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<Appendix Label="FileOverview">
<Heading>Overview of the &MatricesForHomalg; Package Source Code</Heading>

<Section Label="RingsMapsMatrices">

<Heading>Rings, Ring Maps, Matrices, Ring Relations</Heading>

<Table Align="l|l">
<Caption><E>The &MatricesForHomalg; package files</E></Caption>

<Row><Item>Filename <F>.gd</F>/<F>.gi</F></Item><Item>Content</Item></Row>

<HorLine/>

<Row>
  <Item><F>homalg</F></Item>
  <Item>definitions of the basic &GAP4; categories</Item>
</Row>
<Row>
  <Item></Item>
  <Item>and some tool functions (e.g. <C>homalgMode</C>)</Item>
</Row>

<Row><Item></Item><Item></Item></Row>

<Row>
  <Item><F>homalgTable</F></Item>
  <Item>dictionaries between &MatricesForHomalg;</Item>
</Row>
<Row>
  <Item></Item>
  <Item>and the computing engines</Item>
</Row>

<Row>
  <Item><F>HomalgRing</F></Item>
  <Item>internal and external rings</Item>
</Row>

<Row><Item></Item><Item></Item></Row>

<Row>
  <Item><F>HomalgRingMap</F></Item>
  <Item>ring maps</Item>
</Row>

<Row><Item></Item><Item></Item></Row>

<Row>
  <Item><F>HomalgMatrix</F></Item>
  <Item>internal and external matrices</Item>
</Row>

<Row><Item></Item><Item></Item></Row>

<Row>
  <Item><F>HomalgRingRelations</F></Item>
  <Item>a set of ring relations</Item>
</Row>

</Table>

</Section>

<Section Label="Low Level Algorithms">

<Heading>The Low Level Algorithms</Heading>

In the following CAS or CASystem mean computer algebra systems.

<Table Align="l|l">
<Caption><E>The &MatricesForHomalg; package files (continued)</E></Caption>

<Row><Item>Filename <F>.gd</F>/<F>.gi</F></Item><Item>Content</Item></Row>

<HorLine/>

<Row>
  <Item><F>Tools</F></Item>
  <Item>the elementary matrix operations that can be</Item>
</Row>

<Row>
  <Item></Item>
  <Item>overwritten using the homalgTable</Item>
</Row>

<Row>
  <Item></Item>
  <Item>(and hence delegable even to other CASystems)</Item>
</Row>

<Row><Item></Item><Item></Item></Row>

<Row>
  <Item><F>Service</F></Item>
  <Item>the three operations: basis, reduction, and syzygies;</Item>
</Row>

<Row>
  <Item></Item>
  <Item>they can also be overwritten using the homalgTable</Item>
</Row>

<Row>
  <Item></Item>
  <Item>(and hence delegable even to other CASystems)</Item>
</Row>

<Row><Item></Item><Item></Item></Row>

<Row>
  <Item><F>Basic</F></Item>
  <Item>higher level operations for matrices</Item>
</Row>

<Row>
  <Item></Item>
  <Item>(cannot be overwritten using the homalgTable)</Item>
</Row>

</Table>

</Section>

<Section Label="Logical Implications">

<Heading>Logical Implications for &MatricesForHomalg; Objects</Heading>

<Table Align="l|l">
<Caption><E>The &MatricesForHomalg; package files (continued)</E></Caption>
<Row><Item>Filename <F>.gd</F>/<F>.gi</F></Item><Item>Content</Item></Row>

<HorLine/>

<Row>
  <Item><F>LIRNG</F></Item>
  <Item>logical implications for rings</Item>
</Row>

<Row><Item></Item><Item></Item></Row>

<Row>
  <Item><F>LIMAP</F></Item>
  <Item>logical implications for ring maps</Item>
</Row>

<Row><Item></Item><Item></Item></Row>

<Row>
  <Item><F>LIMAT</F></Item>
  <Item>logical implications for matrices</Item>
</Row>

<Row><Item></Item><Item></Item></Row>

<Row>
  <Item><F>COLEM</F></Item>
  <Item>clever operations for lazy evaluated matrices</Item>
</Row>

<Row><Item></Item><Item></Item></Row>

</Table>

</Section>

<Section Label="ResidueClassRingForHomalg_subpackage">

<Heading>The subpackage &ResidueClassRingForHomalg;</Heading>

<Table Align="l|l">
<Caption><E>The &MatricesForHomalg; package files (continued)</E></Caption>
<Row><Item>Filename <F>.gd</F>/<F>.gi</F></Item><Item>Content</Item></Row>

<HorLine/>

<Row>
  <Item><F>ResidueClassRingForHomalg</F></Item>
  <Item>some global variables</Item>
</Row>

<Row><Item></Item><Item></Item></Row>

<Row>
  <Item><F>ResidueClassRing</F></Item>
  <Item>residue class rings, their elements, and matrices,</Item>
</Row>

<Row>
  <Item></Item>
  <Item>together with their constructors and operations</Item>
</Row>

<Row><Item></Item><Item></Item></Row>

<Row>
  <Item><F>ResidueClassRingTools</F></Item>
  <Item>the elementary matrix operations for matrices</Item>
</Row>

<Row>
  <Item></Item>
  <Item>over residue class rings</Item>
</Row>

<Row><Item></Item><Item></Item></Row>

<Row>
  <Item><F>ResidueClassRingBasic</F></Item>
  <Item>the three operations: basis, reduction, and syzygies</Item>
</Row>

<Row>
  <Item></Item>
  <Item>for matrices over residue class rings</Item>
</Row>

<Row><Item></Item><Item></Item></Row>

</Table>

</Section>

<Section Label="The built-in rings">

<Heading>The homalgTable for &GAP4; built-in rings</Heading>

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

<Table Align="l|l">
<Caption><E>The &MatricesForHomalg; package files (continued)</E></Caption>
<Row><Item>Filename .gi</Item><Item>Content</Item></Row>

<HorLine/>

<Row>
  <Item><F>Integers</F></Item>
  <Item>the homalgTable for the ring of integers</Item>
</Row>

</Table>

</Section>

</Appendix>

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