Quelle semiffmat.xml Sprache: XML

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#W  semiffmat.xml
#Y  Copyright (C) 2014                                    Markus Pfeiffer
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##  Licensing information can be found in the README file of this package.
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<#GAPDoc Label="IsMatrixOverFiniteFieldSemigroup">
<ManSection>
  <Prop Name="IsMatrixOverFiniteFieldSemigroup" Arg="S"/>
  <Prop Name="IsMatrixOverFiniteFieldMonoid" Arg="S"/>
  <Returns><K>true</K> or <K>false</K>.</Returns>
  <Description>
    A <E>matrix semigroup</E> is simply a semigroup consisting of
    matrices over a finite field. An object in &GAP; is a matrix semigroup if
    it satisfies <Ref Prop="IsSemigroup" BookName="ref"/> and
    <Ref Filt = "IsMatrixOverFiniteFieldCollection"/>. <P/>

    A <E>matrix monoid</E> is simply a monoid consisting of
    matrices over a finite field. An object in &GAP; is a matrix monoid if
    it satisfies <Ref Prop="IsMonoid" BookName="ref"/> and
    <Ref Filt = "IsMatrixOverFiniteFieldCollection"/>. <P/>

    Note that it is possible for a matrix semigroup to have a
    multiplicative neutral element (i.e. an identity element) but not to
    satisfy <C>IsMatrixOverFiniteFieldMonoid</C>.
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="MatrixSemigroup">
<ManSection>
  <Func Name="MatrixSemigroup" Arg="list [, F]"/>
  <Returns>A matrix semigroup.</Returns>
  <Description>
    This is a helper function to create matrix semigroups from &GAP; matrices.
    The argument <A>list</A> is a homogeneous list of &GAP; matrices over a
    finite field, and the optional argument <A>F</A> is a finite field.<P/>

    The specification of the field <A>F</A> can be necessary to prevent
    &GAP; from trying to find a smaller common field for the entries in
    <A>list</A>.
    <Example><![CDATA[
gap> S := Semigroup([
> Matrix(GF(9), Z(3) * [[1, 0, 0], [1, 1, 0], [0, 1, 0]]),
> Matrix(GF(9), Z(3) * [[0, 0, 0], [0, 0, 1], [0, 1, 0]])]);
<semigroup of 3x3 matrices over GF(3^2) with 2 generators>
gap> S := Semigroup([
>  Matrix(GF(3), Z(3) * [[1, 0, 0], [1, 1, 0], [0, 1, 0]]),
>  Matrix(GF(3), Z(3) * [[0, 0, 0], [0, 0, 1], [0, 1, 0]])]);
<semigroup of 3x3 matrices over GF(3) with 2 generators>
gap> S := Semigroup([
>  Matrix(GF(4), Z(4) * [[1, 0, 0], [1, 1, 0], [0, 1, 0]]),
>  Matrix(GF(4), Z(4) * [[0, 0, 0], [0, 0, 1], [0, 1, 0]])]);
<semigroup of 3x3 matrices over GF(2^2) with 2 generators>]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc>
<ManSection>
  <Attr Name="DegreeOfMatrixSemigroup" Arg="S"/>
  <Returns>An integer</Returns>
  <Description>
    This attribute is the number of rows or columns of any matrix
    in the matrix semigroup <A>S</A>.
  </Description>
</ManSection>
<#/GAPDoc>



<#GAPDoc Label="IsFullMatrixMonoid">
  <ManSection>
    <Prop Name="IsFullMatrixMonoid" Arg="S"/>
    <Prop Name="IsGeneralLinearMonoid" Arg="S"/>
    <Description>
      <C>IsFullMatrixMonoid</C> and <C>IsGeneralLinearMonoid</C> return
      <K>true</K> if the semigroup <C>S</C> was created using either of the
      commands <Ref Oper="FullMatrixMonoid"/> or
      <Ref Oper="GeneralLinearMonoid"/> and <K>false</K> otherwise.

      <Example><![CDATA[
gap> S := RandomSemigroup(IsTransformationSemigroup, 4, 4);;
gap> IsFullMatrixMonoid(S);
false
gap> S := GeneralLinearMonoid(3, 3);
<general linear monoid 3x3 over GF(3)>
gap> IsFullMatrixMonoid(S);
true]]></Example>
    </Description>
  </ManSection>
<#/GAPDoc>

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