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<title>GAP (MatricesForHomalg) - Appendix F: Overview of the MatricesForHomalg Package Source Code</title>
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<p><a id="X863882737DAD95A3" name="X863882737DAD95A3"></a></p>
<div class="ChapSects"><a href="chapF_mj.html#X863882737DAD95A3">F <span class="Heading">Overview of the <strong class="pkg">MatricesForHomalg</strong> Package Source Code</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapF_mj.html#X87E0F36680867FA2">F.1 <span class="Heading">Rings, Ring Maps, Matrices, Ring Relations</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapF_mj.html#X7C4917CE80359953">F.2 <span class="Heading">The Low Level Algorithms</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapF_mj.html#X7B2BFFB8876C548C">F.3 <span class="Heading">Logical Implications for <strong class="pkg">MatricesForHomalg</strong> Objects</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapF_mj.html#X85A3964E7A98C065">F.4 <span class="Heading">The subpackage <strong class="pkg">ResidueClassRingForHomalg</strong></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapF_mj.html#X860DCFF383750919">F.5 <span class="Heading">The homalgTable for <strong class="pkg">GAP4</strong> built-in rings</span></a>
</span>
</div>
</div>

<h3>F <span class="Heading">Overview of the <strong class="pkg">MatricesForHomalg</strong> Package Source Code</span></h3>

<p><a id="X87E0F36680867FA2" name="X87E0F36680867FA2"></a></p>

<h4>F.1 <span class="Heading">Rings, Ring Maps, Matrices, Ring Relations</span></h4>

<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable"><b>Table: </b><em>The <strong class="pkg">MatricesForHomalg</strong> package files</em></caption>
<tr>
<td class="tdleft">Filename <code class="file">.gd</code>/<code class="file">.gi</code></td>
<td class="tdleft">Content</td>
</tr>
<tr>
<td class="tdleft"><code class="file">homalg</code></td>
<td class="tdleft">definitions of the basic <strong class="pkg">GAP4</strong> categories</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft">and some tool functions (e.g. <code class="code">homalgMode</code>)</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft"></td>
</tr>
<tr>
<td class="tdleft"><code class="file">homalgTable</code></td>
<td class="tdleft">dictionaries between <strong class="pkg">MatricesForHomalg</strong></td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft">and the computing engines</td>
</tr>
<tr>
<td class="tdleft"><code class="file">HomalgRing</code></td>
<td class="tdleft">internal and external rings</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft"></td>
</tr>
<tr>
<td class="tdleft"><code class="file">HomalgRingMap</code></td>
<td class="tdleft">ring maps</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft"></td>
</tr>
<tr>
<td class="tdleft"><code class="file">HomalgMatrix</code></td>
<td class="tdleft">internal and external matrices</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft"></td>
</tr>
<tr>
<td class="tdleft"><code class="file">HomalgRingRelations</code></td>
<td class="tdleft">a set of ring relations</td>
</tr>
</table><br />
</div>

<p><a id="X7C4917CE80359953" name="X7C4917CE80359953"></a></p>

<h4>F.2 <span class="Heading">The Low Level Algorithms</span></h4>

<p>In the following CAS or CASystem mean computer algebra systems.</p>

<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable"><b>Table: </b><em>The <strong class="pkg">MatricesForHomalg</strong> package files (continued)</em></caption>
<tr>
<td class="tdleft">Filename <code class="file">.gd</code>/<code class="file">.gi</code></td>
<td class="tdleft">Content</td>
</tr>
<tr>
<td class="tdleft"><code class="file">Tools</code></td>
<td class="tdleft">the elementary matrix operations that can be</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft">overwritten using the homalgTable</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft">(and hence delegable even to other CASystems)</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft"></td>
</tr>
<tr>
<td class="tdleft"><code class="file">Service</code></td>
<td class="tdleft">the three operations: basis, reduction, and syzygies;</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft">they can also be overwritten using the homalgTable</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft">(and hence delegable even to other CASystems)</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft"></td>
</tr>
<tr>
<td class="tdleft"><code class="file">Basic</code></td>
<td class="tdleft">higher level operations for matrices</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft">(cannot be overwritten using the homalgTable)</td>
</tr>
</table><br />
</div>

<p><a id="X7B2BFFB8876C548C" name="X7B2BFFB8876C548C"></a></p>

<h4>F.3 <span class="Heading">Logical Implications for <strong class="pkg">MatricesForHomalg</strong> Objects</span></h4>

<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable"><b>Table: </b><em>The <strong class="pkg">MatricesForHomalg</strong> package files (continued)</em></caption>
<tr>
<td class="tdleft">Filename <code class="file">.gd</code>/<code class="file">.gi</code></td>
<td class="tdleft">Content</td>
</tr>
<tr>
<td class="tdleft"><code class="file">LIRNG</code></td>
<td class="tdleft">logical implications for rings</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft"></td>
</tr>
<tr>
<td class="tdleft"><code class="file">LIMAP</code></td>
<td class="tdleft">logical implications for ring maps</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft"></td>
</tr>
<tr>
<td class="tdleft"><code class="file">LIMAT</code></td>
<td class="tdleft">logical implications for matrices</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft"></td>
</tr>
<tr>
<td class="tdleft"><code class="file">COLEM</code></td>
<td class="tdleft">clever operations for lazy evaluated matrices</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft"></td>
</tr>
</table><br />
</div>

<p><a id="X85A3964E7A98C065" name="X85A3964E7A98C065"></a></p>

<h4>F.4 <span class="Heading">The subpackage <strong class="pkg">ResidueClassRingForHomalg</strong></span></h4>

<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable"><b>Table: </b><em>The <strong class="pkg">MatricesForHomalg</strong> package files (continued)</em></caption>
<tr>
<td class="tdleft">Filename <code class="file">.gd</code>/<code class="file">.gi</code></td>
<td class="tdleft">Content</td>
</tr>
<tr>
<td class="tdleft"><code class="file">ResidueClassRingForHomalg</code></td>
<td class="tdleft">some global variables</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft"></td>
</tr>
<tr>
<td class="tdleft"><code class="file">ResidueClassRing</code></td>
<td class="tdleft">residue class rings, their elements, and matrices,</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft">together with their constructors and operations</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft"></td>
</tr>
<tr>
<td class="tdleft"><code class="file">ResidueClassRingTools</code></td>
<td class="tdleft">the elementary matrix operations for matrices</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft">over residue class rings</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft"></td>
</tr>
<tr>
<td class="tdleft"><code class="file">ResidueClassRingBasic</code></td>
<td class="tdleft">the three operations: basis, reduction, and syzygies</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft">for matrices over residue class rings</td>
</tr>
<tr>
<td class="tdleft"></td>
<td class="tdleft"></td>
</tr>
</table><br />
</div>

<p><a id="X860DCFF383750919" name="X860DCFF383750919"></a></p>

<h4>F.5 <span class="Heading">The homalgTable for <strong class="pkg">GAP4</strong> built-in rings</span></h4>

<p>For the purposes of <strong class="pkg">homalg</strong>, the ring of integers is, at least up till now, the only ring which is properly supported in <strong class="pkg">GAP4</strong>. The <strong class="pkg">GAP4</strong> built-in cababilities for polynomial rings (also univariate) and group rings do not statisfy the minimum requirements of <strong class="pkg">homalg</strong>. The <strong class="pkg">GAP4</strong> package <strong class="pkg">Gauss</strong> enables <strong class="pkg">GAP</strong> to fullfil the <strong class="pkg">homalg</strong> requirements for prime fields, and <span class="SimpleMath">\(ℤ / p^n\)</span>.</p>

<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable"><b>Table: </b><em>The <strong class="pkg">MatricesForHomalg</strong> package files (continued)</em></caption>
<tr>
<td class="tdleft">Filename .gi</td>
<td class="tdleft">Content</td>
</tr>
<tr>
<td class="tdleft"><code class="file">Integers</code></td>
<td class="tdleft">the homalgTable for the ring of integers</td>
</tr>
</table><br />
</div>


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