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<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>

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<p id="mathjaxlink" class="pcenter"><a href="chap6.html">[MathJax off]</a></p>
<p><a id="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></a>
</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>

<h3>6 <span class="Heading">Generators</span></h3>

<p>To present a left/right module it suffices to take a matrix <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>

<p>In <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>

<p>The data structure of a module in <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>

<p><a id="X827B67D27E3B91FC" name="X827B67D27E3B91FC"></a></p>

<h4>6.1 <span class="Heading">Generators: Categories and Representations</span></h4>

<p><a id="X79A6BA1280510584" name="X79A6BA1280510584"></a></p>

<h5>6.1-1 IsHomalgGenerators</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsHomalgGenerators</code>( <var class="Arg">rel</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>

<p>The <strong class="pkg">GAP</strong> category of <strong class="pkg">homalg</strong> generators.</p>

<p><a id="X83E88425797FFC9C" name="X83E88425797FFC9C"></a></p>

<h5>6.1-2 IsHomalgGeneratorsOfLeftModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsHomalgGeneratorsOfLeftModule</code>( <var class="Arg">rel</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>

<p>The <strong class="pkg">GAP</strong> category of <strong class="pkg">homalg</strong> generators of a left module.</p>

<p>(It is a subcategory of the <strong class="pkg">GAP</strong> category <code class="code">IsHomalgGenerators</code>.)</p>

<p><a id="X86E9029487FE58DF" name="X86E9029487FE58DF"></a></p>

<h5>6.1-3 IsHomalgGeneratorsOfRightModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsHomalgGeneratorsOfRightModule</code>( <var class="Arg">rel</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>

<p>The <strong class="pkg">GAP</strong> category of <strong class="pkg">homalg</strong> generators of a right module.</p>

<p>(It is a subcategory of the <strong class="pkg">GAP</strong> category <code class="code">IsHomalgGenerators</code>.)</p>

<p><a id="X8671AA997D666F04" name="X8671AA997D666F04"></a></p>

<h5>6.1-4 IsGeneratorsOfModuleRep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsGeneratorsOfModuleRep</code>( <var class="Arg">rel</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>

<p>The <strong class="pkg">GAP</strong> representation of a finite set of generators of a <strong class="pkg">homalg</strong> module.</p>

<p>(It is a representation of the <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>
DeclareRepresentation( "IsGeneratorsOfModuleRep",
        IsHomalgGenerators,
        [ "generators" ] );
</pre></div>

<p><a id="X78512B8A8613FBF1" name="X78512B8A8613FBF1"></a></p>

<h5>6.1-5 IsGeneratorsOfFinitelyGeneratedModuleRep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsGeneratorsOfFinitelyGeneratedModuleRep</code>( <var class="Arg">rel</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>

<p>The <strong class="pkg">GAP</strong> representation of a finite set of generators of a finitely generated <strong class="pkg">homalg</strong> module.</p>

<p>(It is a representation of the <strong class="pkg">GAP</strong> representation <code class="func">IsGeneratorsOfModuleRep</code> (<a href="chap6_mj.html#X8671AA997D666F04"><span class="RefLink">6.1-4</span></a>))</p>


<div class="example"><pre>
DeclareRepresentation( "IsGeneratorsOfFinitelyGeneratedModuleRep",
        IsGeneratorsOfModuleRep,
        [ "generators", "relations_of_hullmodule" ] );
</pre></div>

<p><a id="X8289206C81622597" name="X8289206C81622597"></a></p>

<h4>6.2 <span class="Heading">Generators: Constructors</span></h4>

<p><a id="X8576E1368448066B" name="X8576E1368448066B"></a></p>

<h4>6.3 <span class="Heading">Generators: Properties</span></h4>

<p><a id="X7ED359D87D6B2F79" name="X7ED359D87D6B2F79"></a></p>

<h5>6.3-1 IsReduced</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsReduced</code>( <var class="Arg">gen</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="code">true</code> or <code class="code">false</code></p>

<p>Check if the <strong class="pkg">homalg</strong> set of generators <var class="Arg">gen</var> is marked reduced. <br /> (no method installed)</p>

<p><a id="X7E136BCD7F22B571" name="X7E136BCD7F22B571"></a></p>

<h4>6.4 <span class="Heading">Generators: Attributes</span></h4>

<p><a id="X7B6F787085536F90" name="X7B6F787085536F90"></a></p>

<h5>6.4-1 ProcedureToReadjustGenerators</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ProcedureToReadjustGenerators</code>( <var class="Arg">gen</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a function</p>

<p>A function that takes the rows/columns of <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>

<p><a id="X7AC876EC8137AEA4" name="X7AC876EC8137AEA4"></a></p>

<h4>6.5 <span class="Heading">Generators: Operations and Functions</span></h4>


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