<?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 6: Generators</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="chap6" 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="chap5_mj.html" >[Previous Chapter]</a> <a href="chap7_mj.html" >[Next Chapter]</a> </div >"mathjaxlink" class="pcenter" ><a href="chap6.html" >[MathJax off]</a></p>"X7BD5B55C802805B4" name="X7BD5B55C802805B4" ></a></p>div class="ChapSects" ><a href="chap6_mj.html#X7BD5B55C802805B4" >6 <span class="Heading" >Generators</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X827B67D27E3B91FC" >6.1 <span class="Heading" >Generators: Categories and Representations</span span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X79A6BA1280510584" >6.1-1 IsHomalgGenerators</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X83E88425797FFC9C" >6.1-2 IsHomalgGeneratorsOfLeftModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X86E9029487FE58DF" >6.1-3 IsHomalgGeneratorsOfRightModule</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X8671AA997D666F04" >6.1-4 IsGeneratorsOfModuleRep</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X78512B8A8613FBF1" >6.1-5 IsGeneratorsOfFinitelyGeneratedModuleRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X8289206C81622597" >6.2 <span class="Heading" >Generators: Constructors</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X8576E1368448066B" >6.3 <span class="Heading" >Generators: Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7ED359D87D6B2F79" >6.3-1 IsReduced</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X7E136BCD7F22B571" >6.4 <span class="Heading" >Generators: Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6_mj.html#X7B6F787085536F90" >6.4-1 ProcedureToReadjustGenerators</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6_mj.html#X7AC876EC8137AEA4" >6.5 <span class="Heading" >Generators: Operations and Functions</span ></a>span >div >div >span class="Heading" >Generators</span ></h3>var class="Arg" >rel</var > and interpret its rows/columns as relations among <span class="SimpleMath" >\(n\)</span > <em >abstract</em > generators, where <span class="SimpleMath" >\(n\)</span > is the number of columns/rows of <var class="Arg" >rel</var >. Only that these abstract generators are useless when it comes to specific modules like modules of homomorphisms, where one expects the generators to be maps between modules. For this reason a presentation of a module in <strong class="pkg" >homalg</strong > is not merely a matrix of relations, but together with a set of generators.</p>strong class="pkg" >homalg</strong > a set of generators of a left/right module is given by a matrix <var class="Arg" >gen</var > with rows/columns being interpreted as the generators.</p>strong class="pkg" >homalg</strong > is designed to contain not only one but several sets of generators (together with their sets of relations (--> Chapter <a href="chap5_mj.html#X838651287FCCEFD8" ><span class="RefLink" >5</span ></a>)). The different sets of generators are linked with so-called transition matrices (--> Chapter <a href="chap7_mj.html#X8183A6857B0C3633" ><span class="RefLink" >7</span ></a>).</p>"X827B67D27E3B91FC" name="X827B67D27E3B91FC" ></a></p>span class="Heading" >Generators: Categories and Representations</span ></h4>"X79A6BA1280510584" name="X79A6BA1280510584" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsHomalgGenerators</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 > generators.</p>"X83E88425797FFC9C" name="X83E88425797FFC9C" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsHomalgGeneratorsOfLeftModule</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 > generators of a left module.</p>strong class="pkg" >GAP</strong > category <code class="code" >IsHomalgGenerators</code >.)</p>"X86E9029487FE58DF" name="X86E9029487FE58DF" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsHomalgGeneratorsOfRightModule</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 > generators of a right module.</p>strong class="pkg" >GAP</strong > category <code class="code" >IsHomalgGenerators</code >.)</p>"X8671AA997D666F04" name="X8671AA997D666F04" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsGeneratorsOfModuleRep</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 generators of a <strong class="pkg" >homalg</strong > module.</p>strong class="pkg" >GAP</strong > category <code class="func" >IsHomalgGenerators</code > (<a href="chap6_mj.html#X79A6BA1280510584" ><span class="RefLink" >6.1-1</span ></a>))</p>div class="example" ><pre >"IsGeneratorsOfModuleRep" ,"generators" ] );pre ></div >"X78512B8A8613FBF1" name="X78512B8A8613FBF1" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsGeneratorsOfFinitelyGeneratedModuleRep</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 generators of a finitely generated <strong class="pkg" >homalg</strong > module.</p>strong class="pkg" >GAP</strong > representation <code class="func" >IsGeneratorsOfModuleRep</code > (<a href="chap6_mj.html#X8671AA997D666F04" ><span class="RefLink" span ></a>))</p>div class="example" ><pre >"IsGeneratorsOfFinitelyGeneratedModuleRep" ,"generators" , "relations_of_hullmodule" ] );pre ></div >"X8289206C81622597" name="X8289206C81622597" ></a></p>span class="Heading" >Generators: Constructors</span ></h4>"X8576E1368448066B" name="X8576E1368448066B" ></a></p>span class="Heading" >Generators: Properties</span ></h4>"X7ED359D87D6B2F79" name="X7ED359D87D6B2F79" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsReduced</code >( <var class="Arg" >gen</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 generators <var class="Arg" >gen</var > is marked reduced. <br /> (no method installed)</p>"X7E136BCD7F22B571" name="X7E136BCD7F22B571" ></a></p>span class="Heading" >Generators: Attributes</span ></h4>"X7B6F787085536F90" name="X7B6F787085536F90" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ ProcedureToReadjustGenerators</code >( <var class="Arg" >gen</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >var class="Arg" >gen</var > and returns an object (e.g. a matrix) that can be interpreted as a generator (this is important for modules of homomorphisms).</p>"X7AC876EC8137AEA4" name="X7AC876EC8137AEA4" ></a></p>span class="Heading" >Generators: Operations and Functions</span ></h4>div class="chlinkprevnextbot" > <a href="chap0_mj.html" >[Top of Book]</a> <a 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