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#W ideals.xml
#Y Copyright (C) 2014 James D. Mitchell
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## Licensing information can be found in the README file of this package.
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<#GAPDoc Label="MinimalIdealGeneratingSet">
<ManSection>
<Attr Name = "MinimalIdealGeneratingSet" Arg = "I"/>
<Returns>A minimal set ideal generators of an ideal.</Returns>
<Description>
This function returns a minimal set of elements of the parent of the
semigroup ideal <A>I</A> required to generate <A>I</A> as an ideal. <P/>
The notion of the generators of an ideal is distinct from the notion of
the generators of a semigroup or monoid. In particular, the semigroup
generated by the generators of an ideal is not, in general, equal to that
ideal. Use <Ref Attr = "GeneratorsOfSemigroup" BookName = "ref"/> to obtain a
semigroup generating set for an ideal, but beware that this can be very
costly.
<#GAPDoc Label="GeneratorsOfSemigroupIdeal">
<ManSection>
<Attr Name = "GeneratorsOfSemigroupIdeal" Arg = "I"/>
<Returns>The generators of an ideal of a semigroup.</Returns>
<Description>
This function returns the generators of the two-sided ideal <A>I</A>,
which were used to defined <A>I</A> when it was created. <P/>
If <A>I</A> is an ideal of a semigroup, then <A>I</A> is defined to be the
least 2-sided ideal of a semigroup <C>S</C> containing a set <C>J</C> of
elements of <C>S</C>. The set <C>J</C> is said to <E>generate</E> <A>I</A>.
<P/>
The notion of the generators of an ideal is distinct from the notion of
the generators of a semigroup or monoid. In particular, the semigroup
generated by the generators of an ideal is not, in general, equal to that
ideal. Use <Ref Attr = "GeneratorsOfSemigroup" BookName = "ref"/> to obtain a
semigroup generating set for an ideal, but beware that this can be very
costly.
<#GAPDoc Label="SemigroupIdeal">
<ManSection>
<Func Name = "SemigroupIdeal" Arg = "S, obj1, obj2, .. . "/>
<Returns>
An ideal of a semigroup.
</Returns>
<Description>
If <A>obj1</A>, <A>obj2</A>, .. . are (any combination) of elements of the
semigroup <A>S</A> or collections of elements of <A>S</A> (including
subsemigroups and ideals of <A>S</A>), then <C>SemigroupIdeal</C> returns the
2-sided ideal of the semigroup <A>S</A> generated by the union of
<A>obj1</A>, <A>obj2</A>, .. .. <P/>
The <Ref Func = "Parent" BookName = "ref"/> of the ideal returned by this
function is <A>S</A>.
<Example><![CDATA[
gap> S := SymmetricInverseMonoid(10);
<symmetric inverse monoid of degree 10>
gap> I := SemigroupIdeal(S, PartialPerm([1, 2]));
<inverse partial perm semigroup ideal of rank 10 with 1 generator>
gap> Size(I);
4151
gap> I := SemigroupIdeal(S, I, Idempotents(S));
<inverse partial perm semigroup ideal of rank 10 with 1025 generators>]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="SupersemigroupOfIdeal">
<ManSection>
<Attr Name = "SupersemigroupOfIdeal" Arg = "I"/>
<Returns>
An ideal of a semigroup.
</Returns>
<Description>
The <Ref Func = "Parent" BookName = "ref"/> of an ideal is the semigroup in
which the ideal was created, i.e. the first argument of <Ref
Func = "SemigroupIdeal"/> or <C>SemigroupIdealByGenerators</C>. This
function returns the semigroup containing the generators of the semigroup
(i.e. <Ref Attr = "GeneratorsOfSemigroup" BookName = "ref"/>) which are
used to compute the ideal.
<P/>
For a regular semigroup ideal, <C>SupersemigroupOfIdeal</C> will always be
the top most semigroup used to create any of the predecessors of the
current ideal. For example, if <C>S</C> is a semigroup, <C>I</C> is a
regular ideal of <C>S</C>, and <C>J</C> is an ideal of <C>I</C>, then
<C>Parent(J)</C> is <C>I</C> and <C>SupersemigroupOfIdeal(J)</C> is
<C>S</C>. This is to avoid computing a generating set for <C>I</C>,
in this example, which is expensive and unnecessary since <C>I</C> is
regular (in which case the Green's relations of I are just
restrictions of the Green's relations on S).
<P/>
If <C>S</C> is a semigroup, <C>I</C> is a non-regular ideal of <C>S</C>,
<C>J</C> is an ideal of <C>I</C>, then <C>SupersemigroupOfIdeal(J)</C> is
<C>I</C>, since we currently have to use <C>GeneratorsOfSemigroup(I)</C>
to compute anything about <C>I</C> other than its size and membership.
<Example><![CDATA[
gap> S := FullTransformationSemigroup(8);
<full transformation monoid of degree 8>
gap> x := Transformation([2, 6, 7, 2, 6, 1, 1, 5]);;
gap> D := DClass(S, x);
<Green's D-class: Transformation( [ 2, 6, 7, 2, 6, 1, 1, 5 ] )>
gap> R := PrincipalFactor(D);
<Rees 0-matrix semigroup 1050x56 over Group([ (2,8,7,4,3), (3,4) ])>
gap> S := Semigroup(List([1 .. 10], x -> Random(R)));
<subsemigroup of 1050x56 Rees 0-matrix semigroup with 10 generators>
gap> I := SemigroupIdeal(S, MultiplicativeZero(S));
<regular Rees 0-matrix semigroup ideal with 1 generator>
gap> SupersemigroupOfIdeal(I);
<subsemigroup of 1050x56 Rees 0-matrix semigroup with 10 generators>
gap> J := SemigroupIdeal(I, Representative(MinimalDClass(S)));
<regular Rees 0-matrix semigroup ideal with 1 generator>
gap> Parent(J) = I;
true
gap> SupersemigroupOfIdeal(J) = I;
false]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="Ideals">
<ManSection>
<Attr Name = "Ideals" Arg = "S" Label = "for a semigroup"/>
<Returns>
An list of ideals.
</Returns>
<Description>
If <A>S</A> is a finite non-empty semigroup, then this attribute returns a
list of the non-empty two-sided ideals of <A>S</A>. <P/>
The ideals are returned in no particular order, and each ideal uses the
minimum possible number of generators
(see <Ref Attr = "GeneratorsOfSemigroupIdeal"/>).
<Example><![CDATA[
gap> S := Semigroup([Transformation([4, 3, 4, 1]),
> Transformation([4, 3, 2, 2])]);
<transformation semigroup of degree 4 with 2 generators>
gap> Ideals(S);
[ <non-regular transformation semigroup ideal of degree 4 with
1 generator>,
<non-regular transformation semigroup ideal of degree 4 with
1 generator>,
<non-regular transformation semigroup ideal of degree 4 with
2 generators>,
<regular transformation semigroup ideal of degree 4 with 1 generator>,
<non-regular transformation semigroup ideal of degree 4 with
1 generator>,
<regular transformation semigroup ideal of degree 4 with 1 generator>
]
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
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