Spracherkennung für: .gd vermutete Sprache: Unknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen]
#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Andrew Solomon and Isabel Araújo.
##
## 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 declarations for finitely
## presented semigroups.
#############################################################################
##
#C IsElementOfFpSemigroup(<elm>)
##
## <#GAPDoc Label="IsElementOfFpSemigroup">
## <ManSection>
## <Filt Name="IsElementOfFpSemigroup" Arg='elm' Type='Category'/>
## <Filt Name="IsElementOfFpMonoid" Arg='elm' Type='Category'/>
##
## <Description>
## returns true if <A>elm</A> is an element of a finitely presented
## semigroup or monoid.
## <P/>
## <Example><![CDATA[
## gap> f := FreeSemigroup( "a", "b" );;
## gap> IsFpSemigroup( f );
## false
## gap> s := f / [ [ f.1^2, f.2^2 ] ];;
## gap> IsFpSemigroup( s );
## true
## gap> t := Semigroup( [ s.1^2 ] );
## <commutative semigroup with 1 generator>
## gap> IsSubsemigroupFpSemigroup( t );
## true
## gap> IsSubsemigroupFpSemigroup( s );
## true
## gap> IsSubsemigroupFpSemigroup( f );
## false
## gap> IsElementOfFpSemigroup( t.1^3 );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsElementOfFpSemigroup",
IsMultiplicativeElement and IsAssociativeElement );
#############################################################################
##
#O FpSemigroupOfElementOfFpSemigroup( <elm> )
##
## <ManSection>
## <Oper Name="FpSemigroupOfElementOfFpSemigroup" Arg='elm'/>
##
## <Description>
## returns the finitely presented semigroup to which <A>elm</A> belongs to
## </Description>
## </ManSection>
##
DeclareOperation( "FpSemigroupOfElementOfFpSemigroup",
[IsElementOfFpSemigroup]);
#############################################################################
##
#C IsElementOfFpSemigroupCollection(<e>)
##
## <ManSection>
## <Filt Name="IsElementOfFpSemigroupCollection" Arg='e' Type='Category'/>
##
## <Description>
## Created now so that lists of things in the category IsElementOfFpSemigroup
## are given the category CategoryCollections(IsElementOfFpSemigroup)
## Otherwise these lists (and other collections) won't create the
## collections category. See CollectionsCategory in the manual.
## </Description>
## </ManSection>
##
DeclareCategoryCollections("IsElementOfFpSemigroup");
#############################################################################
##
#A IsSubsemigroupFpSemigroup( <t> )
##
## <#GAPDoc Label="IsSubsemigroupFpSemigroup">
## <ManSection>
## <Filt Name="IsSubsemigroupFpSemigroup" Arg='t'/>
## <Filt Name="IsSubmonoidFpMonoid" Arg='t'/>
##
## <Description>
## The first function returns true if <A>t</A> is a finitely presented
## semigroup or a subsemigroup of a finitely presented semigroup.
## The second function does the equivalent thing for monoids.
## (Generally speaking, such a subsemigroup or monoid can be constructed
## with <C>Semigroup(<A>gens</A>)</C> or <C>Monoid(<A>gens</A>)</C>,
## where <A>gens</A> is a list of elements
## of a finitely presented semigroup or monoid.)
## <P/>
## A submonoid of a monoid has the same identity as the monoid.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonymAttr( "IsSubsemigroupFpSemigroup",
IsSemigroup and IsElementOfFpSemigroupCollection );
#############################################################################
##
#C IsElementOfFpSemigroupFamily
##
## <ManSection>
## <Filt Name="IsElementOfFpSemigroupFamily" Arg='obj' Type='Category'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareCategoryFamily( "IsElementOfFpSemigroup" );
#############################################################################
##
#F FactorFreeSemigroupByRelations( <f>, <rels> )
#F FactorFreeMonoidByRelations( <f>, <rels> )
##
## <#GAPDoc Label="FactorFreeSemigroupByRelations">
## <ManSection>
## <Func Name="FactorFreeSemigroupByRelations" Arg='f, rels'/>
## <Func Name="FactorFreeMonoidByRelations" Arg='f, rels'/>
##
## <Description>
## for a free semigroup or free monoid <A>f</A>
## and a list <A>rels</A> of pairs of elements of <A>f</A>.
## Returns the finitely presented semigroup or monoid
## which is the quotient of <A>f</A> by the least congruence on <A>f</A>
## generated by the pairs in <A>rels</A>.
## <P/>
## Users should be aware that much of the code described in this chapter
## is in need of substantial revision.
## In particular, the two functions described here are <E>not</E>
## called by the operation <C>\/</C> of the previous subsection,
## and so are liable to be removed in due course.
## <P/>
## <Example><![CDATA[
## gap> fm := FreeMonoid( 3 );;
## gap> y := GeneratorsOfMonoid( fm );;
## gap> m := FactorFreeMonoidByRelations( fm,
## > [ [ y[1] * y[2] * y[1], y[1] ],[ y[2]^4, y[1] ] ] );
## <fp monoid on the generators [ m1, m2, m3 ]>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("FactorFreeSemigroupByRelations");
#############################################################################
##
#O ElementOfFpSemigroup( <fam>, <word> )
#O ElementOfFpMonoid( <fam>, <word> )
##
## <#GAPDoc Label="ElementOfFpSemigroup">
## <ManSection>
## <Oper Name="ElementOfFpSemigroup" Arg='fam, word'/>
## <Oper Name="ElementOfFpMonoid" Arg='fam, word'/>
##
## <Description>
## for a family <A>fam</A> of elements of a finitely presented semigroup
## or monoid and a word <A>word</A> in the free generators underlying this
## finitely presented semigroup or monoid.
## Returns the element of the finitely presented semigroup or monoid
## with the representative <A>word</A> in the free semigroup or free monoid.
## These operations are inverse to <C>UnderlyingElement</C>.
## <P/>
## <Example><![CDATA[
## gap> fam := FamilyObj( genm[1] );;
## gap> w := y[1]^3 * y[2]^4 * y[3]^5;
## m1^3*m2^4*m3^5
## gap> ew := ElementOfFpMonoid( fam, w );
## m1^3*m2^4*m3^5
## gap> ew in fm;
## false
## gap> ew in m;
## true
## gap> w = UnderlyingElement( ew );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "ElementOfFpSemigroup",
[ IsElementOfFpSemigroupFamily, IsAssocWord ] );
#############################################################################
##
#P IsFpSemigroup(<s>)
##
## <#GAPDoc Label="IsFpSemigroup">
## <ManSection>
## <Filt Name="IsFpSemigroup" Arg='s'/>
## <Filt Name="IsFpMonoid" Arg='m'/>
##
## <Description>
## The first function is a synonym for
## <C>IsSubsemigroupFpSemigroup(<A>s</A>)</C> and
## <C>IsWholeFamily(<A>s</A>)</C> (this is because a subsemigroup
## of a finitely presented semigroup is not necessarily finitely presented).
## <P/>
## Similarly, the second function is a synonym for
## <C>IsSubmonoidFpMonoid(<A>m</A>)</C> and <C>IsWholeFamily(<A>m</A>)</C>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsFpSemigroup",IsSubsemigroupFpSemigroup and IsWholeFamily);
#############################################################################
##
#A FreeGeneratorsOfFpSemigroup( <s> )
#A FreeGeneratorsOfFpMonoid( <m> )
##
## <#GAPDoc Label="FreeGeneratorsOfFpSemigroup">
## <ManSection>
## <Attr Name="FreeGeneratorsOfFpSemigroup" Arg='s'/>
## <Attr Name="FreeGeneratorsOfFpMonoid" Arg='m'/>
##
## <Description>
## returns the underlying free generators corresponding to the generators of
## the finitely presented semigroup <A>s</A> or monoid <A>m</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("FreeGeneratorsOfFpSemigroup", IsFpSemigroup );
#############################################################################
##
#A FreeSemigroupOfFpSemigroup( <s> )
#A FreeMonoidOfFpMonoid( <m> )
##
## <#GAPDoc Label="FreeSemigroupOfFpSemigroup">
## <ManSection>
## <Attr Name="FreeSemigroupOfFpSemigroup" Arg='s'/>
## <Attr Name="FreeMonoidOfFpMonoid" Arg='m'/>
##
## <Description>
## returns the underlying free semigroup or free monoid
## for the finitely presented semigroup <A>s</A> or monoid <A>m</A>,
## i.e. the free semigroup or free monoid over which <A>s</A> or <A>m</A>
## is defined as a quotient.
## (This is the free semigroup or free monoid generated by the free generators
## provided by <C>FreeGeneratorsOfFpSemigroup(<A>s</A>)</C>
## or <C>FreeGeneratorsOfFpMonoid(<A>m</A>)</C>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("FreeSemigroupOfFpSemigroup", IsFpSemigroup);
############################################################################
##
#A RelationsOfFpSemigroup(<s>)
#A RelationsOfFpMonoid(<m>)
##
## <#GAPDoc Label="RelationsOfFpSemigroup">
## <ManSection>
## <Attr Name="RelationsOfFpSemigroup" Arg='s'/>
## <Attr Name="RelationsOfFpMonoid" Arg='m'/>
##
## <Description>
## returns the relations of the finitely presented semigroup <A>s</A>
## or monoid <A>m</A> as pairs of words in the free generators provided by
## <C>FreeGeneratorsOfFpSemigroup(<A>s</A>)</C> or
## <C>FreeGeneratorsOfFpMonoid(<A>m</A>)</C>.
## <P/>
## <Example><![CDATA[
## gap> fs = FreeSemigroupOfFpSemigroup( s );
## true
## gap> FreeGeneratorsOfFpMonoid( m );
## [ m1, m2, m3 ]
## gap> RelationsOfFpSemigroup( s );
## [ [ s1*s2*s1, s1 ], [ s2^4, s1 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("RelationsOfFpSemigroup",IsFpSemigroup);
############################################################################
##
#A IsomorphismFpSemigroup( <m> )
#A IsomorphismFpMonoid( <g> )
##
## <#GAPDoc Label="IsomorphismFpSemigroup">
## <ManSection>
## <Attr Name="IsomorphismFpSemigroup" Arg='m'/>
## <Attr Name="IsomorphismFpMonoid" Arg='g'/>
##
## <Description>
## for a finitely presented monoid <A>m</A>
## or a finitely presented group <A>g</A>.
## Returns an isomorphism from <A>m</A> or <A>g</A>
## to a finitely presented semigroup or monoid.
## <P/>
## <Example><![CDATA[
## gap> phis := IsomorphismFpSemigroup( m );
## MappingByFunction( <fp monoid on the generators
## [ m1, m2, m3 ]>, <fp semigroup on the generators [ <identity ...>, m1, m2, m3
## ]>, function( x ) ... end, function( x ) ... end )
## gap> fg := FreeGroup( 2 );;
## gap> g := fg / [ fg.1^4, fg.2^5 ];
## <fp group on the generators [ f1, f2 ]>
## gap> phim := IsomorphismFpMonoid( g );
## MappingByFunction( <fp group on the generators
## [ f1, f2 ]>, <fp monoid on the generators [ f1, f1^-1, f2, f2^-1
## ]>, function( x ) ... end, function( x ) ... end )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("IsomorphismFpSemigroup",IsSemigroup);
############################################################################
##
#O FpGrpMonSmgOfFpGrpMonSmgElement( <elm> )
##
## <#GAPDoc Label="FpGrpMonSmgOfFpGrpMonSmgElement">
## <ManSection>
## <Oper Name="FpGrpMonSmgOfFpGrpMonSmgElement" Arg='elm'/>
##
## <Description>
## returns the finitely presented group, monoid or semigroup to which
## <A>elm</A> belongs.
## <P/>
## <Example><![CDATA[
## gap> s = FpGrpMonSmgOfFpGrpMonSmgElement( s.1 );
## true
## gap> s = FpGrpMonSmgOfFpGrpMonSmgElement( t.1 );
## true
## gap> f := FreeMonoid( 2 );;
## gap> m := f / [ [ f.1^2, f.2^2 ] ];
## <fp monoid on the generators [ m1, m2 ]>
## gap> m = FpGrpMonSmgOfFpGrpMonSmgElement( m.1 * m.2 );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation("FpGrpMonSmgOfFpGrpMonSmgElement",[IsMultiplicativeElement]);
