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<h1>QPA</h1>
<h2>Quivers and Path Algebras</h2>
<p>Version 1.35</p>
<p>January 2024</p>
</div>
<p>
</p>
<p><b>The QPA-team
</b>
<br />Email: <span class="URL"><a href="mailto:oyvind.solberg@ntnu.no">oyvind.solberg@ntnu.no</a></span>
<br />Homepage: <span class="URL"><a href="https://folk.ntnu.no/oyvinso/QPA/">https://folk.ntnu.no/oyvinso/QPA/</a></span>
<br />Address: <br />Department of Mathematical Sciences<br /> NTNU<br /> N-7491 Trondheim<br /> Norway
</p>
<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p>The GAP4 deposited package <strong class="pkg">QPA</strong> extends the GAP functionality for computations with finite dimensional quotients of path algebras. <strong class="pkg">QPA</strong> has data structures for quivers, quotients of path algebras, representations of quivers with relations and complexes of modules. Basic operations on representations of quivers are implemented as well as constructing minimal projective resolutions of modules (using using linear algebra). A not necessarily minimal projective resolution constructed by using Groebner basis theory and a paper by Green-Solberg-Zacharia, "Minimal projective resolutions", has been implemented. A goal is to have a test for finite representation type. This work has started, but there is a long way left. Part of this work is to implement/port the functionality and data structures that was available in CREP.</p>
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2014-2024 by The QPA-team.</p>
<p><strong class="pkg">QPA</strong> is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. For details, see the FSF’s own site (<span class="URL"><a href="https://www.gnu.org/licenses/gpl.html">https://www.gnu.org/licenses/gpl.html</a></span>).</p>
<p>If you obtained <strong class="pkg">QPA</strong>, we would be grateful for a short notification sent to one of members of the QPA-team. If you publish a result which was partially obtained with the usage of <strong class="pkg">QPA</strong>, please cite it in the following form:</p>
<p>The QPA-team, <strong class="pkg">QPA</strong> - Quivers, path algebras and representations, Version 1.35; 2024 (<span class="URL"><a href="https://folk.ntnu.no/oyvinso/QPA/">https://folk.ntnu.no/oyvinso/QPA/</a></span>)</p>
<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>The system design of <strong class="pkg">QPA</strong> was initiated by Edward L. Green, Lenwood S. Heath, and Craig A. Struble. It was continued and completed by Randall Cone and Edward Green. We would like to thank the following people for their contributions:</p>
<div class="pcenter"><table class="GAPDocTablenoborder">
<tr>
<td class="tdleft">Chain complexes</td>
<td class="tdleft">Kristin Krogh Arnesen and Øystein Skartsæterhagen</td>
</tr>
<tr>
<td class="tdleft">Degeneration order for modules in finite type</td>
<td class="tdleft">Andrzej Mroz</td>
</tr>
<tr>
<td class="tdleft">GBNP interface (for Groebner bases)</td>
<td class="tdleft">Randall Cone</td>
</tr>
<tr>
<td class="tdleft">Homomorphisms of modules</td>
<td class="tdleft">Øyvind Solberg and Anette Wraalsen</td>
</tr>
<tr>
<td class="tdleft">Koszul duals</td>
<td class="tdleft">Stephen Corwin</td>
</tr>
<tr>
<td class="tdleft">Matrix representations of path algebras</td>
<td class="tdleft">Øyvind Solberg and George Yuhasz</td>
</tr>
<tr>
<td class="tdleft">Opposite algebra and tensor products of algebras</td>
<td class="tdleft">Øystein Skartsæterhagen</td>
</tr>
<tr>
<td class="tdleft">Predefined classes of algebras</td>
<td class="tdleft">Andrzej Mroz and Øyvind Solberg</td>
</tr>
<tr>
<td class="tdleft">Projective resolutions (using Groebnar basis)</td>
<td class="tdleft">Randall Cone and Øyvind Solberg</td>
</tr>
<tr>
<td class="tdleft">Projective resolutions (using linear algebra)</td>
<td class="tdleft">Øyvind Solberg</td>
</tr>
<tr>
<td class="tdleft">Quickstart</td>
<td class="tdleft">Kristin Krogh Arnesen</td>
</tr>
<tr>
<td class="tdleft">Quivers, path algebras</td>
<td class="tdleft">Gerard Brunick</td>
</tr>
<tr>
<td class="tdleft">The bounded derived category</td>
<td class="tdleft">Kristin Krogh Arnesen and Øystein Skartsæterhagen</td>
</tr>
<tr>
<td class="tdleft">Unitforms</td>
<td class="tdleft">Øyvind Solberg</td>
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<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>
<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X8557083378F2A3B2">1.1 <span class="Heading">General aims</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7DB566D5785B7DBC">1.2 <span class="Heading">Installation and system requirements</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2.html#X7F83DF528480AEA3">2 <span class="Heading">Quickstart</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7D3488D984288697">2.1 <span class="Heading">Example 1 -- quivers, path
algebras and quotients of path algebras</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7D0E555F79FFD1EE">2.2 <span class="Heading">Example 2 -- Introducing modules</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X790BB3A1815A9B4D">2.3 <span class="Heading">Example 3 -- Constructing modules and module homomorphisms</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap3.html#X7FA7E6B581D41A94">3 <span class="Heading">Quivers</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7C14F4617F7E9F09">3.1 <span class="Heading">Information class, Quivers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X786A65F07FA1BB78">3.1-1 InfoQuiver</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X860B15D57EAB46D7">3.2 <span class="Heading">Constructing Quivers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7BD8455A7F2C5CA3">3.2-1 Quiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7EE08F058702A717">3.2-2 DynkinQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X83E498008233E2DD">3.2-3 OrderedBy</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X80BB6B6183134D88">3.3 <span class="Heading">Categories and Properties of Quivers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7E9C03497FD7778B">3.3-1 IsQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X82E99C5F8624BD86">3.3-2 IsAcyclicQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X846C44937B3AF09A">3.3-3 IsUAcyclicQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7909BF627C5D0D4A">3.3-4 IsConnectedQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7A2B55BD7B6F0360">3.3-5 IsTreeQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X85EC85B58688CCCC">3.3-6 IsDynkinQuiver</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X78BBB63B828EB9FB">3.4 <span class="Heading">Orderings of paths in a
quiver</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X80CF69E37B54F3C1">3.5 <span class="Heading">Attributes and Operations for Quivers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8198B2897FF5AC4B">3.5-1 .</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7C82A4BC7FB329D8">3.5-2 VerticesOfQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X82C42D7D820D5F9B">3.5-3 ArrowsOfQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7AF572F081AEFE98">3.5-4 AdjacencyMatrixOfQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7B4A7F0F813E63FC">3.5-5 GeneratorsOfQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X822BD7F37F8AF016">3.5-6 NumberOfVertices</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7AC77C9C7D069663">3.5-7 NumberOfArrows</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X84D1D1AA82689B03">3.5-8 OrderingOfQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X84B82F6F84A442AB">3.5-9 OppositeQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X87B9B85483CBB238">3.5-10 FullSubquiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X824D7C768708F006">3.5-11 ConnectedComponentsOfQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7C4BB3E5872FD483">3.5-12 DoubleQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7B4EA2D6869BBAF3">3.5-13 SeparatedQuiver</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X862A804B80C5A47D">3.6 <span class="Heading">Categories and Properties of Paths</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X78B503DB83F2B6DE">3.6-1 IsPath</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X791474408297F7A0">3.6-2 IsQuiverVertex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8266C9B8840C12EB">3.6-3 IsArrow</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8624A0B0795149CF">3.6-4 IsZeroPath</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7C8294338676C80E">3.7 <span class="Heading">Attributes and Operations of Paths</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X84D8493C7AAF4ACC">3.7-1 SourceOfPath</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X86827D3F78F51815">3.7-2 TargetOfPath</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7FF179D17C4F9FAC">3.7-3 LengthOfPath</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X781A8E06850E47B4">3.7-4 WalkOfPath</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7857704878577048">3.7-5 *</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X806A4814806A4814">3.7-6 =</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7DAD2700853E8C21">3.7-7 <</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X8158F0D27C4628FB">3.8 <span class="Heading">Attributes of Vertices</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X85E53C177F80E77E">3.8-1 IncomingArrowsOfVertex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8345D79381E46D73">3.8-2 OutgoingArrowsOfVertex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7C9CD0527CB9E6EF">3.8-3 InDegreeOfVertex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7A09EB648070276D">3.8-4 OutDegreeOfVertex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7A557B4C83B7C601">3.8-5 NeighborsOfVertex</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X79540DAB85902432">3.9 <span class="Heading">Posets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X80854BB778E3833E">3.9-1 Poset</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X858ADA3B7A684421">3.9-2 Size</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7AF9E0CD850F8B03">3.9-3 UnderlyingSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8468B5F77EBA547A">3.9-4 PartialOrderOfPoset</a></span>
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<div class="ContChap"><a href="chap4.html#X7E8A43A484CE0BA8">4 <span class="Heading">Path Algebras</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7DFB63A97E67C0A1">4.1 <span class="Heading">Introduction</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X848A225A84A15B1E">4.2 <span class="Heading">Constructing Path Algebras</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7CA1C87B8202C2E9">4.2-1 PathAlgebra</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X85A3A8767E7C11AD">4.3 <span class="Heading">Categories and Properties of Path Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8255FDF78315E1B3">4.3-1 IsPathAlgebra</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7DE2F2A48492041A">4.4 <span class="Heading">Attributes and Operations for Path Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7DF51D297E0E6A8B">4.4-1 AssociatedMonomialAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X83FBA499856580B0">4.4-2 QuiverOfPathAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8279084B828E5FD7">4.4-3 OrderingOfAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X86CDD46F7F05ADE9">4.4-4 .</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7CEF60107CE4616B">4.5 <span class="Heading">Operations on Path Algebra Elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X841C00E87E19528E">4.5-1 ElementOfPathAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E3FAB1F803E26FF">4.5-2 <</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X853C8B0B8665BFBB">4.5-3 IsLeftUniform</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7C06BE7483992634">4.5-4 IsRightUniform</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8735FBE180797557">4.5-5 IsUniform</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84C98E687A3A84D8">4.5-6 LeadingTerm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X80710E9B7D8340BD">4.5-7 LeadingCoefficient</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B3EAE41795598A5">4.5-8 LeadingMonomial</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8172B40181E1B7D2">4.5-9 MakeUniformOnRight</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X796249A682818750">4.5-10 MappedExpression</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D6DDDF178B0F2D9">4.5-11 SupportOfQuiverAlgebraElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X849AC0F67A131929">4.5-12 VertexPosition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X86795D8E7ED73048">4.5-13 RelationsOfAlgebra</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7F0D555379C97A6E">4.6 <span class="Heading">Constructing Quotients of Path Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X814203E281F3272E">4.6-1 AssignGeneratorVariables</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X812C0F8D7E4B1134">4.7 <span class="Heading">Ideals and operations
on ideals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82ACACDD7D8E9B25">4.7-1 Ideal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7EACC0D285D18E19">4.7-2 IdealOfQuotient</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X859C987B7C5F0D8D">4.7-3 PathsOfLengthTwo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X830187497E0BD4F0">4.7-4 NthPowerOfArrowIdeal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X824D550E8371098C">4.7-5 AddNthPowerToRelations</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X83E8D45B82356D8E"><code>4.7-6 \in</code></a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X82A1683E7A402E73">4.8 <span class="Heading">Categories and properties of ideals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F40193B877D76BC">4.8-1 IsAdmissibleIdeal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X818EE2B9789BB175">4.8-2 IsIdealInPathAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7C40D53785D67A9E">4.8-3 IsMonomialIdeal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X841058F8850FA9D3">4.8-4 IsQuadraticIdeal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7C45A01B7A587D9E">4.9 <span class="Heading">Operations on ideals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7C1D2A2481599348">4.9-1 ProductOfIdeals</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B0035AF7B030BDF">4.9-2 QuadraticPerpOfPathAlgebraIdeal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X85D4E72B787B1C49">4.10 <span class="Heading">Attributes of ideals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D2896F27C976231">4.10-1 GroebnerBasisOfIdeal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7A3CA333873389AD">4.11 <span class="Heading">Categories and Properties of Quotients of Path Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7ACDD33087F98B88">4.11-1 IsAdmissibleQuotientOfPathAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X790DB9BF831B577D">4.11-2 IsQuotientOfPathAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X802DB9FB824B0167">4.11-3 IsFiniteDimensional</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8523C98A870CF7B5">4.11-4 IsCanonicalAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X78C9AA2085058DFA">4.11-5 IsDistributiveAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7EF4868B84CC749E">4.11-6 IsFiniteGlobalDimensionAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X796625487F5F92A7">4.11-7 IsGentleAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E6BE2187B48691D">4.11-8 IsGorensteinAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X859FF8F2865D0A3A">4.11-9 IsHereditaryAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7809E044817388D1">4.11-10 IsKroneckerAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7952044F8303A688">4.11-11 IsMonomialAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7CB110A7873F7942">4.11-12 IsNakayamaAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8798B8BA7A145A2D">4.11-13 IsQuiverAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7A8D13FE8379776E">4.11-14 IsRadicalSquareZeroAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82F25BFD7D43AB10">4.11-15 IsSchurianAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8555FC6B85FE9C6D">4.11-16 IsSelfinjectiveAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8558D44A79AA16CD">4.11-17 IsSemicommutativeAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X85A8EC2287F35DC1">4.11-18 IsSemisimpleAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D7AC1D07A9607DF">4.11-19 IsSpecialBiserialAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X86F0E4AF7C9916CB">4.11-20 IsStringAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X79DA912C82D01EE8">4.11-21 IsSymmetricAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87EC06D18021AD76">4.11-22 IsTriangularReduced</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7EF281F980319375">4.11-23 IsWeaklySymmetricAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X81C569797E900AE9">4.11-24 BongartzTest</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8140D2557A23CDAC">4.11-25 IsFiniteTypeAlgebra</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X861E6670814290D0">4.12 <span class="Heading"> Operations on String Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X85F2FFFD78355788">4.12-1 IsValidString</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X85B82A4086AA53D6">4.12-2 StringsLessThan</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F5B0A1A7AAF2C18">4.12-3 IsABand</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X85FFA183800621EA">4.12-4 BandsLessThan</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X870DB8577F0ABF0E">4.12-5 BandRepresentativesLessThan</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7C42DD687CE572DF">4.12-6 IsDomesticStringAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X85D0616C82375B5C">4.12-7 BridgeQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82DBBB737885C73B">4.12-8 LocalARQuiver</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X86647D317A961513">4.13 <span class="Heading">Attributes and Operations (for Quotients) of Path
Algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84E3FEF587CB66C3">4.13-1 CartanMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F8084A67A3BE874">4.13-2 Centre/Center</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X835A161E8524797A">4.13-3 ComplexityOfAlgebra</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F6F526C86052150">4.13-5 CoxeterPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E6926C6850E7C4E">4.13-6 Dimension</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7A3EACE782DC2198">4.13-7 FrobeniusForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X789D1DB97C1B9A0D">4.13-8 FrobeniusLinearFunctional</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D511B3E7A50AB2A">4.13-9 GlobalDimension</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7BEA44FB819910B6">4.13-10 LoewyLength</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X818AF5A979F8E539">4.13-11 NakayamaAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7C068FE379FBCE18">4.13-12 NakayamaPermutation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X78E378EB83BA3D8A">4.13-13 OrderOfNakayamaAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X782EFE477EC0C1C6">4.13-14 RadicalSeriesOfAlgebra</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7BD7DB497917893C">4.14 <span class="Heading">Attributes and Operations on Elements
of Quotients of Path Algebra</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87495684791B5742">4.14-1 IsElementOfQuotientOfPathAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X80B32F667BF6AFD8">4.14-2 Coefficients</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8271E6F27C2C826E">4.14-3 IsNormalForm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X837DD99B7A233FB5">4.14-4 <</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F2527747A3D0D6D">4.14-5 ElementOfQuotientOfPathAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8088721187BA8D82">4.14-6 OriginalPathAlgebra</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7B209E0A7DD93C08">4.15 <span class="Heading">Predefined classes and classes
of (quotients of) path algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X820AE0208636B9AA">4.15-1 BrauerConfigurationAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E63A4F37856A075">4.15-2 CanonicalAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X83498D3D856CC08A">4.15-3 KroneckerAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7C678A08836F77CC">4.15-4 NakayamaAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8369BB398212101C">4.15-5 AdmissibleSequenceGenerator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X78C675F2836D1B18">4.15-6 PosetAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F877C5F839A3AA9">4.15-7 PosetOfPosetAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7EF1AE62790D7486">4.15-8 TruncatedPathAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87E17E137E2B0FC4">4.15-9 IsSpecialBiserialQuiver</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X840BAB827C62AA4C">4.16 <span class="Heading">Opposite algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X85794BE082B632B9">4.16-1 OppositePath</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87A86AFB782211D6">4.16-2 OppositePathAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82C303CE808D54C1">4.16-3 OppositePathAlgebraElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X86B3EDE679B2493E">4.16-4 OppositeAlgebraHomomorphism</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X842527EC7F90C8C5">4.17 <span class="Heading">Tensor products of path algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X835BBBE18104654A">4.17-1 QuiverProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X858517C18242C2F1">4.17-2 QuiverProductDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X80E3731882B80106">4.17-3 IsQuiverProductDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X856E8B5B7F550647">4.17-4 IncludeInProductQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8455692C7E282C6C">4.17-5 ProjectFromProductQuiver</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7A9026937BDDFA6C">4.17-6 TensorProductOfAlgebras</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7EE8921D787C8377">4.17-7 TensorAlgebrasInclusion</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B31F4F680135E72">4.17-8 SimpleTensor</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F0EBF88866A537D">4.17-9 TensorProductDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X820195C47E2BE7E0">4.17-10 EnvelopingAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8210F6627AB95229">4.17-11 EnvelopingAlgebraHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7AE13B567B5F72FC">4.17-12 IsEnvelopingAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X80D827747ACD76FA">4.17-13 AlgebraAsModuleOverEnvelopingAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7A4C262287D74AB0">4.17-14 DualOfAlgebraAsModuleOverEnvelopingAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8101415F7FFB34CF">4.17-15 TrivialExtensionOfQuiverAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X81D98182822E8911">4.17-16 TrivialExtensionOfQuiverAlgebraProjection</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X79B8B91F8097BB80">4.18 <span class="Heading">Operations on quiver algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7EE5A11883B86971">4.18-1 QuiverAlgebraOfAmodAeA</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B2D7385829F5EC6">4.18-2 QuiverAlgebraOfeAe</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8561BCB6835D561F">4.19 <span class="Heading">Finite dimensional algebras over finite fields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X810A29FB7E6EA24D">4.19-1 AlgebraAsQuiverAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E9078077EE8B51B">4.19-2 BlocksOfAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84B423137F933795">4.19-3 IsBasicAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D30E9C878221B42">4.19-4 IsElementaryAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B35109B8176FE56">4.19-5 PreprojectiveAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X80C0C6C37C4A2ABD">4.19-6 PrimitiveIdempotents</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7DDBF6F47A2E021C">4.20 <span class="Heading">Algebras</span></a>
</span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X821B7B047871B42D">4.20-1 LiftingCompleteSetOfOrthogonalIdempotents</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X83041BDF78BF3CCA">4.20-2 LiftingIdempotent</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X850B9F12806FF76B">4.21 <span class="Heading">Saving and reading quotients of path algebras to and from a
file</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82A638C77FA75549">4.21-1 ReadAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E60DDCE848CB739">4.21-2 SaveAlgebra</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap5.html#X8371E66387CB2E49">5 <span class="Heading">Groebner Basis</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X850B47047FD4D709">5.1 <span class="Heading">Constructing a Groebner Basis</span></a>
</span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8451936885F68598">5.1-1 InfoGroebnerBasis</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7A43611E876B7560">5.1-2 GroebnerBasis</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X79C7DC5D873A14D0">5.2 <span class="Heading">Categories and Properties of Groebner
Basis</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X85C0C1CD87C70AAA">5.2-1 IsCompletelyReducedGroebnerBasis</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X86E7D0AE87CA048D">5.2-2 IsCompleteGroebnerBasis</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7BFD28E687AADFBB">5.2-3 IsGroebnerBasis</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X799FD421784D1FFC">5.2-4 IsHomogeneousGroebnerBasis</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8592E4C87E41A15A">5.2-5 IsTipReducedGroebnerBasis</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X84EF455882169920">5.3 <span class="Heading">Attributes and Operations for Groebner Basis</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8722C1577C236116">5.3-1 AdmitsFinitelyManyNontips</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X878AC1107E9671BA">5.3-2 CompletelyReduce</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7FF28B7B80759D24">5.3-3 CompletelyReduceGroebnerBasis</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7EF8910F82B45EC7">5.3-4 Enumerator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8137C99A7934C1CA">5.3-5 IsPrefixOfTipInTipIdeal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X83ADF8287ED0668E">5.3-6 Iterator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7EAA029F8071ACC6">5.3-7 Nontips</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7840F54D8240C288">5.3-8 NontipSize</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7C6CD739788E7F59">5.3-9 TipReduce</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7B4F38D6852DF8B6">5.3-10 TipReduceGroebnerBasis</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X82C1C09486934532">5.4 <span class="Heading">Right Groebner Basis</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X86EC39527F33EABE">5.4-1 IsRightGroebnerBasis</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7B29B9207D20EA9E">5.4-2 RightGroebnerBasis</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X812BFF79867FF73A">5.4-3 RightGroebnerBasisOfIdeal</a></span>
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<div class="ContChap"><a href="chap6.html#X87EFC38F7BC77B27">6 <span class="Heading">Right Modules over Path
Algebras</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X86DD15DF877834FD">6.1 <span class="Heading">Modules of matrix type</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X85E5097D82D9BE62">6.1-1 RightModuleOverPathAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X844DD9B386A6AC56">6.1-2 RightAlgebraModuleToPathAlgebraMatModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X823D5739809A2D9A"><code>6.1-3 \=</code></a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X869F4DD2877A99BA">6.2 <span class="Heading">Categories Of Matrix Modules</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8710CC447F1F7B17">6.2-1 IsPathAlgebraMatModule</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X862F510485ADBC67">6.3 <span class="Heading">Acting on Module Elements</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7BB4066E7D5B15B8">6.3-1 ^</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7E5B84B1832D839E">6.4 <span class="Heading">Operations on representations</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X830B834D7F2F0FAD">6.4-1 AnnihilatorOfModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X83404C0B7C15E7D0">6.4-2 BasicVersionOfModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7CD05524803C7777">6.4-3 BlockDecompositionOfModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X85AC0E7C7E3E697D">6.4-4 BlockSplittingIdempotents</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X81EF6AE97A4F77FA">6.4-8 DecomposeModuleProbabilistic</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7A4A2B0F7BB16D0D">6.4-9 DecomposeModuleViaCharPoly</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X848EAC6C833C95B1">6.4-10 DecomposeModuleViaTop</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7E957BFE7897F504">6.4-11 DecomposeModuleWithMultiplicities</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7E38D5A48344C173">6.4-12 Dimension</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B5EA4B0820DE28C">6.4-13 DimensionVector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7BDD77707A013FBE">6.4-14 DirectSumOfQPAModules</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X857807CF8560B3C4">6.4-15 DirectSumInclusions</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X80CFB7E47A785E12">6.4-16 DirectSumProjections</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X837AE1BF7F31AD7C">6.4-17 FromIdentityToDoubleStarHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7FE61CFE7A138755">6.4-18 IntersectionOfSubmodules</a></ | |
