="Heading">Properties of a Lie Algebra</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7F97D08F7B738ADE">64.5-1 IsLieAbelian</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X78452F4E875A62A8">64.5-2 IsLieNilpotent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X859FF1B3812B8FCC">64.5-3 IsLieSolvable</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap64.html#X83F829017D46C544">64.6 <span class="Heading">Semisimple Lie Algebras and Root Systems</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X8401CDC2859F8A85">64.6-1 SemiSimpleType</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X82EBF10A7B3B6F6E">64.6-2 ChevalleyBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X79B5D27681193625">64.6-3 IsRootSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7D64D49479CBB203">64.6-4 IsRootSystemFromLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X80D15C027BB8029B">64.6-5 RootSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7CA021E28527763E">64.6-6 UnderlyingLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7B6B0BBD8035D7E5">64.6-7 PositiveRoots</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X81F9E0E67DD2688F">64.6-8 NegativeRoots</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X829C78427A442C23">64.6-9 PositiveRootVectors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7AB374DC87A39349">64.6-10 NegativeRootVectors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7DBD179E7CCF6699">64.6-11 SimpleSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X84E3FEF587CB66C3">64.6-12 CartanMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X878644D68571BF44">64.6-13 BilinearFormMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7FAE45B37C5779A0">64.6-14 CanonicalGenerators</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap64.html#X7945D07786D1C4BB">64.7 <span class="Heading">Semisimple Lie Algebras and Weyl Groups of Root Systems</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X82AA29DD7969A935">64.7-1 IsWeylGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X81EF01E57E5DC18A">64.7-2 SparseCartanMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X86BED5098322EBEF">64.7-3 WeylGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7829BC4D7F253649">64.7-4 ApplySimpleReflection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X80A7204F7D40D80F">64.7-5 LongestWeylWordPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7D4E213F82F73857">64.7-6 ConjugateDominantWeight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7E000FA97949BFD5">64.7-7 WeylOrbitIterator</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap64.html#X878080BB79BE3F2E">64.8 <span class="Heading">Restricted Lie algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X81F28B1D830F28EB">64.8-1 IsRestrictedLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7D7BD5908016461B">64.8-2 PthPowerImages</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X879BB01782E7D7A9">64.8-3 PthPowerImage</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X8692ADD581359CA1">64.8-4 JenningsLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X785251E879E1BFC6">64.8-5 PCentralLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X781ADBEC850C7DE7">64.8-6 NaturalHomomorphismOfLieAlgebraFromNilpotentGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap64.html#X7C419FFA835EBE12">64.9 <span class="Heading">The Adjoint Representation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X786886D882795F78">64.9-1 AdjointMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X873A64307AC6C63E">64.9-2 AdjointAssociativeAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X877CCFD5832E035D">64.9-3 KillingMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X8234046083B60F6E">64.9-4 KappaPerp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7A00601387A060CF">64.9-5 IsNilpotentElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X86EF3E6F7BC0A8AD">64.9-6 NonNilpotentElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7A912D9E7B3BA874">64.9-7 FindSl2</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap64.html#X7875070C85DD4E8E">64.10 <span class="Heading">Universal Enveloping Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X8226CD1680207A5F">64.10-1 UniversalEnvelopingAlgebra</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap64.html#X7B8C71E07F50B286">64.11 <span class="Heading">Finitely Presented Lie Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X780A5B457A051110">64.11-1 FpLieAlgebraByCartanMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X79FD70C487EA9438">64.11-2 NilpotentQuotientOfFpLieAlgebra</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap64.html#X7FBCB43C86BDD9C2">64.12 <span class="Heading">Modules over Lie Algebras and Their Cohomology</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X82CC31CF79F59FEE">64.12-1 IsCochain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X79F3DF0D8791C2E3">64.12-2 Cochain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7CF2919081600A3D">64.12-3 CochainSpace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7D6760DA84683011">64.12-4 ValueCochain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X851F5EF47FA90CBC">64.12-5 LieCoboundaryOperator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7FB815F38143939E">64.12-6 Cocycles</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7C4F372C7AE2F739">64.12-7 Coboundaries</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap64.html#X78A201238137E822">64.13 <span class="Heading">Modules over Semisimple Lie Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7D8522E37ED1024A">64.13-1 DominantWeights</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X79AAC71E8267E9F8">64.13-2 DominantCharacter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7BE7129384B012DF">64.13-3 DecomposeTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7D67A9BC7E4714D9">64.13-4 DimensionOfHighestWeightModule</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap64.html#X840E5FAE7D2C2702">64.14 <span class="Heading">Admissible Lattices in UEA</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X86E6722379576746">64.14-1 IsUEALatticeElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X79F4F58B7888B0A5">64.14-2 LatticeGeneratorsInUEA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X875FD1627F3B72DB">64.14-3 ObjByExtRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X8248DB547B02B0FA">64.14-4 IsWeightRepElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7FB14F7F80EFF33F">64.14-5 HighestWeightModule</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap64.html#X8750BDBF7EA5E868">64.15 <span class="Heading">Tensor Products and Exterior and Symmetric Powers of Algebra Modules</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7A1E0AC4800E7FDA">64.15-1 TensorProductOfAlgebraModules</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X7F4AB6A1863E8FB2">64.15-2 ExteriorPowerOfAlgebraModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap64.html#X842DF85687D61A56">64.15-3 SymmetricPowerOfAlgebraModule</a></span>
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</div>
<div class="ContChap"><a href="chap65.html#X825897DC7A16E07D">65 <span class="Heading">Magma Rings</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap65.html#X8398F87F8231A163">65.1 <span class="Heading">Free Magma Rings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X7B9AF0A47F44E4B4">65.1-1 FreeMagmaRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X86D2CA90847C091B">65.1-2 GroupRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X7A24B95C8210BD09">65.1-3 IsFreeMagmaRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X8382ED697A28CE67">65.1-4 IsFreeMagmaRingWithOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X82C63644805EB1EE">65.1-5 IsGroupRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X848D60417DFF7947">65.1-6 UnderlyingMagma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X7B21DB3E7CD80983">65.1-7 AugmentationIdeal</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap65.html#X8402D3897F2C5955">65.2 <span class="Heading">Elements of Free Magma Rings</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X827B2D7D7E41780C">65.2-1 IsMagmaRingObjDefaultRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X7D9C684A81E66310">65.2-2 IsElementOfFreeMagmaRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X869768AF7B444BF8">65.2-3 IsElementOfFreeMagmaRingFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X843D1D8578C33513">65.2-4 CoefficientsAndMagmaElements</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X78C3DB417E353390">65.2-5 ZeroCoefficient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X8671DE0A81BEEFB0">65.2-6 ElementOfMagmaRing</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap65.html#X80366F1480ACD8DF">65.3 <span class="Heading">Natural Embeddings related to Magma Rings</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap65.html#X81B002EE799B5E77">65.4 <span class="Heading">Magma Rings modulo Relations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X869D54847E881848">65.4-1 IsElementOfMagmaRingModuloRelations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X875BEB1A840FFAA4">65.4-2 IsElementOfMagmaRingModuloRelationsFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X85956ED27FA6AC68">65.4-3 NormalizedElementOfMagmaRingModuloRelations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X804B5AAB813E184D">65.4-4 IsMagmaRingModuloRelations</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap65.html#X7D859DBF81DFA751">65.5 <span class="Heading">Magma Rings modulo the Span of a Zero Element</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X7B3D45A6802B695C">65.5-1 IsElementOfMagmaRingModuloSpanOfZeroFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X872713EE84DA9B72">65.5-2 IsMagmaRingModuloSpanOfZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap65.html#X7A7F880D7D7D3722">65.5-3 MagmaRingModuloSpanOfZero</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap65.html#X79889F017F2EB7ED">65.6 <span class="Heading">Technical Details about the Implementation of Magma Rings</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap66.html#X7A14A6588268810A">66 <span class="Heading">Polynomials and Rational Functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X7A8FADCD875826DA">66.1 <span class="Heading">Indeterminates</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X79D0380D7FA39F7D">66.1-1 <span class="Heading">Indeterminate</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X816C8D797C804380">66.1-2 IndeterminateNumberOfUnivariateRationalFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7A2FA46885EF403D">66.1-3 IndeterminateOfUnivariateRationalFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7FD4AC807A1C8E27">66.1-4 IndeterminateName</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X791A06E67F784328">66.1-5 CIUnivPols</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X86A68FD582F4F757">66.2 <span class="Heading">Operations for Rational Functions</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X824B6D328643CE04">66.3 <span class="Heading">Comparison of Rational Functions</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X7D871EA180E9486C">66.4 <span class="Heading">Properties and Attributes of Rational Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X86C92F677DA9347F">66.4-1 IsPolynomialFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7D7D2667803D8D8A">66.4-2 NumeratorOfRationalFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X78DC1B5B866ADB6C">66.4-3 DenominatorOfRationalFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7974B0707C8DAB6C">66.4-4 IsPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7914771F7C6013EF">66.4-5 AsPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X8738F73583273FCA">66.4-6 IsUnivariateRationalFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7F1F67527A35A9CE">66.4-7 CoefficientsOfUnivariateRationalFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X86A2546685D0016B">66.4-8 IsUnivariatePolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X78C9524D7F2708C2">66.4-9 CoefficientsOfUnivariatePolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X79138FF28213B6EC">66.4-10 IsLaurentPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7F2A49208341C2A8">66.4-11 IsConstantRationalFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X834B54947FAADEA4">66.4-12 IsPrimitivePolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X87531E03849391C1">66.4-13 SplittingField</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X82E2F1707FC2E553">66.5 <span class="Heading">Univariate Polynomials</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X8379F8CB7D0076BA">66.5-1 UnivariatePolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X85178A3E7B4F11E0">66.5-2 UnivariatePolynomialByCoefficients</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X78AF77C383245254">66.5-3 DegreeOfLaurentPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7CBB760C87B04F75">66.5-4 RootsOfPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X80CEB10D7879767F">66.5-5 RootsOfUPol</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7887FBC78149BB0C">66.5-6 QuotRemLaurpols</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7DDADF157879EFBF">66.5-7 UnivariatenessTestRationalFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7A3BC96B7A50DE98">66.5-8 InfoPoly</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X81499B5A823E6EA3">66.6 <span class="Heading">Polynomials as Univariate Polynomials in one Indeterminate</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X826B99B17ABD11BE">66.6-1 DegreeIndeterminate</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X85646FD07F9C60F5">66.6-2 PolynomialCoefficientsOfPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X80710E9B7D8340BD">66.6-3 LeadingCoefficient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7B3EAE41795598A5">66.6-4 LeadingMonomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7B57CEE2780D0E0B">66.6-5 Derivative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7C7D790A7D6E11AD">66.6-6 Discriminant</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X857AD5587EF49029">66.6-7 Resultant</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X85ABC4687DF05777">66.7 <span class="Heading">Multivariate Polynomials</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7A70769C7F52CD2D">66.7-1 <span class="Heading">Value</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X7ED3E7D17C7AC732">66.8 <span class="Heading">Minimal Polynomials</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X8643915A8424DAF8">66.8-1 MinimalPolynomial</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X837B8E55832CDFEB">66.9 <span class="Heading">Cyclotomic Polynomials</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X827FC7FE81EE4C02">66.9-1 CyclotomicPolynomial</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X8551EF5187509D69">66.10 <span class="Heading">Polynomial Factorization</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X83511D517B544F36">66.10-1 Factors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7F5A4ACB7AF9E329">66.10-2 FactorsSquarefree</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X7F45E9E47EA2C18B">66.11 <span class="Heading">Polynomials over the Rationals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7E66494B7B05A055">66.11-1 PrimitivePolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7A73A3877EB73566">66.11-2 PolynomialModP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7AB9A6257ED694EC">66.11-3 GaloisType</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7EB610D37D156DC6">66.11-4 ProbabilityShapes</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X7C178AB9866FDDE5">66.12 <span class="Heading">Factorization of Polynomials over the Rationals</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X8723075C81D2CCA6">66.12-1 BombieriNorm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X856D769D878AF7AE">66.12-2 MinimizedBombieriNorm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X8139BB0F87399F2C">66.12-3 HenselBound</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X79CC9C8D7C9F6B6A">66.12-4 OneFactorBound</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X844B3C6C87A0E7E0">66.13 <span class="Heading">Laurent Polynomials</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X8467263B7EFA013E">66.13-1 LaurentPolynomialByCoefficients</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X86D58AB67F86469F">66.13-2 CoefficientsOfLaurentPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X8381E1B582F38C85">66.13-3 IndeterminateNumberOfLaurentPolynomial</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X7C1708D27F97B05F">66.14 <span class="Heading">Univariate Rational Functions</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X83DD411179888783">66.14-1 UnivariateRationalFunctionByCoefficients</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7B1EE4E07A9631C6">66.14-2 TaylorSeriesRationalFunction</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X7C59471783C3FEDC">66.15 <span class="Heading">Polynomial Rings and Function Fields</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7D2F16E480060330">66.15-1 <span class="Heading">PolynomialRing</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X80D585E1793D4552">66.15-2 IndeterminatesOfPolynomialRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X8235D10781BE8003">66.15-3 CoefficientsRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7D631ACC86C584B7">66.15-4 IsPolynomialRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X86F391237A76D804">66.15-5 IsFiniteFieldPolynomialRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X782D07F77BCF67C1">66.15-6 IsAbelianNumberFieldPolynomialRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7D45213A8642033B">66.15-7 IsRationalsPolynomialRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X812E801484E3624E">66.15-8 <span class="Heading">FunctionField</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X8090C9EC85201AAC">66.15-9 IsFunctionField</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X85CA757B844F12AE">66.16 <span class="Heading">Univariate Polynomial Rings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X84DC2A59797A26DE">66.16-1 <span class="Heading">UnivariatePolynomialRing</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7A43D74B812401CA">66.16-2 IsUnivariatePolynomialRing</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X86E2ADEA784AD163">66.17 <span class="Heading">Monomial Orderings</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X79D4CBBF820EA204">66.17-1 IsMonomialOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7D052A017A73E91E">66.17-2 LeadingMonomialOfPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7B6231137BA8B95F">66.17-3 LeadingTermOfPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X798E707D86141087">66.17-4 LeadingCoefficientOfPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7EDE941781BA7F8B">66.17-5 MonomialComparisonFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7EDC3A457E7B591E">66.17-6 MonomialExtrepComparisonFun</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X852D7BB37ECE98E1">66.17-7 MonomialLexOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X786C866C824D2688">66.17-8 MonomialGrlexOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X8094C733808D1799">66.17-9 MonomialGrevlexOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X84AC871283A74EC0">66.17-10 EliminationOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7C99593584D478D7">66.17-11 PolynomialReduction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7DE7D4467EBAD916">66.17-12 PolynomialReducedRemainder</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7C8239057FD4EC03">66.17-13 PolynomialDivisionAlgorithm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7A30E10B820311D1">66.17-14 MonomialExtGrlexLess</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X79BAB2937E6085D6">66.18 <span class="Heading">Groebner Bases</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7A43611E876B7560">66.18-1 <span class="Heading">GroebnerBasis</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7DEF286384967C9E">66.18-2 <span class="Heading">ReducedGroebnerBasis</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7FC1EFE78498C17C">66.18-3 StoredGroebnerBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7C55702786D284A7">66.18-4 InfoGroebner</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X8113DD9B781CA6C1">66.19 <span class="Heading">Rational Function Families</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X855DD73C78A90BC3">66.19-1 RationalFunctionsFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X86E097307D188D3B">66.19-2 IsPolynomialFunctionsFamily</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7AADCA45826866FB">66.19-3 CoefficientsFamily</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X7E360788785DE530">66.20 <span class="Heading">The Representations of Rational Functions</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X7F44CF87801DB965">66.21 <span class="Heading">The Defining Attributes of Rational Functions</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X791E16C67A352263">66.21-1 IsRationalFunctionDefaultRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7DF955C87CBFC12B">66.21-2 ExtRepNumeratorRatFun</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X8059E74D7DCABDBC">66.21-3 ExtRepDenominatorRatFun</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X84F546F87B5ECFE0">66.21-4 ZeroCoefficientRatFun</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X833CE16579AB26E0">66.21-5 IsPolynomialDefaultRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X8406EE2E8775FBAB">66.21-6 ExtRepPolynomialRatFun</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7E1B98CC7BADAF56">66.21-7 IsLaurentPolynomialDefaultRep</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X791FADD278A2F32F">66.22 <span class="Heading">Creation of Rational Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X81297E4587A9F2A6">66.22-1 RationalFunctionByExtRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X79E445AF7849F48A">66.22-2 PolynomialByExtRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7E2A46D68472F492">66.22-3 LaurentPolynomialByExtRep</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X809028CD7C0EA7CE">66.23 <span class="Heading">Arithmetic for External Representations of Polynomials</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X855094857A78ABF9">66.23-1 ZippedSum</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7B911136782F0F6D">66.23-2 ZippedProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X87E5EB8985AF04CD">66.23-3 QuotientPolynomialsExtRep</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap66.html#X811B7D8E79E4BD46">66.24 <span class="Heading">Cancellation Tests for Rational Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X878A1AC87B492E3D">66.24-1 RationalFunctionByExtRepWithCancellation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X7BFB55887A153003">66.24-2 TryGcdCancelExtRepPolynomials</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap66.html#X8477D7337C4A98AB">66.24-3 HeuristicCancelPolynomialsExtRep</a></span>
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</div>
<div class="ContChap"><a href="chap67.html#X85732CEF7ECFCA68">67 <span class="Heading">Algebraic extensions of fields</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap67.html#X7AD9B24E78ADC27F">67.1 <span class="Heading">Creation of Algebraic Extensions</span></a>
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