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<p><a id="X838651287FCCEFD8" name="X838651287FCCEFD8"></a></p>
<div class="ChapSects"><a href="chap5.html#X838651287FCCEFD8">5 <span class="Heading">Relations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X87DADB7B7CB126DC">5.1 <span class="Heading">Relations: Categories and Representations</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X80AD050F7999B7C0">5.1-1 IsHomalgRelations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X790F68B17A4846DC">5.1-2 IsHomalgRelationsOfLeftModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7FF5A3B180614698">5.1-3 IsHomalgRelationsOfRightModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X8322A26C84E80303">5.1-4 IsRelationsOfFinitelyPresentedModuleRep</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7CF74FB785F90889">5.2 <span class="Heading">Relations: Constructors</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X859231317954D702">5.3 <span class="Heading">Relations: Properties</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X798D893B7FBFCF07">5.3-1 CanBeUsedToDecideZeroEffectively</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7B9398827AEEA2E6">5.3-2 IsInjectivePresentation</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7EF7BBCA85851EF1">5.4 <span class="Heading">Relations: Attributes</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7890B5EA80774AB5">5.5 <span class="Heading">Relations: Operations and Functions</span></a>
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<h3>5 <span class="Heading">Relations</span></h3>

<p>A finite presentation of a module is given by a finite set of generators and a finite set of relations 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>

<p>The data structure of a module in <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.html#X7BD5B55C802805B4"><span class="RefLink">6</span></a>)). The different sets of relations are linked with so-called transition matrices (--> Chapter <a href="chap7.html#X8183A6857B0C3633"><span class="RefLink">7</span></a>).</p>

<p>The relations of a <strong class="pkg">homalg</strong> module are evaluated in a lazy way. This avoids unnecessary computations.</p>

<p><a id="X87DADB7B7CB126DC" name="X87DADB7B7CB126DC"></a></p>

<h4>5.1 <span class="Heading">Relations: Categories and Representations</span></h4>

<p><a id="X80AD050F7999B7C0" name="X80AD050F7999B7C0"></a></p>

<h5>5.1-1 IsHomalgRelations</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsHomalgRelations</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> relations.</p>

<p><a id="X790F68B17A4846DC" name="X790F68B17A4846DC"></a></p>

<h5>5.1-2 IsHomalgRelationsOfLeftModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsHomalgRelationsOfLeftModule</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> relations of a left module.</p>

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

<p><a id="X7FF5A3B180614698" name="X7FF5A3B180614698"></a></p>

<h5>5.1-3 IsHomalgRelationsOfRightModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsHomalgRelationsOfRightModule</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> relations of a right module.</p>

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

<p><a id="X8322A26C84E80303" name="X8322A26C84E80303"></a></p>

<h5>5.1-4 IsRelationsOfFinitelyPresentedModuleRep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsRelationsOfFinitelyPresentedModuleRep</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 relations of a finitely presented <strong class="pkg">homalg</strong> module.</p>

<p>(It is a representation of the <strong class="pkg">GAP</strong> category <code class="func">IsHomalgRelations</code> (<a href="chap5.html#X80AD050F7999B7C0"><span class="RefLink">5.1-1</span></a>))</p>

<p><a id="X7CF74FB785F90889" name="X7CF74FB785F90889"></a></p>

<h4>5.2 <span class="Heading">Relations: Constructors</span></h4>

<p><a id="X859231317954D702" name="X859231317954D702"></a></p>

<h4>5.3 <span class="Heading">Relations: Properties</span></h4>

<p><a id="X798D893B7FBFCF07" name="X798D893B7FBFCF07"></a></p>

<h5>5.3-1 CanBeUsedToDecideZeroEffectively</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CanBeUsedToDecideZeroEffectively</code>( <var class="Arg">rel</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 relations <var class="Arg">rel</var> can be used for normal form reductions. <br /> (no method installed)</p>

<p><a id="X7B9398827AEEA2E6" name="X7B9398827AEEA2E6"></a></p>

<h5>5.3-2 IsInjectivePresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsInjectivePresentation</code>( <var class="Arg">rel</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 relations <var class="Arg">rel</var> has zero syzygies.</p>

<p><a id="X7EF7BBCA85851EF1" name="X7EF7BBCA85851EF1"></a></p>

<h4>5.4 <span class="Heading">Relations: Attributes</span></h4>

<p><a id="X7890B5EA80774AB5" name="X7890B5EA80774AB5"></a></p>

<h4>5.5 <span class="Heading">Relations: Operations and Functions</span></h4>


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