#############################################################################
##
#W semiffmat.xml
#Y Copyright (C) 2014 Markus Pfeiffer
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##
<#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>
<!-- FIXME(later) this will need to be updated once the method is fixed
since this is not the behaviour that we want -->
<#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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