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<h1>ResClasses</h1>


<h2>Set-Theoretic Computations with Residue Classes</h2>

<p>
    4.7.4</p>

<p>
    24 September 2025
  </p>

</div>
<p><b>
    Stefan Kohl



  </b>
<br />Email: <span class="URL"><a href="mailto:sk239@st-andrews.ac.uk">sk239@st-andrews.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="https://stefan-kohl.github.io/">https://stefan-kohl.github.io/</a></span>
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<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p><strong class="pkg">ResClasses</strong> is a package for <strong class="pkg">GAP</strong> 4 which provides a fully-featured and easy-to-use implementation of set-theoretic unions of residue classes of the integers and of a few other rings.</p>

<p>The class of sets which <strong class="pkg">ResClasses</strong> can deal with includes the open and the closed sets in the topology on the respective ring which is induced by taking the set of all residue classes as a basis, as far as the usual restrictions imposed by the finiteness of computing resources permit this.</p>

<p>The package further provides slightly more specialized functionality for unions of residue classes with distinguished representatives and signed moduli.</p>

<p>The <strong class="pkg">ResClasses</strong> package is used in a group theoretical context by the <strong class="pkg">RCWA</strong> package <a href="chapBib.html#biBRCWA">[Koh16]</a>.</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2003 - 2017 by Stefan Kohl.</p>

<p><strong class="pkg">ResClasses</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.</p>

<p><strong class="pkg">ResClasses</strong> is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.</p>

<p>For a copy of the GNU General Public License, see the file <code class="file">GPL</code> in the <code class="file">etc</code> directory of the <strong class="pkg">GAP</strong> distribution or see <span class="URL"><a href="https://www.gnu.org/licenses/gpl.html">https://www.gnu.org/licenses/gpl.html</a></span>.</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#X815A3DDE7C0BC44A">1 <span class="Heading">Set-Theoretic Unions of Residue Classes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7E16A64485A7AB79">1.1 <span class="Heading">Entering residue classes and set-theoretic unions thereof</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8753CC098447BE0D">1.1-1 ResidueClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X85327C777F32DA8F">1.1-2 ResidueClassUnion</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8326D6F285081E0F">1.1-3 AllResidueClassesModulo</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7A3FA13187CEADED">1.2 <span class="Heading">Methods for residue class unions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X854315A2877B69A7">1.2-1 SplittedClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X87C166FE7FA17325">1.2-2 AsUnionOfFewClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8079E174813646DA">1.2-3 PartitionsIntoResidueClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8653860F7ADA4D38">1.2-4 RandomPartitionIntoResidueClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7BAF36BF7A276D46">1.2-5 CoverByResidueClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X791C16E77BE97FE3">1.2-6 Density</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7921D4368632E15C">1.3 <span class="Heading">On residue class unions of <span class="SimpleMath">ℤ^2</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7B26BB1C7C8495A5">1.4 <span class="Heading">The categories and families of residue class unions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7EA29BBD82552352">1.4-1 IsResidueClassUnion</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7F9BCB1A797F8F48">1.4-2 ResidueClassUnionsFamily</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap2.html#X87871B5A86254179">2 <span class="Heading">Unions of Residue Classes with Fixed Representatives</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X78BE0FA38691B1B6">2.1 <span class="Heading">
  Entering unions of residue classes with fixed representatives
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X831C976D876FDAD3">2.1-1 ResidueClassWithFixedRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X871ED1D17AE52A95">2.1-2 UnionOfResidueClassesWithFixedReps</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8198386A809A6B17">2.1-3 AllResidueClassesWithFixedRepsModulo</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7C26C7E682C2FD2E">2.2 <span class="Heading">
  Methods for unions of residue classes with fixed representatives
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X808442EC7F3F5748">2.2-1 Multiplicity</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8296A97779FE8B72">2.2-2 Union</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7FC166A57F1F5601">2.2-3 Intersection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X82A4EE727E67AB98">2.2-4 Difference</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7A1CBA1B7E6B93D0">2.3 <span class="Heading">
  The invariant Delta
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X78DCAB2C7C4E37E8">2.3-1 Delta</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7E109A6E81A0465C">2.3-2 RepresentativeStabilizingRefinement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7BA9DE337DF742E9">2.4 <span class="Heading">
  The categories of unions of residue classes with fixed rep's
</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap3.html#X80CB0518869B1818">3 <span class="Heading">Semilocalizations of the Integers</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7B3B22AC7E6247A4">3.1 <span class="Heading">Entering semilocalizations of the integers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7861432E7F221610">3.1-1 Z_pi</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7E75C4217DCA45D0">3.2 <span class="Heading">Methods for semilocalizations of the integers</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap4.html#X859F6BF88754E5CC">4 <span class="Heading">Installation and Auxiliary Functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X85A08CF187A6D986">4.1 <span class="Heading">Requirements</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8360C04082558A12">4.2 <span class="Heading">Installation</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X865D6A49826B92EC">4.3 <span class="Heading">The testing routines</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8733F00082901C2C">4.3-1 ResClassesTest</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X802034807F5D6C47">4.3-2 ResClassesTestExamples</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7B0129E185088420">4.4 <span class="Heading">Creating timestamped logfiles</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7B023FD184CC118D">4.4-1 LogToDatedFile</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X81DD0163859E9EF5">4.5 <span class="Heading">DownloadFile, SendEmail and EmailLogFile</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C1543408472A82B">4.5-1 DownloadFile</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F27919085001EA8">4.5-2 SendEmail</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8000E5CF7DF4339C">4.5-3 EmailLogFile</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X855F2CE284BFB06B">4.6 <span class="Heading">Creating bitmap pictures</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X81DE1E838615C214">4.6-1 SaveAsBitmapPicture</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83D7129581B8CF21">4.6-2 DrawLineNC</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X808A30E37A293D19">4.7 <span class="Heading">Some general utility functions</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chapBib.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd.html"><span class="Heading">Index</span></a></div>
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