############################################################################# ## ## 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>
## <#/GAPDoc> ##
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 2
############################################################################## ## #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> ##
## <## <#/GAPDoc>##
IsSemigroupIsMultiplicativeElement;
############################################################################# ## #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> ##
## <Oper Name="AsSemigroup" Arg='C'/>
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
############################################################################# ## #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(
############################################################################# ## #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(## gap> IsGeneratorsOfSemigroup([Transformation([2,3,1])]);
############################################################################# ## #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 ] );
############################################################################# ## #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## several different ways.
InstallTrueMethod## and do not necessarily distinguish generators;
InstallTrueMethod(IsGeneratorsOfSemigroup## <List>
# the following covers the case of elements of a quotient semigroup
InstallTrueMethod## <Item>
############################################################################# ## #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 ## <free semigroup on the generators [ s1, s2, s3, s4, s5 ]> ## <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## If the semigroup does not find its zero, then a break-loop is entered.
## <Description> ## is <K>true</K> if and only if the semigroup <A>S</A> is a group with zero ## adjoined. ## </Description> ## </ManSection> ## <#/GAPDoc> ##
DeclareProperty( ##
## <Description>
############################################################################# ## #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( "java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 0
InstallTrueMethod## <Prop Name="IsReesCongruenceSemigroup" Arg='S'/>
############################################################################# ## #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## <ManSection>
InstallTrueMethod( ## <Oper Name="HomomorphismFactorSemigroupByClosure" Arg='java.lang.StringIndexOutOfBoundsException: Range [0, 60) out of bounds for length 45
############################################################################ ## #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",## <C>Range( HomomorphismFactorSemigroup(<A>S</A>, <A>C</A>) )</C>.
############################################################################ ## #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( "java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 0
InstallTrueMethod( IsSemigroup, ;
############################################################################# ## #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( "# <ManSection>
DeclareOperation( "HomomorphismFactorSemigroupByClosure",
[ java.lang.StringIndexOutOfBoundsException: Range [0, 17) out of bounds for length 2
DeclareOperation## A semigroup <A>S</A> is an <E>inverse semigroup</E> if every element
[ IsSemigroup, IsSemigroupCongruence ] );
############################################################################# ## #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("java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 2
## <Oper Name="DisplaySemigroup" Arg='S##
############################################################################# ## #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> ## ";
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 ## <Description> ## 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## <#/GAPDoc>
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("DeclareProperty("IsCommutativeSemigroup", 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> ##
java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 63
DeclareProperty("IsBand", IsSemigroup);
java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0
InstallTrueMethod(IsSemigroup,(IsSemigroup java.lang.StringIndexOutOfBoundsException: Range [52, 50) out of bounds for length 52
up)
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
DeclarePropertyjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
InstallTrueMethod(IsSemigroup,IsCompletelyRegularSemigroupjava.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
DeclareProperty("IsCompletelySimpleSemigroup", IsSemigroup);
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
DeclareProperty(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
( java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 51
",IsSemigroup)java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
InstallTrueMethod(sBrandtSemigroup and);
InstallTrueMethod,IsSemilatticeand IsSemigroup)java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
DeclareProperty("IsNilpotentSemigroup", IsSemigroup);
java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 79
java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 52
java.lang.StringIndexOutOfBoundsException: Range [52, 50) out of bounds for length 52
DeclareProperty("IsZeroSemigroup", InstallTrueMethod(IsSemiband, IsIdempotentGenerated and 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 andInstallTrueMethod(,IsSemigroupand );
InstallTrueMethod( ;
InstallTrueMethod( and)
InstallTrueMethod(IsCompletelyRegularSemigroup# ...to here was added by JDM / WW.
InstallTrueMethod# the following allow us to only use a # semigroups of transformations and partial perms.
InstallTrueMethodDeclareOperation(",[];
InstallTrueMethod(IsIdempotentGenerated, IsBand and IsSemigroup);
InstallTrueMethod(",I])
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.
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