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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the declaration of operations for semigroups.
##
#############################################################################
##
#P IsSemigroup( <D> )
##
## <#GAPDoc Label="IsSemigroup">
## <ManSection>
## <Filt Name="IsSemigroup" Arg='D' Type='Synonym'/>
##
## <Description>
## returns <K>true</K> if the object <A>D</A> is a semigroup.
## <Index>semigroup</Index>
## A <E>semigroup</E> is a magma (see <Ref Chap="Magmas"/>) with
## associative multiplication.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonymAttr( "IsSemigroup", IsMagma and IsAssociative );
##############################################################################
##
#O InversesOfSemigroupElement( <S>, <x> )
##
## <#GAPDoc Label="InversesOfSemigroupElement">
## <ManSection>
## <Oper Name="InversesOfSemigroupElement" Arg="S, x"/>
## <Returns>A list of the inverses of an element of a semigroup.</Returns>
## <Description>
## <C>InversesOfSemigroupElement</C> returns a list of the inverses of
## the element <A>x</A> in the semigroup <A>S</A>.<P/>
##
## An element <A>y</A> in <A>S</A> is an <E>inverse</E> of <A>x</A> if
## <C><A>x</A>*y*<A>x</A>=<A>x</A></C> and <C>y*<A>x</A>*y=y</C>.
## The element <A>x</A> has an inverse if and only if <A>x</A> is a
## regular element of <A>S</A>.
## <Example><![CDATA[
## gap> S := Semigroup([
## > Transformation([3, 1, 4, 2, 5, 2, 1, 6, 1]),
## > Transformation([5, 7, 8, 8, 7, 5, 9, 1, 9]),
## > Transformation([7, 6, 2, 8, 4, 7, 5, 8, 3])]);
## <transformation semigroup of degree 9 with 3 generators>
## gap> x := Transformation([3, 1, 4, 2, 5, 2, 1, 6, 1]);;
## gap> InversesOfSemigroupElement(S, x);
## [ ]
## gap> IsRegularSemigroupElement(S, x);
## false
## gap> x := Transformation([1, 9, 7, 5, 5, 1, 9, 5, 1]);;
## gap> Set(InversesOfSemigroupElement(S, x));
## [ Transformation( [ 1, 2, 3, 5, 5, 1, 3, 5, 2 ] ),
## Transformation( [ 1, 5, 1, 1, 5, 1, 3, 1, 2 ] ),
## Transformation( [ 1, 5, 1, 2, 5, 1, 3, 2, 2 ] ) ]
## gap> IsRegularSemigroupElement(S, x);
## true
## gap> S := ReesZeroMatrixSemigroup(Group((1,2,3)),
## > [[(), ()], [(), 0], [(), (1,2,3)]]);;
## gap> x := ReesZeroMatrixSemigroupElement(S, 2, (1,2,3), 3);;
## gap> InversesOfSemigroupElement(S, x);
## [ (1,(1,2,3),3), (1,(1,3,2),1), (2,(),3), (2,(1,2,3),1) ]]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation("InversesOfSemigroupElement",
[IsSemigroup, IsMultiplicativeElement]);
#############################################################################
##
#F Semigroup( <gen1>, <gen2> ... ) . . . . semigroup generated by collection
#F Semigroup( <gens> ) . . . . . . . . . . semigroup generated by collection
##
## <#GAPDoc Label="Semigroup">
## <ManSection>
## <Heading>Semigroup</Heading>
## <Func Name="Semigroup" Arg='gen1, gen2 ...'
## Label="for various generators"/>
## <Func Name="Semigroup" Arg='gens' Label="for a list"/>
##
## <Description>
## In the first form, <Ref Func="Semigroup" Label="for various generators"/>
## returns the semigroup generated by the arguments <A>gen1</A>,
## <A>gen2</A>, <M>\ldots</M>,
## that is, the closure of these elements under multiplication.
## In the second form, <Ref Func="Semigroup" Label="for a list"/> returns
## the semigroup generated by the elements in the homogeneous list
## <A>gens</A>;
## a square matrix as only argument is treated as one generator,
## not as a list of generators.
## <P/>
## It is <E>not</E> checked whether the underlying multiplication is
## associative, use <Ref Func="Magma"/> and <Ref Prop="IsAssociative"/>
## if you want to check whether a magma is in fact a semigroup.
## <P/>
## <Example><![CDATA[
## gap> a:= Transformation( [ 2, 3, 4, 1 ] );
## Transformation( [ 2, 3, 4, 1 ] )
## gap> b:= Transformation( [ 2, 2, 3, 4 ] );
## Transformation( [ 2, 2 ] )
## gap> s:= Semigroup(a, b);
## <transformation semigroup of degree 4 with 2 generators>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "Semigroup" );
#############################################################################
##
#F Subsemigroup( <S>, <gens> ) . . . subsemigroup of <S> generated by <gens>
#F SubsemigroupNC( <S>, <gens> ) . . subsemigroup of <S> generated by <gens>
##
## <#GAPDoc Label="Subsemigroup">
## <ManSection>
## <Func Name="Subsemigroup" Arg='S, gens'/>
## <Func Name="SubsemigroupNC" Arg='S, gens'/>
##
## <Description>
## are just synonyms of <Ref Func="Submagma"/> and <Ref Func="SubmagmaNC"/>,
## respectively.
## <P/>
## <Example><![CDATA[
## gap> a:=GeneratorsOfSemigroup(s)[1];
## Transformation( [ 2, 3, 4, 1 ] )
## gap> t:=Subsemigroup(s,[a]);
## <commutative transformation semigroup of degree 4 with 1 generator>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "Subsemigroup", Submagma );
DeclareSynonym( "SubsemigroupNC", SubmagmaNC );
#############################################################################
##
#O SemigroupByGenerators( <gens> ) . . . . . . semigroup generated by <gens>
##
## <#GAPDoc Label="SemigroupByGenerators">
## <ManSection>
## <Oper Name="SemigroupByGenerators" Arg='gens'/>
##
## <Description>
## is the underlying operation
## of <Ref Func="Semigroup" Label="for various generators"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "SemigroupByGenerators", [ IsCollection ] );
#############################################################################
##
#A AsSemigroup( <C> ) . . . . . . . . collection <C> regarded as semigroup
##
## <#GAPDoc Label="AsSemigroup">
## <ManSection>
## <Oper Name="AsSemigroup" Arg='C'/>
##
## <Description>
## If <A>C</A> is a collection whose elements form a semigroup
## under <Ref Oper="\*"/>
## (see <Ref Filt="IsSemigroup"/>) then <Ref Oper="AsSemigroup"/>
## returns this semigroup.
## Otherwise <K>fail</K> is returned.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "AsSemigroup", [IsCollection]);
#############################################################################
##
#O AsSubsemigroup( <D>, <C> )
##
## <#GAPDoc Label="AsSubsemigroup">
## <ManSection>
## <Oper Name="AsSubsemigroup" Arg='D, C'/>
##
## <Description>
## Let <A>D</A> be a domain and <A>C</A> a collection.
## If <A>C</A> is a subset of <A>D</A> that forms a semigroup then
## <Ref Oper="AsSubsemigroup"/>
## returns this semigroup, with parent <A>D</A>.
## Otherwise <K>fail</K> is returned.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "AsSubsemigroup", [ IsDomain, IsCollection ] );
#############################################################################
##
#A GeneratorsOfSemigroup( <S> ) . . . semigroup generators of semigroup <S>
##
## <#GAPDoc Label="GeneratorsOfSemigroup">
## <ManSection>
## <Attr Name="GeneratorsOfSemigroup" Arg='S'/>
##
## <Description>
## Semigroup generators of a semigroup <A>D</A> are the same as magma
## generators, see <Ref Attr="GeneratorsOfMagma"/>.
## <Example><![CDATA[
## gap> GeneratorsOfSemigroup(s);
## [ Transformation( [ 2, 3, 4, 1 ] ), Transformation( [ 2, 2 ] ) ]
## gap> GeneratorsOfSemigroup(t);
## [ Transformation( [ 2, 3, 4, 1 ] ) ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonymAttr( "GeneratorsOfSemigroup", GeneratorsOfMagma );
#############################################################################
##
#P IsGeneratorsOfSemigroup( <C> ) . . . list or collection of generators
##
## <#GAPDoc Label="IsGeneratorsOfSemigroup">
## <ManSection>
## <Prop Name="IsGeneratorsOfSemigroup" Arg='C'/>
##
## <Description>
## This property reflects whether the list or collection <A>C</A> generates
## a semigroup.
## <Ref Filt="IsAssociativeElementCollection"/> implies
## <Ref Prop="IsGeneratorsOfSemigroup"/>,
## but is not used directly in semigroup code, because of conflicts
## with matrices.
##
## <Example><![CDATA[
## gap> IsGeneratorsOfSemigroup([Transformation([2,3,1])]);
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty("IsGeneratorsOfSemigroup", IsListOrCollection);
InstallTrueMethod(IsGeneratorsOfSemigroup, IsSemigroup);
InstallTrueMethod(IsGeneratorsOfSemigroup, IsAssociativeElementCollection);
# the following covers the case of elements of a quotient semigroup
InstallTrueMethod(IsGeneratorsOfSemigroup, IsAssociativeElementCollColl);
#############################################################################
##
#A CayleyGraphSemigroup( <S> )
#A CayleyGraphDualSemigroup( <S> )
##
## <ManSection>
## <Attr Name="CayleyGraphSemigroup" Arg='S'/>
## <Attr Name="CayleyGraphDualSemigroup" Arg='S'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareAttribute("CayleyGraphSemigroup",IsSemigroup);
DeclareAttribute("CayleyGraphDualSemigroup",IsSemigroup);
#############################################################################
##
#F FreeSemigroup( [<wfilt>, ]<rank>[, <name>] )
#F FreeSemigroup( [<wfilt>, ]<name1>[, <name2>[, ...]] )
#F FreeSemigroup( [<wfilt>, ]<names> )
#F FreeSemigroup( [<wfilt>, ]infinity[, <name>][, <init>] )
##
## <#GAPDoc Label="FreeSemigroup">
## <ManSection>
## <Heading>FreeSemigroup</Heading>
## <Func Name="FreeSemigroup" Arg='[wfilt, ]rank[, name]'
## Label="for given rank"/>
## <Func Name="FreeSemigroup" Arg='[wfilt, ]name1[, name2[, ...]]'
## Label="for various names"/>
## <Func Name="FreeSemigroup" Arg='[wfilt, ]names'
## Label="for a list of names"/>
## <Func Name="FreeSemigroup" Arg='[wfilt, ]infinity[, name][, init]'
## Label="for infinitely many generators"/>
##
## <Description>
## <C>FreeSemigroup</C> returns a free semigroup. The number of
## generators, and the labels given to the generators, can be specified in
## several different ways.
## Warning: the labels of generators are only an aid for printing,
## and do not necessarily distinguish generators;
## see the examples at the end for more information.
## <List>
## <Mark>
## 1: For a given rank, and an optional generator name prefix
## </Mark>
## <Item>
## Called with a positive integer <A>rank</A>,
## <Ref Func="FreeSemigroup" Label="for given rank"/> returns
## a free semigroup on <A>rank</A> generators.
## The optional argument <A>name</A> must be a string;
## its default value is <C>"s"</C>. <P/>
##
## If <A>name</A> is not given but the <C>generatorNames</C> option is,
## then this option is respected as described in
## Section <Ref Sect="Generator Names"/>. <P/>
##
## Otherwise, the generators of the returned free semigroup are labelled
## <A>name</A><C>1</C>, ..., <A>name</A><C>k</C>,
## where <C>k</C> is the value of <A>rank</A>. <P/>
## </Item>
## <Mark>2: For given generator names</Mark>
## <Item>
## Called with various (at least one) nonempty strings,
## <Ref Func="FreeSemigroup" Label="for various names"/> returns
## a free semigroup on as many generators as arguments, which are labelled
## <A>name1</A>, <A>name2</A>, etc.
## </Item>
## <Mark>3: For a given list of generator names</Mark>
## <Item>
## Called with a nonempty finite list <A>names</A> of
## nonempty strings,
## <Ref Func="FreeSemigroup" Label="for a list of names"/> returns
## a free semigroup on <C>Length(<A>names</A>)</C> generators, whose
## <C>i</C>-th generator is labelled <A>names</A><C>[i]</C>.
## </Item>
## <Mark>
## 4: For the rank <K>infinity</K>,
## an optional default generator name prefix,
## and an optional finite list of generator names
## </Mark>
## <Item>
## Called in the fourth form,
## <Ref Func="FreeSemigroup" Label="for infinitely many generators"/>
## returns a free semigroup on infinitely many generators.
## The optional argument <A>name</A> must be a string; its default value is
## <C>"s"</C>,
## and the optional argument <A>init</A> must be a finite list of
## nonempty strings; its default value is an empty list.
## The generators are initially labelled according to the list <A>init</A>,
## followed by
## <A>name</A><C>i</C> for each <C>i</C> in the range from
## <C>Length(<A>init</A>)+1</C> to <K>infinity</K>; such a label is not
## allowed to appear in <A>init</A>.
## </Item>
## </List>
##
## If the optional first argument <A>wfilt</A> is given, then it must be either
## <C>IsSyllableWordsFamily</C>, <C>IsLetterWordsFamily</C>,
## <C>IsWLetterWordsFamily</C>, or <C>IsBLetterWordsFamily</C>.
## This filter specifies the representation used for the elements of
## the free semigroup
## (see <Ref Sect="Representations for Associative Words"/>).
## If no such filter is given, a letter representation is used.
## <P/>
##
## For more on associative words see
## Chapter <Ref Chap="Associative Words"/>. <P/>
##
## <Example><![CDATA[
## gap> f1 := FreeSemigroup( 3 );
## <free semigroup on the generators [ s1, s2, s3 ]>
## gap> f2 := FreeSemigroup( 3 , "generator" );
## <free semigroup on the generators
## [ generator1, generator2, generator3 ]>
## gap> f3 := FreeSemigroup( "gen1" , "gen2" );
## <free semigroup on the generators [ gen1, gen2 ]>
## gap> f4 := FreeSemigroup( ["gen1" , "gen2"] );
## <free semigroup on the generators [ gen1, gen2 ]>
## gap> FreeSemigroup( 3 : generatorNames := "boom" );
## <free semigroup on the generators [ boom1, boom2, boom3 ]>
## gap> FreeSemigroup( 2 : generatorNames := [ "u", "v", "w" ] );
## <free semigroup on the generators [ u, v ]>
## gap> FreeSemigroup( infinity ) ;
## <free semigroup on the generators [ s1, s2, ... ]>
## gap> F := FreeSemigroup( infinity, "g", [ "a", "b" ]);
## <free semigroup on the generators [ a, b, ... ]>
## gap> GeneratorsOfSemigroup( F ){[1..4]};
## [ a, b, g3, g4 ]
## gap> GeneratorsOfSemigroup( FreeSemigroup( infinity, "gen" ) ){[1..3]};
## [ gen1, gen2, gen3 ]
## gap> GeneratorsOfSemigroup( FreeSemigroup( infinity, [ "f" ] ) ){[1..3]};
## [ f, s2, s3 ]
## gap> FreeSemigroup(IsSyllableWordsFamily, 5);
## <free semigroup on the generators [ s1, s2, s3, s4, s5 ]>
## ]]></Example>
## <P/>
## Each free object defines a unique alphabet (and a unique family of words).
## Its generators are the letters of this alphabet,
## thus words of length one. <P/>
## <Example><![CDATA[
## gap> FreeSemigroup( 5 );
## <free semigroup on the generators [ s1, s2, s3, s4, s5 ]>
## gap> FreeMonoid( "a", "b" );
## <free monoid on the generators [ a, b ]>
## gap> FreeGroup( infinity );
## <free group with infinity generators>
## gap> FreeSemigroup( "x", "y" );
## <free semigroup on the generators [ x, y ]>
## gap> FreeMonoid( 7 );
## <free monoid on the generators [ m1, m2, m3, m4, m5, m6, m7 ]>
## ]]></Example>
## <P/>
## Remember that names are just a help for printing and do not necessarily
## distinguish letters.
## It is possible to create arbitrarily weird situations by choosing strange
## names for the letters.
## <P/>
## <Example><![CDATA[
## gap> f := FreeGroup( "x", "x" );
## <free group on the generators [ x, x ]>
## gap> gens := GeneratorsOfGroup( f );
## [ x, x ]
## gap> gens[1] = gens[2];
## false
## gap> f:= FreeGroup( "f1*f2", "f2^-1", "Group( [ f1, f2 ] )" );
## <free group on the generators [ f1*f2, f2^-1, Group( [ f1, f2 ] ) ]>
## gap> gens:= GeneratorsOfGroup( f );;
## gap> gens[1] * gens[2];
## f1*f2*f2^-1
## gap> gens[1] / gens[3];
## f1*f2*Group( [ f1, f2 ] )^-1
## gap> gens[3] / gens[1] / gens[2];
## Group( [ f1, f2 ] )*f1*f2^-1*f2^-1^-1
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "FreeSemigroup" );
#############################################################################
##
#P IsZeroGroup( <S> )
##
## <#GAPDoc Label="IsZeroGroup">
## <ManSection>
## <Prop Name="IsZeroGroup" Arg='S'/>
##
## <Description>
## is <K>true</K> if and only if the semigroup <A>S</A> is a group with zero
## adjoined.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsZeroGroup", IsSemigroup );
InstallTrueMethod( IsSemigroup, IsZeroGroup );
#############################################################################
##
#P IsSimpleSemigroup( <S> )
##
## <#GAPDoc Label="IsSimpleSemigroup">
## <ManSection>
## <Prop Name="IsSimpleSemigroup" Arg='S'/>
##
## <Description>
## is <K>true</K> if and only if the semigroup <A>S</A> has no proper
## ideals.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsSimpleSemigroup", IsSemigroup );
InstallTrueMethod( IsSemigroup, IsSimpleSemigroup );
#############################################################################
##
#P IsZeroSimpleSemigroup( <S> )
##
## <#GAPDoc Label="IsZeroSimpleSemigroup">
## <ManSection>
## <Prop Name="IsZeroSimpleSemigroup" Arg='S'/>
##
## <Description>
## is <K>true</K> if and only if the semigroup has no proper ideals except
## for 0, where <A>S</A> is a semigroup with zero.
## If the semigroup does not find its zero, then a break-loop is entered.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsZeroSimpleSemigroup", IsSemigroup );
InstallTrueMethod( IsSemigroup, IsZeroSimpleSemigroup );
############################################################################
##
#A ANonReesCongruenceOfSemigroup( <S> )
##
## <ManSection>
## <Attr Name="ANonReesCongruenceOfSemigroup" Arg='S'/>
##
## <Description>
## for a semigroup <A>S</A>, returns a non-Rees congruence if one exists
## or otherwise returns <K>fail</K>.
## </Description>
## </ManSection>
##
DeclareAttribute("ANonReesCongruenceOfSemigroup",IsSemigroup);
############################################################################
##
#P IsReesCongruenceSemigroup( <S> )
##
## <#GAPDoc Label="IsReesCongruenceSemigroup">
## <ManSection>
## <Prop Name="IsReesCongruenceSemigroup" Arg='S'/>
##
## <Description>
## returns <K>true</K> if <A>S</A> is a Rees Congruence semigroup, that is,
## if all congruences of <A>S</A> are Rees Congruences.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsReesCongruenceSemigroup", IsSemigroup );
InstallTrueMethod( IsSemigroup, IsReesCongruenceSemigroup );
#############################################################################
##
#O HomomorphismFactorSemigroup( <S>, <C> )
#O HomomorphismFactorSemigroupByClosure( <S>, <L> )
#O FactorSemigroup( <S>, <C> )
#O FactorSemigroupByClosure( <S>, <L> )
##
## <ManSection>
## <Oper Name="HomomorphismFactorSemigroup" Arg='S, C'/>
## <Oper Name="HomomorphismFactorSemigroupByClosure" Arg='S, L'/>
## <Oper Name="FactorSemigroup" Arg='S, C'/>
## <Oper Name="FactorSemigroupByClosure" Arg='S, L'/>
##
## <Description>
## each find the quotient of <A>S</A> by a congruence.
## <P/>
## In the first form <A>C</A> is a congruence and
## <Ref Func="HomomorphismFactorSemigroup"/>
## returns a homomorphism <M><A>S</A> \rightarrow <A>S</A>/<A>C</A></M>.
## <P/>
## In the second form, <A>L</A> is a list of pairs of elements of <A>S</A>.
## Returns a homomorphism <M><A>S</A> \rightarrow <A>S</A>/<A>C</A></M>,
## where <A>C</A> is the congruence generated by <A>L</A>.
## <P/>
## <C>FactorSemigroup(<A>S</A>, <A>C</A>)</C> returns
## <C>Range( HomomorphismFactorSemigroup(<A>S</A>, <A>C</A>) )</C>.
## <P/>
## <C>FactorSemigroupByClosure(<A>S</A>, <A>L</A>)</C> returns
## <C>Range( HomomorphismFactorSemigroupByClosure(<A>S</A>, <A>L</A>) )</C>.
## </Description>
## </ManSection>
##
DeclareOperation( "HomomorphismFactorSemigroup",
[ IsSemigroup, IsSemigroupCongruence ] );
DeclareOperation( "HomomorphismFactorSemigroupByClosure",
[ IsSemigroup, IsList ] );
DeclareOperation( "FactorSemigroup",
[ IsSemigroup, IsSemigroupCongruence ] );
DeclareOperation( "FactorSemigroupByClosure",
[ IsSemigroup, IsList ] );
#############################################################################
##
#O IsRegularSemigroupElement( <S>, <x> )
##
## <#GAPDoc Label="IsRegularSemigroupElement">
## <ManSection>
## <Oper Name="IsRegularSemigroupElement" Arg='S, x'/>
##
## <Description>
## returns <K>true</K> if <A>x</A> has a general inverse in <A>S</A>, i.e.,
## there is an element <M>y \in <A>S</A></M>
## such that <M><A>x</A> y <A>x</A> = <A>x</A></M> and
## <M>y <A>x</A> y = y</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation("IsRegularSemigroupElement", [IsSemigroup,
IsMultiplicativeElement]);
#############################################################################
##
#P IsRegularSemigroup( <S> )
##
## <#GAPDoc Label="IsRegularSemigroup">
## <ManSection>
## <Prop Name="IsRegularSemigroup" Arg='S'/>
##
## <Description>
## returns <K>true</K> if <A>S</A> is regular, i.e.,
## if every &D;-class of <A>S</A> is regular.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty("IsRegularSemigroup", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsRegularSemigroup);
#############################################################################
##
#P IsInverseSemigroup( <S> )
##
## <#GAPDoc Label="IsInverseSemigroup">
## <ManSection>
## <Prop Name="IsInverseSemigroup" Arg="S"/>
## <Filt Name="IsInverseMonoid" Arg="S" Type="Category"/>
## <Returns><K>true</K> or <K>false</K>.</Returns>
## <Description>
## A semigroup <A>S</A> is an <E>inverse semigroup</E> if every element
## <C>x</C> in <A>S</A> has a unique semigroup inverse, that is, a unique
## element <C>y</C> in <A>S</A> such that <C>x*y*x=x</C> and
## <C>y*x*y=y</C>.<P/>
##
## A monoid that happens to be an inverse semigroup is called an
## <E>inverse monoid</E>; see <Ref Filt="IsMonoid"/>.
## <Example>
## gap> S := Semigroup([
## > Transformation([1, 2, 4, 5, 6, 3, 7, 8]),
## > Transformation([3, 3, 4, 5, 6, 2, 7, 8]),
## > Transformation([1, 2, 5, 3, 6, 8, 4, 4])]);;
## gap> IsInverseSemigroup(S);
## true</Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty("IsInverseSemigroup", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsInverseSemigroup);
#############################################################################
##
#O DisplaySemigroup( <S> )
##
## <#GAPDoc Label="DisplaySemigroup">
## <ManSection>
## <Oper Name="DisplaySemigroup" Arg='S'/>
##
## <Description>
## Produces a convenient display of a transformation semigroup's D-Class
## structure. Let <A>S</A> be a transformation semigroup of degree
## <M>n</M>. Then for each <M>r\leq n</M>, we show all D-classes of
## rank <M>r</M>.
## <P/>
## A regular D-class with a single H-class of size 120 appears as
## <Log><![CDATA[
## *[H size = 120, 1 L-class, 1 R-class]
## ]]></Log>
## (the <C>*</C> denoting regularity).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation("DisplaySemigroup", [IsSemigroup]);
# Everything from here...
DeclareAttribute("NilpotencyDegree", IsSemigroup);
##############################################################################
##
#O IsSubsemigroup( <S>, <T> )
##
## <#GAPDoc Label="IsSubsemigroup">
## <ManSection>
## <Oper Name="IsSubsemigroup" Arg="S, T"/>
## <Returns><K>true</K> or <K>false</K>.</Returns>
## <Description>
##
## This operation returns <K>true</K> if the semigroup <A>T</A> is a
## subsemigroup of the semigroup <A>S</A> and <K>false</K> if it is not.
## <Example>
## gap> f := Transformation([5, 6, 7, 1, 4, 3, 2, 7]);
## Transformation( [ 5, 6, 7, 1, 4, 3, 2, 7 ] )
## gap> T := Semigroup(f);;
## gap> IsSubsemigroup(FullTransformationSemigroup(4), T);
## false
## gap> S := Semigroup(f);;
## gap> T := Semigroup(f ^ 2);;
## gap> IsSubsemigroup(S, T);
## true</Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation("IsSubsemigroup", [IsSemigroup, IsSemigroup]);
DeclareProperty("IsBand", IsSemigroup);
DeclareProperty("IsBrandtSemigroup", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsBrandtSemigroup);
DeclareProperty("IsCliffordSemigroup", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsCliffordSemigroup);
DeclareProperty("IsCommutativeSemigroup", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsCommutativeSemigroup);
DeclareProperty("IsCompletelyRegularSemigroup", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsCompletelyRegularSemigroup);
DeclareProperty("IsCompletelySimpleSemigroup", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsCompletelySimpleSemigroup);
DeclareProperty("IsGroupAsSemigroup", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsGroupAsSemigroup);
DeclareProperty("IsIdempotentGenerated", IsSemigroup);
DeclareProperty("IsLeftZeroSemigroup", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsLeftZeroSemigroup);
DeclareProperty("IsMonogenicSemigroup", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsMonogenicSemigroup);
DeclareProperty("IsMonoidAsSemigroup", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsMonoidAsSemigroup);
DeclareProperty("IsNilpotentSemigroup", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsNilpotentSemigroup);
DeclareProperty("IsOrthodoxSemigroup", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsOrthodoxSemigroup);
DeclareProperty("IsRectangularBand", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsRectangularBand);
DeclareProperty("IsRightZeroSemigroup", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsRightZeroSemigroup);
DeclareProperty("IsSemiband", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsSemiband);
DeclareProperty("IsSemilattice", IsSemigroup);
DeclareProperty("IsZeroSemigroup", IsSemigroup);
InstallTrueMethod(IsSemigroup, IsZeroSemigroup);
InstallTrueMethod(IsMonoidAsSemigroup, IsMagmaWithOne and IsSemigroup);
InstallTrueMethod(IsGroupAsSemigroup, IsMagmaWithInverses and IsSemigroup);
InstallTrueMethod(IsGroupAsSemigroup, IsInverseSemigroup and IsSimpleSemigroup and IsFinite);
InstallTrueMethod(IsGroupAsSemigroup, IsCommutative and IsSimpleSemigroup);
InstallTrueMethod(IsBand, IsSemilattice and IsSemigroup);
InstallTrueMethod(IsBrandtSemigroup, IsInverseSemigroup and IsZeroSimpleSemigroup);
InstallTrueMethod(IsCliffordSemigroup, IsSemilattice and IsSemigroup);
InstallTrueMethod(IsCompletelyRegularSemigroup, IsCliffordSemigroup);
InstallTrueMethod(IsCompletelyRegularSemigroup, IsSimpleSemigroup);
InstallTrueMethod(IsCompletelySimpleSemigroup, IsSimpleSemigroup and IsFinite);
InstallTrueMethod(IsIdempotentGenerated, IsBand and IsSemigroup);
InstallTrueMethod(IsInverseSemigroup, IsSemilattice and IsSemigroup);
InstallTrueMethod(IsInverseSemigroup, IsCliffordSemigroup);
InstallTrueMethod(IsInverseSemigroup, IsGroupAsSemigroup);
InstallTrueMethod(IsLeftZeroSemigroup, IsInverseSemigroup and IsTrivial);
InstallTrueMethod(IsRegularSemigroup, IsInverseSemigroup);
InstallTrueMethod(IsRegularSemigroup, IsSimpleSemigroup);
InstallTrueMethod(IsOrthodoxSemigroup, IsInverseSemigroup);
InstallTrueMethod(IsRightZeroSemigroup, IsInverseSemigroup and IsTrivial);
InstallTrueMethod(IsSemiband, IsIdempotentGenerated and IsSemigroup);
InstallTrueMethod(IsSemilattice, IsSemigroup and IsCommutative and IsBand);
InstallTrueMethod(IsSimpleSemigroup, IsGroupAsSemigroup);
InstallTrueMethod(IsZeroSemigroup, IsInverseSemigroup and IsTrivial);
# ...to here was added by JDM / WW.
# the following allow us to only use a single method for ViewString for
# semigroups of transformations and partial perms.
DeclareOperation("SemigroupViewStringPrefix", [IsSemigroup]);
DeclareOperation("SemigroupViewStringSuffix", [IsSemigroup]);
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