#############################################################################
##
#W dual.xml
#Y Copyright (C) 2018 Finn Smith
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##
<#GAPDoc Label="DualSemigroup">
<ManSection>
<Attr Name = "DualSemigroup" Arg = "S"/>
<Returns>The dual semigroup of the given semigroup.</Returns>
<Description>
The dual semigroup of a semigroup <A>S</A> is the semigroup with the
same underlying set as <A>S</A>, but with multiplication reversed; this
is anti-isomorphic to <A>S</A>. This attribute returns a semigroup
isomorphic to the dual semigroup of <A>S</A>.
<Example>
<![CDATA[
gap> S := Semigroup([Transformation([1, 4, 3, 2, 2]),
> Transformation([5, 4, 4, 1, 2])]);;
gap> D := DualSemigroup(S);
<dual semigroup of <transformation semigroup of degree 5 with 2
generators>>
gap> Size(S) = Size(D);
true
gap> NrDClasses(S) = NrDClasses(D);
true]]></Example> </Description> </ManSection>
<#/GAPDoc>
<#GAPDoc Label="AntiIsomorphismDualSemigroup">
<ManSection>
<Attr Name= "AntiIsomorphismDualSemigroup" Arg = "S"/>
<Returns>
An anti-isomorphism from <A>S</A> to the corresponding dual semigroup.
</Returns>
<Description>
The dual semigroup of <A>S</A> mathematically has the same underlying
set as <A>S</A>, but is represented with a different set of elements in
&SEMIGROUPS;. This function returns a mapping which is an anti-isomorphism from
<A>S</A> to its dual.
<Example>
<![CDATA[
gap> S := PartitionMonoid(3);
<regular bipartition *-monoid of size 203, degree 3 with 4 generators>
gap> map := AntiIsomorphismDualSemigroup(S);
MappingByFunction( <regular bipartition *-monoid of size 203,
degree 3 with 4 generators>, <dual semigroup of
<regular bipartition *-monoid of size 203, degree 3 with 4 generators>
>, function( x ) ... end, function( x ) ... end )
gap> inv := InverseGeneralMapping(map);;
gap> x := Bipartition([[1, -2], [2, -3], [3, -1]]);
<block bijection: [ 1, -2 ], [ 2, -3 ], [ 3, -1 ]>
gap> y := Bipartition([[1], [2, -2], [3, -3], [-1]]);
<bipartition: [ 1 ], [ 2, -2 ], [ 3, -3 ], [ -1 ]>
gap> (x ^ map) * (y ^ map) = (y * x) ^ map;
true
gap> x ^ map;
<<block bijection: [ 1, -2 ], [ 2, -3 ], [ 3, -1 ]>
in the dual semigroup>]]></Example> </Description> </ManSection>
<#/GAPDoc>
<#GAPDoc Label="IsDualSemigroupElement">
<ManSection>
<Filt Name = "IsDualSemigroupElement"Type = "Category" Arg="elt"/>
<Returns>Returns <K>true</K> if <A>elt</A> has the representation of a dual
semigroup element.</Returns>
<Description>
Elements of a dual semigroup obtained using
<Ref Attr = "AntiIsomorphismDualSemigroup"/> normally lie in this
category. The exception is elements obtained by applying
the map <Ref Attr = "AntiIsomorphismDualSemigroup"/> to elements already
in this category. That is, the elements of a semigroup lie in the
category <Ref Filt="IsDualSemigroupElement"/> if and only if the
elements of the corresponding dual semigroup do not.
<Example>
<![CDATA[
gap> S := SingularPartitionMonoid(4);;
gap> D := DualSemigroup(S);;
gap> s := GeneratorsOfSemigroup(S)[1];;
gap> map := AntiIsomorphismDualSemigroup(S);;
gap> t := s ^ map;
<<block bijection: [ 1, 2, -1, -2 ], [ 3, -3 ], [ 4, -4 ]>
in the dual semigroup>
gap> IsDualSemigroupElement(t);
true
gap> inv := InverseGeneralMapping(map);;
gap> x := t ^ inv;
<block bijection: [ 1, 2, -1, -2 ], [ 3, -3 ], [ 4, -4 ]>
gap> IsDualSemigroupElement(x);
false]]></Example> </Description> </ManSection>
<#/GAPDoc>
<#GAPDoc Label="IsDualSemigroupRep">
<ManSection>
<Filt Name = "IsDualSemigroupRep"Type = "Category" Arg="sgrp"/>
<Returns>Returns <K>true</K> if <A>sgrp</A> lies in the category of
dual semigroups.</Returns>
<Description>
Semigroups created using <Ref Func="DualSemigroup"/>
normally lie in this category. The exception is semigroups
which are the dual of semigroups already lying in this category.
That is, a semigroup lies in the category
<Ref Filt="IsDualSemigroupRep"/> if and only if the corresponding
dual semigroup does not. Note that this is not a Representation in the
GAP sense, and will likely be renamed in a future major release of the
package.
<Example>
<![CDATA[
gap> S := Semigroup([Transformation([3, 5, 1, 1, 2]),
> Transformation([1, 2, 4, 4, 3])]);
<transformation semigroup of degree 5 with 2 generators>
gap> D := DualSemigroup(S);
<dual semigroup of <transformation semigroup of degree 5 with 2
generators>>
gap> IsDualSemigroupRep(D);
true
gap> R := DualSemigroup(D);
<transformation semigroup of degree 5 with 2 generators>
gap> IsDualSemigroupRep(R);
false
gap> R = S;
true
gap> T := Range(IsomorphismTransformationSemigroup(D));
<transformation semigroup of size 16, degree 17 with 2 generators>
gap> IsDualSemigroupRep(T);
false
gap> x := Representative(D);
<Transformation( [ 3, 5, 1, 1, 2 ] ) in the dual semigroup>
gap> V := Semigroup(x);
<dual semigroup of <commutative transformation semigroup of degree 5
with 1 generator>>
gap> IsDualSemigroupRep(V);
true]]></Example> </Description> </ManSection>
<#/GAPDoc>
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