<?xml version="1.0" encoding="UTF-8" ?>DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd " >html xmlns="http://www.w3.org/1999/xhtml " xml:lang="en" >head >script type="text/javascript" "https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML " >script >title >GAP (Modules) - Chapter 5: Relations</title >meta http-equiv="content-type" content="text/html; charset=UTF-8" />meta name="generator" content="GAPDoc2HTML" />link rel="stylesheet" type="text/css" href="manual.css" />script src="manual.js" type="text/javascript" ></script >script type="text/javascript" >overwriteStyle();</script >head >body class="chap5" onload="jscontent()" >div class="chlinktop" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0_mj.html" >Top</a> <a href="chap1_mj.html" >1</a> <a href="chap2_mj.html" >2</a> <a href="chap3_mj.html" >3</a> <a href="chap4_mj.html" >4</a> <a href="chap5_mj.html" >5</a> <a href="chap6_mj.html" >6</a> <a href="chap7_mj.html" >7</a> <a href="chap8_mj.html" >8</a> <a href="chap9_mj.html" >9</a> <a href="chap10_mj.html" >10</a> <a href="chap11_mj.html" >11</a> <a href="chap12_mj.html" >12</a> <a href="chap13_mj.html" >13</a> <a href="chapA_mj.html" >A</a> <a href="chapB_mj.html" >B</a> <a href="chapC_mj.html" >C</a> <a href="chapBib_mj.html" >Bib</a> <a href="chapInd_mj.html" >Ind</a> </div >div class="chlinkprevnexttop" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chap4_mj.html" >[Previous Chapter]</a> <a href="chap6_mj.html" >[Next Chapter]</a> </div >"mathjaxlink" class="pcenter" ><a href="chap5.html" >[MathJax off]</a></p>"X838651287FCCEFD8" name="X838651287FCCEFD8" ></a></p>div class="ChapSects" ><a href="chap5_mj.html#X838651287FCCEFD8" >5 <span class="Heading" >Relations</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X87DADB7B7CB126DC" >5.1 <span class="Heading" >Relations: Categories and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X80AD050F7999B7C0" >5.1-1 IsHomalgRelations</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X790F68B17A4846DC" >5.1-2 IsHomalgRelationsOfLeftModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7FF5A3B180614698" >5.1-3 IsHomalgRelationsOfRightModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X8322A26C84E80303" >5.1-4 IsRelationsOfFinitelyPresentedModuleRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X7CF74FB785F90889" >5.2 <span class="Heading" >Relations: Constructors</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X859231317954D702" >5.3 <span class="Heading" >Relations: Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X798D893B7FBFCF07" >5.3-1 CanBeUsedToDecideZeroEffectively</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap5_mj.html#X7B9398827AEEA2E6" >5.3-2 IsInjectivePresentation</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X7EF7BBCA85851EF1" >5.4 <span class="Heading" >Relations: Attributes</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap5_mj.html#X7890B5EA80774AB5" >5.5 <span class="Heading" >Relations: Operations and Functions</span ></a>span >div >div >span class="Heading" >Relations</span ></h3>ions among these generators. In <strong class="pkg" >homalg</strong > a set of relations of a left/right module is given by a matrix <var class="Arg" >rel</var >, the rows/columns of which are interpreted as relations among <span class="SimpleMath" >\(n\)</span > generators, <span class="SimpleMath" >\(n\)</span > being the number of columns/rows of the matrix <var class="Arg" >rel</var >.</p>strong class="pkg" >homalg</strong > is designed to contain not only one but several sets of relations (together with corresponding sets of generators (--> Chapter <a href="chap6_mj.html#X7BD5B55C802805B4" ><span class="RefLink" >6</span ></a>)). The different sets of relations are linked with so-called transition matrices (--> Chapter <a href="chap7_mj.html#X8183A6857B0C3633" ><span class="RefLink" >7</span ></a>).</p>strong class="pkg" >homalg</strong > module are evaluated in a lazy way. This avoids unnecessary computations.</p>"X87DADB7B7CB126DC" name="X87DADB7B7CB126DC" ></a></p>span class="Heading" >Relations: Categories and Representations</span ></h4>"X80AD050F7999B7C0" name="X80AD050F7999B7C0" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsHomalgRelations</code >( <var class="Arg" >rel</var > )</td ><td class="tdright" >( category )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >GAP</strong > category of <strong class="pkg" >homalg</strong > relations.</p>"X790F68B17A4846DC" name="X790F68B17A4846DC" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsHomalgRelationsOfLeftModule</code >( <var class="Arg" >rel</var > )</td ><td class="tdright" >( category )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >GAP</strong > category of <strong class="pkg" >homalg</strong > relations of a left module.</p>strong class="pkg" >GAP</strong > category <code class="code" >IsHomalgRelations</code >.)</p>"X7FF5A3B180614698" name="X7FF5A3B180614698" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsHomalgRelationsOfRightModule</code >( <var class="Arg" >rel</var > )</td ><td class="tdright" >( category )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >GAP</strong > category of <strong class="pkg" >homalg</strong > relations of a right module.</p>strong class="pkg" >GAP</strong > category <code class="code" >IsHomalgRelations</code >.)</p>"X8322A26C84E80303" name="X8322A26C84E80303" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsRelationsOfFinitelyPresentedModuleRep</code >( <var class="Arg" >rel</var > )</td ><td class="tdright" >( representation )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >GAP</strong > representation of a finite set of relations of a finitely presented <strong class="pkg" >homalg</strong > module.</p>strong class="pkg" >GAP</strong > category <code class="func" >IsHomalgRelations</code > (<a href="chap5_mj.html#X80AD050F7999B7C0" ><span class="RefLink" >5.1-1</span ></a>))</p>"X7CF74FB785F90889" name="X7CF74FB785F90889" ></a></p>span class="Heading" >Relations: Constructors</span ></h4>"X859231317954D702" name="X859231317954D702" ></a></p>span class="Heading" >Relations: Properties</span ></h4>"X798D893B7FBFCF07" name="X798D893B7FBFCF07" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ CanBeUsedToDecideZeroEffectively</code >( <var class="Arg" >rel</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >homalg</strong > set of relations <var class="Arg" >rel</var > can be used for normal form reductions. <br /> (no method installed)</p>"X7B9398827AEEA2E6" name="X7B9398827AEEA2E6" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsInjectivePresentation</code >( <var class="Arg" >rel</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >homalg</strong > set of relations <var class="Arg" >rel</var > has zero syzygies.</p>"X7EF7BBCA85851EF1" name="X7EF7BBCA85851EF1" ></a></p>span class="Heading" >Relations: Attributes</span ></h4>"X7890B5EA80774AB5" name="X7890B5EA80774AB5" ></a></p>span class="Heading" >Relations: Operations and Functions</span ></h4>div class="chlinkprevnextbot" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chap4_mj.html" >[Previous Chapter]</a> <a href="chap6_mj.html" >[Next Chapter]</a> </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0_mj.html" >Top</a> <a href="chap1_mj.html" >1</a> <a href="chap2_mj.html" >2</a> <a href="chap3_mj.html" >3</a> <a href="chap4_mj.html" >4</a> <a href="chap5_mj.html" >5</a> <a href="chap6_mj.html" >6</a> <a href="chap7_mj.html" >7</a> <a href="chap8_mj.html" >8</a> <a href="chap9_mj.html" >9</a> <a href="chap10_mj.html" >10</a> <a href="chap11_mj.html" >11</a> <a href="chap12_mj.html" >12</a> <a href="chap13_mj.html" >13</a> <a href="chapA_mj.html" >A</a> <a href="chapB_mj.html" >B</a> <a href="chapC_mj.html" >C</a> <a href="chapBib_mj.html" >Bib</a> <a href="chapInd_mj.html" >Ind</a> </div >hr />"foot" >generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc " >GAPDoc2HTML</a></p>body >html >quality 98%
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