<!-- 
  HAPPRIME - internal.xml.in
  Function documentation template file for HAPprime
  Paul Smith

  Copyright (C)  2008
  Paul Smith
  National University of Ireland Galway

  $Id: functions.xml.in 200 2008-02-05 14:29:47Z pas $
-->

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<Chapter  Label="InternalFunctions">
  <Heading>Internal functions</Heading>

  <Section Label="InternalGGenerators">
    <Heading>Matrices as &G;-generators of a &FG;-module vector space</Heading>

    Both <K>FpGModuleGF</K> (Chapter <Ref Chap="FpGModuleGF"/>) and 
    <K>FpGModuleHomomorphismGF</K> 
    (Chapter <Ref Chap="FpGModuleGFHom"/>) store a matrix 
    whose rows are &G;-generators for a module vector space (the module and 
    the homomorphism's image respectively). The internal functions listed here
    provide common operations for dealing with these matrices.

    <!-- GAPDocSourceSuffix="_manGenMatInt" -->

  </Section>
  
  <Section>
    <Heading>&FG;-modules</Heading>

    &FG;-modules in &HAPprime; use the datatype <K>FpGModuleGF</K> 
    (Chapter <Ref Chap="FpGModuleGF"/>). Internally, this uses many of the 
    functions listed in Section <Ref Sect="InternalGGenerators"/>, and
    further internal functions are listed below.

    <!-- GAPDocSourceSuffix="_DTmanFpGModuleInt" -->

  </Section>
  
  <Section>
    <Heading>Resolutions</Heading>

    For details of the main resolution functions in &HAPprime;, see Chapter
    <Ref Chap="Resolution"/> of this datatypes reference manual, and 
    <Ref Sect="Resolutions" BookName="HAPprime"/> in the &HAPprime; user guide.
    This section describes the internal helper functions used by the 
    higher-level functions.

    <!-- GAPDocSourceSuffix="_DTmanResolutionInt" -->

  </Section>
  
  <Section>
    <Heading>Polynomial rings</Heading>

    Ths internal functions in this section implement isomorphisms between 
    polynomial rings via maps from one set of indeterminates to another.

    <!-- GAPDocSourceSuffix="_manDTPolynomialInt" -->

  </Section>
  
  <Section>
    <Heading>Gröbner Bases</Heading>

    Ths internal functions in this section provide a transparent way of using
    Singular's Gröbner basis functions, if they are available. See the 
    &singular; package documentation <Ref Chap="singular" BookName="singular"/>
    for further details.

    <!-- GAPDocSourceSuffix="_manDTGroebnerInt" -->

  </Section>
  
  <Section>
    <Heading>Presentations of graded algebras</Heading>

    For the main functions dealing with presentations of graded algebras, and
    details of the datatype see Chapter 
    <Ref Chap="GradedAlgebraPresentationDatatype"/>. This section details the
    internal functions used by the higher-level functions.

    <!-- GAPDocSourceSuffix="_DTmanGradedAlgebraInt" -->

  </Section>

  <Section>
    <Heading>Derivations</Heading>

    The kernel of a derivation is efficiently computed by writing the ring 
    <M>R</M> as an <M>S</M>-module where <M>S</M> is the subring of squares
    (or some larger ring in the kernel). The internal function in this section
    is used to describe this <M>S</M>-module and allow conversion between 
    elements of <M>R</M> and elements of the <M>S</M>-module.

    <!-- GAPDocSourceSuffix="_DTmanDerivationInt" -->

  </Section>

  <Section>
    <Heading>Test functions</Heading>

    Internal helper functions for testing &HAPprime;.

    <!-- GAPDocSourceSuffix="_manTestInt" -->

  </Section>
  
</Chapter>