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##
#W isomorph.xml
#Y Copyright (C) 2014-20 James D. Mitchell
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##
<#GAPDoc Label="OnMultiplicationTable">
<ManSection>
<Oper Name = "OnMultiplicationTable" Arg = "table, p"/>
<Returns>
A multiplication table.
</Returns>
<Description>
If <A>table</A> is a multiplication table of a semigroup and the second
argument <A>p</A> is a permutation of
<C>[1 .. Length(<A>table</A>)]</C>, then this operation returns a
multiplication table of a semigroup isomorphic to that defined by
<A>table</A> where the elements <C>[1 .. Length(<A>table</A>)]</C> are
relabelled according to <A>p</A>.
<#GAPDoc Label="CanonicalMultiplicationTablePerm">
<ManSection>
<Attr Name = "CanonicalMultiplicationTablePerm" Arg = "S"/>
<Returns>
A permutation.
</Returns>
<Description>
This function returns a permutation <C>p</C> such that
<C>OnMultiplicationTable(MultiplicationTable(<A>S</A>), p);</C>
equals <C>CanonicalMultiplicationTable(<A>S</A>)</C>. <P/>
See <Ref Attr="CanonicalMultiplicationTable"/> for more details.<P/>
<C>CanonicalMultiplicationTablePerm</C> uses the machinery for canonical
labelling of vertex coloured digraphs in &BLISS; via
<Ref Oper="BlissCanonicalLabelling" Label="for a digraph and a list"
BookName="digraphs"/>.<P/>
<#GAPDoc Label="CanonicalMultiplicationTable">
<ManSection>
<Attr Name = "CanonicalMultiplicationTable" Arg = "S"/>
<Returns>
A canonical multiplication table (up to isomorphism) of a semigroup.
</Returns>
<Description>
This function returns a multiplication table of a semigroup
isomorphic to the semigroup <A>S</A>. <C>CanonicalMultiplicationTable</C>
returns a multiplication that is canonical, in the sense that if two
semigroups <C>S</C> and <C>T</C> are isomorphic, then the return values
of <C>CanonicalMultiplicationTable</C> are equal. <P/>
<C>CanonicalMultiplicationTable</C> uses the machinery for canonical
labelling of vertex coloured digraphs in &BLISS; via
<Ref Oper="BlissCanonicalLabelling" Label="for a digraph and a list"
BookName="digraphs"/>.<P/>
The multiplication table returned by this function is the result of
<C>OnMultiplicationTable(MultiplicationTable(<A>S</A>),
CanonicalMultiplicationTablePerm(<A>S</A>));</C><P/>
Note that the performance of <C>CanonicalMultiplicationTable</C> is
vastly superior to that of <C>SmallestMultiplicationTable</C>.<P/>
See also: <Ref Attr= "CanonicalMultiplicationTablePerm"/> and
<Ref Oper="OnMultiplicationTable"/>.
<#GAPDoc Label="SmallestMultiplicationTable">
<ManSection>
<Attr Name = "SmallestMultiplicationTable" Arg = "S"/>
<Returns>
The lex-least multiplication table of a semigroup.
</Returns>
<Description>
This function returns the lex-least multiplication table of a semigroup
isomorphic to the semigroup <A>S</A>. <C>SmallestMultiplicationTable</C>
returns the lex-least multiplication of any semigroup isomorphic to
<A>S</A>.
Due to the high complexity of computing the smallest multiplication table
of a semigroup, this function only performs well for semigroups with
at most approximately 50 elements.<P/>
<C>SmallestMultiplicationTable</C> is based on the function
<Ref Attr = "IdSmallSemigroup" BookName = "Smallsemi"/> by Andreas
Distler.<P/>
From Version 3.3.0 of &SEMIGROUPS; this attribute is computed using
<Ref Func="MinimalImage" BookName = "images"/> from the
the &IMAGES; package.
See also: <Ref Attr= "CanonicalMultiplicationTable"/>.
<#GAPDoc Label="IsIsomorphicSemigroup">
<ManSection>
<Oper Name = "IsIsomorphicSemigroup" Arg = "S, T"/>
<Returns>
<K>true</K> or <K>false</K>.
</Returns>
<Description>
If <A>S</A> and <A>T</A> are semigroups, then this operation attempts
to determine whether <A>S</A> and <A>T</A> are isomorphic semigroups by
using the operation <Ref Oper = "IsomorphismSemigroups" />. If
<C>IsomorphismSemigroups(<A>S</A>, <A>T</A>)</C> returns an isomorphism,
then <C>IsIsomorphicSemigroup(<A>S</A>, <A>T</A>)</C> returns <K>true</K>,
while if <C>IsomorphismSemigroups(<A>S</A>, <A>T</A>)</C> returns
<K>fail</K>, then <C>IsIsomorphicSemigroup(<A>S</A>, <A>T</A>)</C> returns
<K>false</K>.<P/>
Note that in some cases, at present, there is no
method for determining whether <A>S</A> is isomorphic to <A>T</A>, even
if it is obvious to the user whether or not <A>S</A> and <A>T</A> are
isomorphic. There are plans to improve this in the future. <P/>
<#GAPDoc Label="IsomorphismSemigroups">
<ManSection>
<Oper Name = "IsomorphismSemigroups" Arg = "S, T"/>
<Returns>
An isomorphism, or <K>fail</K>.
</Returns>
<Description>
This operation attempts to find an isomorphism from the semigroup <A>S</A>
to the semigroup <A>T</A>. If it finds one, then it is returned, and if
not, then <K>fail</K> is returned. <P/>
<C>IsomorphismSemigroups</C> uses the machinery for finding isomorphisms
between vertex coloured digraphs in &BLISS; via
<Ref Oper="IsomorphismDigraphs"
Label="for digraphs and homogeneous lists"
BookName="digraphs"/> using digraphs constructed from the
multiplication tables of <A>S</A> and <A>T</A>.
<P/>
Note that finding an isomorphism between two semigroups is difficult, and
may not be possible for semigroups whose size exceeds a few hundred
elements. On the other hand, <C>IsomorphismSemigroups</C> may be
able deduce that <A>S</A> and <A>T</A> are not isomorphic by finding
that some of their semigroup-theoretic properties differ.
<Example><![CDATA[
gap> S := RectangularBand(IsTransformationSemigroup, 4, 5);
<regular transformation semigroup of size 20, degree 9 with 5
generators>
gap> T := RectangularBand(IsBipartitionSemigroup, 4, 5);
<regular bipartition semigroup of size 20, degree 3 with 5 generators>
gap> IsomorphismSemigroups(S, T) <> fail;
true
gap> D := DClass(FullTransformationMonoid(5),
> Transformation([1, 2, 3, 4, 1]));;
gap> S := PrincipalFactor(D);;
gap> StructureDescription(UnderlyingSemigroup(S)); "S4"
gap> S;
<Rees 0-matrix semigroup 10x5 over S4>
gap> D := DClass(PartitionMonoid(5),
> Bipartition([[1], [2, -2], [3, -3], [4, -4], [5, -5], [-1]]));;
gap> T := PrincipalFactor(D);;
gap> StructureDescription(UnderlyingSemigroup(T)); "S4"
gap> T;
<Rees 0-matrix semigroup 15x15 over S4>
gap> IsomorphismSemigroups(S, T);
fail
gap> I := SemigroupIdeal(FullTransformationMonoid(5),
> Transformation([1, 1, 2, 3, 4]));;
gap> T := PrincipalFactor(DClass(I, I.1));;
gap> StructureDescription(UnderlyingSemigroup(T)); "S4"
gap> T;
<Rees 0-matrix semigroup 10x5 over S4>
gap> IsomorphismSemigroups(S, T) <> fail;
true]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
<#GAPDoc Label="AutomorphismGroup">
<ManSection>
<Oper Name = "AutomorphismGroup" Arg = "S" Label="for a semigroup"/>
<Returns>
A group.
</Returns>
<Description>
This operation returns the group of automorphisms of the semigroup <A>S</A>.
<C>AutomorphismGroup</C> uses &BLISS; via
<Ref Oper="AutomorphismGroup"
Label="for a digraph and a homogeneous list"
BookName="digraphs"/> using a vertex coloured digraph constructed
from the multiplication table of <A>S</A>. Consequently, this method is
only really feasible for semigroups whose size does not exceed a few hundred elements.
<P/>
<Example><![CDATA[
gap> S := RectangularBand(IsTransformationSemigroup, 4, 5);
<regular transformation semigroup of size 20, degree 9 with 5
generators>
gap> StructureDescription(AutomorphismGroup(S)); "S4 x S5"]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
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