Quelle word.gd Sprache: unbekannt

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##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer, Frank Celler.
##
## 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 declares the categories and operations for words and
## nonassociative words.
##
## 1. Categories of Words and Nonassociative Words
## 2. Comparison of Words
## 3. Operations for Words
## 4. Free Magmas
## 5. External Representation for Nonassociative Words
##

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##
## <#GAPDoc Label="[1]{word}">
## This chapter describes categories of <E>words</E> and
## <E>nonassociative words</E>, and operations for them.
## For information about <E>associative words</E>,
## which occur for example as elements in free groups,
## see Chapter <Ref Chap="Associative Words"/>.
## <#/GAPDoc>
##

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##
## 1. Categories of Words and Nonassociative Words
##

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##
#C IsWord( <obj> )
#C IsWordWithOne( <obj> )
#C IsWordWithInverse( <obj> )
##
## <#GAPDoc Label="IsWord">
## <ManSection>
## <Filt Name="IsWord" Arg='obj' Type='Category'/>
## <Filt Name="IsWordWithOne" Arg='obj' Type='Category'/>
## <Filt Name="IsWordWithInverse" Arg='obj' Type='Category'/>
##
## <Description>
## <Index>abstract word</Index>
## Given a free multiplicative structure <M>M</M> that is freely generated
## by a subset <M>X</M>,
## any expression of an element in <M>M</M> as an iterated product of
## elements in <M>X</M> is called a <E>word</E> over <M>X</M>.
## 
## Interesting cases of free multiplicative structures are those of
## free semigroups, free monoids, and free groups,
## where the multiplication is associative
## (see <Ref Prop="IsAssociative"/>),
## which are described in Chapter <Ref Chap="Associative Words"/>,
## and also the case of free magmas,
## where the multiplication is nonassociative
## (see <Ref Filt="IsNonassocWord"/>).
## 
## Elements in free magmas
## (see <Ref Func="FreeMagma" Label="for given rank"/>)
## lie in the category <Ref Filt="IsWord"/>;
## similarly, elements in free magmas-with-one
## (see <Ref Func="FreeMagmaWithOne" Label="for given rank"/>)
## lie in the category <Ref Filt="IsWordWithOne"/>, and so on.
## 
## <Ref Filt="IsWord"/> is mainly a <Q>common roof</Q> for the two
## <E>disjoint</E> categories
## <Ref Filt="IsAssocWord"/> and <Ref Filt="IsNonassocWord"/>
## of associative and nonassociative words.
## This means that associative words are <E>not</E> regarded as special
## cases of nonassociative words.
## The main reason for this setup is that we are interested in different
## external representations for associative and nonassociative words
## (see <Ref Sect="External Representation for Nonassociative Words"/>
## and <Ref Sect="The External Representation for Associative Words"/>).
## 
## Note that elements in finitely presented groups and also elements in
## polycyclic groups in &GAP; are <E>not</E> in <Ref Filt="IsWord"/>
## although they are usually called words,
## see Chapters <Ref Chap="Finitely Presented Groups"/>
## and <Ref Chap="Pc Groups"/>.
## 
## Words are <E>constants</E>
## (see <Ref Sect="Mutability and Copyability"/>),
## that is, they are not copyable and not mutable.
## 
## The usual way to create words is to form them as products of known words,
## starting from <E>generators</E> of a free structure such as a free magma
## or a free group (see <Ref Func="FreeMagma" Label="for given rank"/>,
## <Ref Func="FreeGroup" Label="for given rank"/>).
## 
## Words are also used to implement free algebras,
## in the same way as group elements are used to implement group algebras
## (see <Ref Sect="Constructing Algebras as Free Algebras"/>
## and Chapter <Ref Chap="Magma Rings"/>).
## 
## <Example><![CDATA[
## gap> m:= FreeMagmaWithOne( 2 );; gens:= GeneratorsOfMagmaWithOne( m );
## [ x1, x2 ]
## gap> w1:= gens[1] * gens[2] * gens[1];
## ((x1*x2)*x1)
## gap> w2:= gens[1] * ( gens[2] * gens[1] );
## (x1*(x2*x1))
## gap> w1 = w2; IsAssociative( m );
## false
## false
## gap> IsWord( w1 ); IsAssocWord( w1 ); IsNonassocWord( w1 );
## true
## false
## true
## gap> s:= FreeMonoid( 2 );; gens:= GeneratorsOfMagmaWithOne( s );
## [ m1, m2 ]
## gap> u1:= ( gens[1] * gens[2] ) * gens[1];
## m1*m2*m1
## gap> u2:= gens[1] * ( gens[2] * gens[1] );
## m1*m2*m1
## gap> u1 = u2; IsAssociative( s );
## true
## true
## gap> IsWord( u1 ); IsAssocWord( u1 ); IsNonassocWord( u1 );
## true
## true
## false
## gap> a:= (1,2,3);; b:= (1,2);;
## gap> w:= a*b*a;; IsWord( w );
## false
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsWord", IsMultiplicativeElement );
DeclareSynonym( "IsWordWithOne", IsWord and IsMultiplicativeElementWithOne );
DeclareSynonym( "IsWordWithInverse",
 IsWord and IsMultiplicativeElementWithInverse );

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##
#C IsWordCollection( <obj> )
##
## <#GAPDoc Label="IsWordCollection">
## <ManSection>
## <Filt Name="IsWordCollection" Arg='obj' Type='Category'/>
##
## <Description>
## <Ref Filt="IsWordCollection"/> is the collections category
## (see <Ref Func="CategoryCollections"/>) of <Ref Filt="IsWord"/>.
## <Example><![CDATA[
## gap> IsWordCollection( m ); IsWordCollection( s );
## true
## true
## gap> IsWordCollection( [ "a", "b" ] );
## false
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategoryCollections( "IsWord" );

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##
#C IsNonassocWord( <obj> )
#C IsNonassocWordWithOne( <obj> )
##
## <#GAPDoc Label="IsNonassocWord">
## <ManSection>
## <Filt Name="IsNonassocWord" Arg='obj' Type='Category'/>
## <Filt Name="IsNonassocWordWithOne" Arg='obj' Type='Category'/>
##
## <Description>
## A <E>nonassociative word</E> in &GAP; is an element in a free magma or
## a free magma-with-one (see <Ref Sect="Free Magmas"/>).
## 
## The default methods for <Ref Oper="ViewObj"/> and <Ref Oper="PrintObj"/>
## show nonassociative words as products of letters,
## where the succession of multiplications is determined by round brackets.
## 
## In this sense each nonassociative word describes a <Q>program</Q> to
## form a product of generators.
## (Also associative words can be interpreted as such programs,
## except that the exact succession of multiplications is not prescribed
## due to the associativity.)
## The function <Ref Oper="MappedWord"/> implements a way to
## apply such a program.
## A more general way is provided by straight line programs
## (see <Ref Sect="Straight Line Programs"/>).
## 
## Note that associative words
## (see Chapter <Ref Chap="Associative Words"/>)
## are <E>not</E> regarded as special cases of nonassociative words
## (see <Ref Filt="IsWord"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsNonassocWord", IsWord );
DeclareSynonym( "IsNonassocWordWithOne", IsNonassocWord and IsWordWithInverse );

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##
#C IsNonassocWordCollection( <obj> )
#C IsNonassocWordWithOneCollection( <obj> )
##
## <#GAPDoc Label="IsNonassocWordCollection">
## <ManSection>
## <Filt Name="IsNonassocWordCollection" Arg='obj' Type='Category'/>
## <Filt Name="IsNonassocWordWithOneCollection" Arg='obj' Type='Category'/>
##
## <Description>
## <Ref Filt="IsNonassocWordCollection"/> is the collections category
## (see <Ref Func="CategoryCollections"/>) of
## <Ref Filt="IsNonassocWord"/>,
## and <Ref Filt="IsNonassocWordWithOneCollection"/> is the collections
## category of <Ref Filt="IsNonassocWordWithOne"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategoryCollections( "IsNonassocWord" );
DeclareCategoryCollections( "IsNonassocWordWithOne" );

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##
#C IsNonassocWordFamily( <obj> )
#C IsNonassocWordWithOneFamily( <obj> )
##
## <ManSection>
## <Filt Name="IsNonassocWordFamily" Arg='obj' Type='Category'/>
## <Filt Name="IsNonassocWordWithOneFamily" Arg='obj' Type='Category'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareCategoryFamily( "IsNonassocWord" );
DeclareCategoryFamily( "IsNonassocWordWithOne" );

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##
## 2. Comparison of Words
##
## <#GAPDoc Label="[2]{word}">
## <ManSection>
## <Oper Name="\=" Label="for nonassociative words" Arg='w1, w2'/>
##
## <Description>
## <Index Subkey="nonassociative words">equality</Index>
## 
## Two words are equal if and only if they are words over the same alphabet
## and with equal external representations
## (see <Ref Sect="External Representation for Nonassociative Words"/>
## and <Ref Sect="The External Representation for Associative Words"/>).
## For nonassociative words, the latter means that the words arise from the
## letters of the alphabet by the same sequence of multiplications.
## </Description>
## </ManSection>
##
## <ManSection>
## <Oper Name="\<" Label="for nonassociative words" Arg='w1, w2'/>
##
## <Description>
## <Index Subkey="nonassociative words">smaller</Index>
## Words are ordered according to their external representation.
## More precisely, two words can be compared if they are words over the same
## alphabet, and the word with smaller external representation is smaller.
## For nonassociative words, the ordering is defined
## in <Ref Sect="External Representation for Nonassociative Words"/>;
## associative words are ordered by the shortlex ordering via <C><</C>
## (see <Ref Sect="The External Representation for Associative Words"/>).
## 
## Note that the alphabet of a word is determined by its family
## (see <Ref Sect="Families"/>),
## and that the result of each call to
## <Ref Func="FreeMagma" Label="for given rank"/>,
## <Ref Func="FreeGroup" Label="for given rank"/> etc. consists of words
## over a new alphabet.
## In particular, there is no <Q>universal</Q> empty word,
## every families of words in <Ref Filt="IsWordWithOne"/> has its own
## empty word.
## 
## <Example><![CDATA[
## gap> m:= FreeMagma( "a", "b" );;
## gap> x:= FreeMagma( "a", "b" );;
## gap> mgens:= GeneratorsOfMagma( m );
## [ a, b ]
## gap> xgens:= GeneratorsOfMagma( x );
## [ a, b ]
## gap> a:= mgens[1];; b:= mgens[2];;
## gap> a = xgens[1];
## false
## gap> a*(a*a) = (a*a)*a; a*b = b*a; a*a = a*a;
## false
## false
## true
## gap> a < b; b < a; a < a*b;
## true
## false
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##

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##
## 3. Operations for Words
## <#GAPDoc Label="[3]{word}">
## Two words can be multiplied via <C>*</C> only if they are words over the
## same alphabet (see <Ref Sect="Comparison of Words"/>).
## <#/GAPDoc>
##

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##
#O MappedWord( <w>, <gens>, <imgs> )
##
## <#GAPDoc Label="MappedWord">
## <ManSection>
## <Oper Name="MappedWord" Arg='w, gens, imgs'/>
##
## <Description>
## <Ref Oper="MappedWord"/> returns the object that is obtained by replacing
## each occurrence in the word <A>w</A> of a generator in the list
## <A>gens</A> by the corresponding object in the list <A>imgs</A>.
## The lists <A>gens</A> and <A>imgs</A> must of course have the same length.
## 
## <Ref Oper="MappedWord"/> needs to do some preprocessing to get internal
## generator numbers etc. When mapping many (several thousand) words, a
## dedicated loop might be faster.
## 
## For example, if the elements in <A>imgs</A> are all
## <E>associative words</E>
## (see Chapter <Ref Chap="Associative Words"/>)
## in the same family as the elements in <A>gens</A>,
## and some of them are equal to the corresponding generators in <A>gens</A>,
## then those may be omitted from <A>gens</A> and <A>imgs</A>.
## In this situation, the special case that the lists <A>gens</A>
## and <A>imgs</A> have only length <M>1</M> is handled more efficiently by
## <Ref Oper="EliminatedWord"/>.
## 
## If the word is from a free group, it is permitted to give inverses of
## (some) of the generators as extra generators. This can speed up the
## execution by removing the need to calculate inverses anew.
## <Example><![CDATA[
## gap> m:= FreeMagma( "a", "b" );; gens:= GeneratorsOfMagma( m );;
## gap> a:= gens[1]; b:= gens[2];
## a
## b
## gap> w:= (a*b)*((b*a)*a)*b;
## (((a*b)*((b*a)*a))*b)
## gap> MappedWord( w, gens, [ (1,2), (1,2,3,4) ] );
## (2,4,3)
## gap> a:= (1,2);; b:= (1,2,3,4);; (a*b)*((b*a)*a)*b;
## (2,4,3)
## gap> f:= FreeGroup( "a", "b" );;
## gap> a:= GeneratorsOfGroup(f)[1];; b:= GeneratorsOfGroup(f)[2];;
## gap> w:= a^5*b*a^2/b^4*a;
## a^5*b*a^2*b^-4*a
## gap> MappedWord( w, [ a, b ], [ (1,2), (1,2,3,4) ] );
## (1,3,4,2)
## gap> MappedWord( w, [ a, b, b^-1 ], [ (1,2), (1,2,3,4), (1,4,3,2) ] );
## (1,3,4,2)
## gap> (1,2)^5*(1,2,3,4)*(1,2)^2/(1,2,3,4)^4*(1,2);
## (1,3,4,2)
## gap> MappedWord( w, [ a ], [ a^2 ] );
## a^10*b*a^4*b^-4*a^2
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "MappedWord", [ IsWord, IsWordCollection, IsList ] );

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##
## 4. Free Magmas
## <#GAPDoc Label="[4]{word}">
## The easiest way to create a family of words is to construct the free
## object generated by these words.
## Each such free object defines a unique alphabet,
## and its generators are simply the words of length one over this alphabet;
## These generators can be accessed via <Ref Attr="GeneratorsOfMagma"/> in
## the case of a free magma,
## and via <Ref Attr="GeneratorsOfMagmaWithOne"/> in the case of a free
## magma-with-one.
## <#/GAPDoc>
##

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##
#C IsFreeMagma( <obj> )
##
## <ManSection>
## <Filt Name="IsFreeMagma" Arg='obj' Type='Category'/>
##
## <Description>
## <Ref Func="IsFreeMagma"/> is just a synonym for
## <C>IsNonassocWordCollection and IsMagma</C>,
## that is, any magma (see <Ref Func="IsMagma"/>) consisting of
## nonassociative words (see <Ref Func="IsNonassocWord"/>) is in this
## category.
## </Description>
## </ManSection>
##
DeclareSynonym( "IsFreeMagma", IsNonassocWordCollection and IsMagma );

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##
## 5. External Representation for Nonassociative Words
## <#GAPDoc Label="[5]{word}">
## The external representation of nonassociative words is defined
## as follows.
## The <M>i</M>-th generator of the family of elements in question has
## external representation <M>i</M>,
## the identity (if exists) has external representation <M>0</M>,
## the inverse of the <M>i</M>-th generator (if exists) has external
## representation <M>-i</M>.
## If <M>v</M> and <M>w</M> are nonassociative words with external
## representations <M>e_v</M> and <M>e_w</M>,
## respectively then the product <M>v * w</M> has external
## representation <M>[ e_v, e_w ]</M>.
## So the external representation of any nonassociative word is either an
## integer or a nested list of integers and lists, where each list has
## length two.
## 
## One can create a nonassociative word from a family of words and the
## external representation of a nonassociative word using
## <Ref Oper="ObjByExtRep"/>.
## 
## <Example><![CDATA[
## gap> m:= FreeMagma( 2 );; gens:= GeneratorsOfMagma( m );
## [ x1, x2 ]
## gap> w:= ( gens[1] * gens[2] ) * gens[1];
## ((x1*x2)*x1)
## gap> ExtRepOfObj( w ); ExtRepOfObj( gens[1] );
## [ [ 1, 2 ], 1 ]
## 1
## gap> ExtRepOfObj( w*w );
## [ [ [ 1, 2 ], 1 ], [ [ 1, 2 ], 1 ] ]
## gap> ObjByExtRep( FamilyObj( w ), 2 );
## x2
## gap> ObjByExtRep( FamilyObj( w ), [ 1, [ 2, 1 ] ] );
## (x1*(x2*x1))
## ]]></Example>
## <#/GAPDoc>
##

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##
#O NonassocWord( <Fam>, <extrep> ) . . construct word from external repr.
##
## <ManSection>
## <Oper Name="NonassocWord" Arg='Fam, extrep'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareSynonym( "NonassocWord", ObjByExtRep );

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