Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 Bib Ind

This chapter describes categories of *words* and *nonassociative words*, and operations for them. For information about *associative words*, which occur for example as elements in free groups, see Chapter 37.

`‣ IsWord` ( obj ) | ( category ) |

`‣ IsWordWithOne` ( obj ) | ( category ) |

`‣ IsWordWithInverse` ( obj ) | ( category ) |

Given a free multiplicative structure \(M\) that is freely generated by a subset \(X\), any expression of an element in \(M\) as an iterated product of elements in \(X\) is called a *word* over \(X\).

Interesting cases of free multiplicative structures are those of free semigroups, free monoids, and free groups, where the multiplication is associative (see `IsAssociative`

(35.4-7)), which are described in Chapter 37, and also the case of free magmas, where the multiplication is nonassociative (see `IsNonassocWord`

(36.1-3)).

Elements in free magmas (see `FreeMagma`

(36.4-1)) lie in the category `IsWord`

; similarly, elements in free magmas-with-one (see `FreeMagmaWithOne`

(36.4-2)) lie in the category `IsWordWithOne`

, and so on.

`IsWord`

is mainly a "common roof" for the two *disjoint* categories `IsAssocWord`

(37.1-1) and `IsNonassocWord`

(36.1-3) of associative and nonassociative words. This means that associative words are *not* 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 36.5 and 37.7).

Note that elements in finitely presented groups and also elements in polycyclic groups in **GAP** are *not* in `IsWord`

although they are usually called words, see Chapters 47 and 46.

Words are *constants* (see 12.6), 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 *generators* of a free structure such as a free magma or a free group (see `FreeMagma`

(36.4-1), `FreeGroup`

(37.2-1)).

Words are also used to implement free algebras, in the same way as group elements are used to implement group algebras (see 62.3 and Chapter 65).

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

`‣ IsWordCollection` ( obj ) | ( category ) |

`IsWordCollection`

is the collections category (see `CategoryCollections`

(30.2-4)) of `IsWord`

(36.1-1).

gap> IsWordCollection( m ); IsWordCollection( s ); true true gap> IsWordCollection( [ "a", "b" ] ); false

`‣ IsNonassocWord` ( obj ) | ( category ) |

`‣ IsNonassocWordWithOne` ( obj ) | ( category ) |

A *nonassociative word* in **GAP** is an element in a free magma or a free magma-with-one (see 36.4).

The default methods for `ViewObj`

(6.3-5) and `PrintObj`

(6.3-5) 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 "program" 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 `MappedWord`

(36.3-1) implements a way to apply such a program. A more general way is provided by straight line programs (see 37.8).

Note that associative words (see Chapter 37) are *not* regarded as special cases of nonassociative words (see `IsWord`

(36.1-1)).

`‣ IsNonassocWordCollection` ( obj ) | ( category ) |

`‣ IsNonassocWordWithOneCollection` ( obj ) | ( category ) |

`IsNonassocWordCollection`

is the collections category (see `CategoryCollections`

(30.2-4)) of `IsNonassocWord`

(36.1-3), and `IsNonassocWordWithOneCollection`

is the collections category of `IsNonassocWordWithOne`

(36.1-3).

`36.2-1 \=`

`‣ \=` ( w1, w2 ) | ( operation ) |

Two words are equal if and only if they are words over the same alphabet and with equal external representations (see 36.5 and 37.7). For nonassociative words, the latter means that the words arise from the letters of the alphabet by the same sequence of multiplications.

`36.2-2 \<`

`‣ \<` ( w1, w2 ) | ( operation ) |

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 36.5; associative words are ordered by the shortlex ordering via `<`

(see 37.7).

Note that the alphabet of a word is determined by its family (see 13.1), and that the result of each call to `FreeMagma`

(36.4-1), `FreeGroup`

(37.2-1) etc. consists of words over a new alphabet. In particular, there is no "universal" empty word, every families of words in `IsWordWithOne`

(36.1-1) has its own empty word.

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

Two words can be multiplied via `*`

only if they are words over the same alphabet (see 36.2).

`‣ MappedWord` ( w, gens, imgs ) | ( operation ) |

`MappedWord`

returns the object that is obtained by replacing each occurrence in the word `w` of a generator in the list `gens` by the corresponding object in the list `imgs`. The lists `gens` and `imgs` must of course have the same length.

`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 `imgs` are all *associative words* (see Chapter 37) in the same family as the elements in `gens`, and some of them are equal to the corresponding generators in `gens`, then those may be omitted from `gens` and `imgs`. In this situation, the special case that the lists `gens` and `imgs` have only length \(1\) is handled more efficiently by `EliminatedWord`

(37.4-6).

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.

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

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 `GeneratorsOfMagma`

(35.4-1) in the case of a free magma, and via `GeneratorsOfMagmaWithOne`

(35.4-2) in the case of a free magma-with-one.

`‣ FreeMagma` ( rank[, name] ) | ( function ) |

`‣ FreeMagma` ( name1[, name2[, ...]] ) | ( function ) |

`‣ FreeMagma` ( names ) | ( function ) |

`‣ FreeMagma` ( infinity[, name][, init] ) | ( function ) |

`FreeMagma`

returns a free magma. 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 of `FreeSemigroup`

(51.1-10) for more information.

**1: For a given rank, and an optional generator name prefix**Called with a positive integer

`rank`,`FreeMagma`

returns a free magma on`rank`generators. The optional argument`name`must be a string; its default value is`"x"`

.If

`name`is not given but the`generatorNames`

option is, then this option is respected as described in Section 50.1-16.Otherwise, the generators of the returned free magma are labelled

`name``1`

, ...,`name``k`

, where`k`

is the value of`rank`.**2: For given generator names**Called with various (at least one) nonempty strings,

`FreeMagma`

returns a free magma on as many generators as arguments, which are labelled`name1`,`name2`, etc.**3: For a given list of generator names**Called with a finite nonempty list

`names`of nonempty strings,`FreeMagma`

returns a free magma on`Length(`

generators, whose`names`)`i`

-th generator is labelled`names``[i]`

.**4: For the rank**`infinity`

, an optional default generator name prefix, and an optional finite list of generator namesCalled in the fourth form,

`FreeMagma`

returns a free magma on infinitely many generators. The optional argument`name`must be a string; its default value is`"x"`

, and the optional argument`init`must be a finite list of nonempty strings; its default value is an empty list. The generators are initially labelled according to the list`init`, followed by`name``i`

for each`i`

in the range from`Length(`

to`init`)+1`infinity`

.

gap> FreeMagma( 4 ); <free magma on the generators [ x1, x2, x3, x4 ]> gap> FreeMagma( 3, "a" ); <free magma on the generators [ a1, a2, a3 ]> gap> FreeMagma( "a", "b" ); <free magma on the generators [ a, b ]> gap> FreeMagma( [ "a", "b" ] ); <free magma on the generators [ a, b ]> gap> FreeMagma( infinity ); <free magma with infinity generators> gap> F := FreeMagma( infinity, "gen" );; gap> GeneratorsOfMagma( F ){[ 1 .. 4 ]}; [ gen1, gen2, gen3, gen4 ] gap> F := FreeMagma( infinity, [ "z", "a" ] );; gap> GeneratorsOfMagma( F ){[ 1 .. 3 ]}; [ z, a, x3 ] gap> F := FreeMagma( infinity, "y", [ "z", "a" ] );; gap> GeneratorsOfMagma( F ){[ 1 .. 4 ]}; [ z, a, y3, y4 ] gap> FreeMagma( 3 : generatorNames := "elt" ); <free magma on the generators [ elt1, elt2, elt3 ]> gap> FreeMagma( 2 : generatorNames := [ "u", "v", "w" ] ); <free magma on the generators [ u, v ]>

`‣ FreeMagmaWithOne` ( rank[, name] ) | ( function ) |

`‣ FreeMagmaWithOne` ( [name1[, name2[, ...]]] ) | ( function ) |

`‣ FreeMagmaWithOne` ( names ) | ( function ) |

`‣ FreeMagmaWithOne` ( infinity[, name][, init] ) | ( function ) |

`FreeMagmaWithOne`

returns a free magma-with-one. 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 of `FreeSemigroup`

(51.1-10) for more information.

**1: For a given rank, and an optional generator name prefix**Called with a nonnegative integer

`rank`,`FreeMagmaWithOne`

returns a free magma-with-one on`rank`generators. The optional argument`name`must be a string; its default value is`"x"`

.If

`name`is not given but the`generatorNames`

option is, then this option is respected as described in Section 50.1-16.Otherwise, the generators of the returned free magma-with-one are labelled

`name``1`

, ...,`name``k`

, where`k`

is the value of`rank`.**2: For given generator names**Called with various nonempty strings,

`FreeMagmaWithOne`

returns a free magma-with-one on as many generators as arguments, which are labelled`name1`,`name2`, etc.**3: For a given list of generator names**Called with a finite list

`names`of nonempty strings,`FreeMagmaWithOne`

returns a free magma-with-one on`Length(`

generators, whose`names`)`i`

-th generator is labelled`names``[i]`

.**4: For the rank**`infinity`

, an optional default generator name prefix, and an optional finite list of generator namesCalled in the fourth form,

`FreeMagmaWithOne`

returns a free magma-with-one on infinitely many generators. The optional argument`name`must be a string; its default value is`"x"`

, and the optional argument`init`must be a finite list of nonempty strings; its default value is an empty list. The generators are initially labelled according to the list`init`, followed by`name``i`

for each`i`

in the range from`Length(`

to`init`)+1`infinity`

.

gap> FreeMagmaWithOne( 4 ); <free magma-with-one on the generators [ x1, x2, x3, x4 ]> gap> FreeMagmaWithOne( 3, "a" ); <free magma-with-one on the generators [ a1, a2, a3 ]> gap> FreeMagmaWithOne( "a", "b" ); <free magma-with-one on the generators [ a, b ]> gap> FreeMagmaWithOne( [ "a", "b" ] ); <free magma-with-one on the generators [ a, b ]> gap> FreeMagmaWithOne( infinity ); <free magma-with-one with infinity generators> gap> F := FreeMagmaWithOne( infinity, "gen" );; gap> GeneratorsOfMagmaWithOne( F ){[ 1 .. 4 ]}; [ gen1, gen2, gen3, gen4 ] gap> F := FreeMagmaWithOne( infinity, [ "z", "a" ] );; gap> GeneratorsOfMagmaWithOne( F ){[ 1 .. 3 ]}; [ z, a, x3 ] gap> F := FreeMagmaWithOne( infinity, "y", [ "z", "a" ] );; gap> GeneratorsOfMagmaWithOne( F ){[ 1 .. 4 ]}; [ z, a, y3, y4 ] gap> FreeMagmaWithOne( 0 ); <free group of rank zero> gap> FreeMagmaWithOne( 3 : generatorNames := "elt" ); <free magma-with-one on the generators [ elt1, elt2, elt3 ]> gap> FreeMagmaWithOne( 2 : generatorNames := [ "u", "v", "w" ] ); <free magma-with-one on the generators [ u, v ]>

The external representation of nonassociative words is defined as follows. The \(i\)-th generator of the family of elements in question has external representation \(i\), the identity (if exists) has external representation \(0\), the inverse of the \(i\)-th generator (if exists) has external representation \(-i\). If \(v\) and \(w\) are nonassociative words with external representations \(e_v\) and \(e_w\), respectively then the product \(v * w\) has external representation \([ e_v, e_w ]\). 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 `ObjByExtRep`

(79.8-1).

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))

generated by GAPDoc2HTML