Associative words are used to represent elements in free groups, semigroups and monoids in GAP (see Free Groups, Monoids and Semigroups). An associative word is just a sequence of letters, where each letter is an element of an alphabet (in the following called a generator) or its inverse. Associative words can be multiplied; in free monoids also the computation of an identity is permitted, in free groups also the computation of inverses (see Operations for Associative Words).
IsAssocWord( obj ) C
IsAssocWordWithOne( obj ) C
IsAssocWordWithInverse( obj ) C
IsAssocWord is the category of associative words in free semigroups,
IsAssocWordWithOne is the category of associative words in free monoids
(which admit the operation One to compute an identity),
IsAssocWordWithInverse is the category of associative words in free
groups (which have an inverse).
See IsWord for more general categories of words.
Different alphabets correspond to different families of associative words. There is no relation whatsoever between words in different families.
gap> f:= FreeGroup( "a", "b", "c" ); <free group on the generators [a,b,c]> gap> gens:= GeneratorsOfGroup(f); [ a, b, c ] gap> w:= gens[1]*gens[2]/gens[3]*gens[2]*gens[1]/gens[1]*gens[3]/gens[2]; a*b*c^-1*b*c*b^-1 gap> w^-1; b*c^-1*b^-1*c*b^-1*a^-1
Words are displayed as products of letters.
The letters are usually printed like f1, f2, ¼,
but it is possible to give user defined names to them,
which can be arbitrary strings.
These names do not necessarily identify a unique letter (generator),
it is possible to have several letters --even in the same family--
that are displayed in the same way.
Note also that
there is no relation between the names of letters and variable names.
Using homomorphisms it is possible to express elements of a group as words in terms of generators, see Expressing group elements as words in generators.
GAP 4 manual