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#W  conguniv.xml
#Y  Copyright (C) 2015                                      Michael Young
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##  Licensing information can be found in the README file of this package.
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<#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.

      <Example><![CDATA[
gap> S := Semigroup([Transformation([3, 2, 3])]);;
gap> U := UniversalSemigroupCongruence(S);;
gap> classes := EquivalenceClasses(U);;
gap> IsUniversalSemigroupCongruenceClass(classes[1]);
true]]></Example>
    </Description>
  </ManSection>
<#/GAPDoc>

<#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>
