<?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 9: Module Elements</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="chap9" 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="chap8_mj.html" >[Previous Chapter]</a> <a href="chap10_mj.html" >[Next Chapter]</a> </div >"mathjaxlink" class="pcenter" ><a href="chap9.html" >[MathJax off]</a></p>"X7E9BCB99816348F2" name="X7E9BCB99816348F2" ></a></p>div class="ChapSects" ><a href="chap9_mj.html#X7E9BCB99816348F2" >9 <span class="Heading" >Module Elements</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X84A51EB87E054D3F" >9.1 <span class="Heading" >Module Elements: Category and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X784BBB2A782DB774" >9.1-1 IsHomalgElement</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X7BF482C77B68ED64" >9.1-2 IsElementOfAModuleGivenByAMorphismRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X7CFD0CF27A3FEB9D" >9.2 <span class="Heading" >Module Elements: Constructors</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X7BCBA7E780FE2B14" >9.3 <span class="Heading" >Module Elements: Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X87FA282579406FC0" >9.3-1 IsElementOfIntegers</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X80AE2D1C82A2059C" >9.4 <span class="Heading" >Module Elements: Attributes</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9_mj.html#X813DF977812C06B6" >9.5 <span class="Heading" >Module Elements: Operations and Functions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9_mj.html#X8769077379997D89" >9.5-1 HomalgRing</a></span >div ></div >div >span class="Heading" >Module Elements</span ></h3>span class="SimpleMath" >\(M\)</span > is internally represented by a module map from the (distinguished) rank 1 free module to the module <span class="SimpleMath" >\(M\)</span >. In particular, the data structure for module elements automatically profits from the intrinsic realization of morphisms in the <strong class="pkg" >homalg</strong > project.</p>"X84A51EB87E054D3F" name="X84A51EB87E054D3F" ></a></p>span class="Heading" >Module Elements: Category and Representations</span ></h4>"X784BBB2A782DB774" name="X784BBB2A782DB774" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsHomalgElement</code >( <var class="Arg" >M</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 module elements.</p>"X7BF482C77B68ED64" name="X7BF482C77B68ED64" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsElementOfAModuleGivenByAMorphismRep</code >( <var class="Arg" >M</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 elements of modules.</p>code class="func" >IsElementOfAnObjectGivenByAMorphismRep</code > (<a href="https://homalg-project.github.io/homalg_project/homalg/doc/chap5_mj.html#X87F3740E85E9AA51 " ><span class="RefLink" >homalg: IsElementOfAnObjectGivenByAMorphismRep</span ></a>).)</p>"X7CFD0CF27A3FEB9D" name="X7CFD0CF27A3FEB9D" ></a></p>span class="Heading" >Module Elements: Constructors</span ></h4>"X7BCBA7E780FE2B14" name="X7BCBA7E780FE2B14" ></a></p>span class="Heading" >Module Elements: Properties</span ></h4>"X87FA282579406FC0" name="X87FA282579406FC0" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ IsElementOfIntegers</code >( <var class="Arg" >m</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>var class="Arg" >m</var > is an element of the integers viewed as a module over itself.</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >zz := HomalgRingOfIntegers( );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >a := HomalgElement( HomalgMap( "[[2]]" , 1 * zz, 1 * zz ) );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >IsElementOfIntegers( a );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Z4 := zz / 4;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >b := HomalgElement( HomalgMap( "[[-1]]" , 1 * Z4, 1 * Z4 ) );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >IsElementOfIntegers( b );</span >pre ></div >"X80AE2D1C82A2059C" name="X80AE2D1C82A2059C" ></a></p>span class="Heading" >Module Elements: Attributes</span ></h4>"X813DF977812C06B6" name="X813DF977812C06B6" ></a></p>span class="Heading" >Module Elements: Operations and Functions</span ></h4>"X8769077379997D89" name="X8769077379997D89" ></a></p>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >‣ HomalgRing</code >( <var class="Arg" >m</var > )</td ><td class="tdright" >( operation )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > ring</p>strong class="pkg" >homalg</strong > ring of the <strong class="pkg" >homalg</strong > module element <var class="Arg" >m</var >.</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >zz := HomalgRingOfIntegers( );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >a := HomalgElement( HomalgMap( "[[2]]" , 1 * zz, 1 * zz ) );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >IsIdenticalObj( zz, HomalgRing( a ) );</span >pre ></div >div class="chlinkprevnextbot" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chap8_mj.html" >[Previous Chapter]</a> <a href="chap10_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 90%
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