#############################################################################
##
#W conguniv.xml
#Y Copyright (C) 2015 Michael Young
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##
<#GAPDoc Label="IsUniversalSemigroupCongruence">
<ManSection>
<Prop Name = "IsUniversalSemigroupCongruence" Arg = "obj"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
This property describes a type of semigroup congruence, which must refer
to the <E>universal semigroup congruence</E> <M>S \times S</M>.
Externally, an object of this type may be used in the same way as any
other object in the category <Ref Prop = "IsSemigroupCongruence"
BookName = "ref"/>.<P/>
An object of this type may be constructed with
<C>UniversalSemigroupCongruence</C> or this representation may be selected
automatically as an alternative to an
<C>IsRZMSCongruenceByLinkedTriple</C> object (since the universal
congruence cannot be represented by a linked triple).
<Example><![CDATA[
gap> S := Semigroup([Transformation([3, 2, 3])]);;
gap> U := UniversalSemigroupCongruence(S);;
gap> IsUniversalSemigroupCongruence(U);
true]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="IsUniversalSemigroupCongruenceClass">
<ManSection>
<Filt Name = "IsUniversalSemigroupCongruenceClass" Arg = "obj"Type = "category"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
This category describes a class of the universal semigroup congruence (see
<Ref Prop = "IsUniversalSemigroupCongruence"/>). A universal semigroup
congruence by definition has precisely one congruence class, which
contains all of the elements of the semigroup in question.
<#GAPDoc Label="UniversalSemigroupCongruence">
<ManSection>
<Oper Name = "UniversalSemigroupCongruence" Arg = "S"/>
<Returns>A universal semigroup congruence.</Returns>
<Description>
This operation returns the universal semigroup congruence for the
semigroup <A>S</A>. It can be used in the same way as any other
semigroup congruence object.
<Example><![CDATA[
gap> S := ReesZeroMatrixSemigroup(SymmetricGroup(3),
> [[(), (1, 3, 2)], [(1, 2), 0]]);;
gap> UniversalSemigroupCongruence(S);
<universal semigroup congruence over
<Rees 0-matrix semigroup 2x2 over Sym( [ 1 .. 3 ] )>>]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="TrivialCongruence">
<ManSection>
<Attr Name = "TrivialCongruence" Arg = "S"/>
<Returns>A trivial semigroup congruence.</Returns>
<Description>
This operation returns the trivial semigroup congruence for the
semigroup <A>S</A>. It can be used in the same way as any other
semigroup congruence object.
<Example><![CDATA[
gap> S := ReesZeroMatrixSemigroup(SymmetricGroup(3),
> [[(), (1, 3, 2)], [(1, 2), 0]]);;
gap> TrivialCongruence(S);
<semigroup congruence over <Rees 0-matrix semigroup 2x2 over
Sym( [ 1 .. 3 ] )> with linked triple (1,2,2)>
gap> S := PartitionMonoid(2);
<regular bipartition *-monoid of size 15, degree 2 with 3 generators>
gap> TrivialCongruence(S);
<2-sided semigroup congruence over <regular bipartition *-monoid
of size 15, degree 2 with 3 generators> with 0 generating pairs>]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
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