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Recognizable subsets of some partially Abelian monoids. (English) Zbl 0559.20040
A free partially abelian monoid S is a monoid which has a finite alphabet A as generating set and in which all laws are of the form ab$$\simeq ba$$ for some a,b$$\in S$$. Let X be a finite set of words over A each containing all the letters of A at least once. Suppose that the graph the vertices of which are the letters of A and for which the edges correspond to non-commuting pairs of letters of A, is connected. Then the main theorem of the paper states that the subset $$[X^*]$$ of the free partially abelian monoid $$A^*/\simeq$$ containing all the words equivalent to a product of words in X, is rational. Also, it is shown that the set of all words $$u\in A^*$$ commuting with a given word $$w\in A^*$$ (i.e. wu$$\simeq uw)$$ is a rational, finitely generated subset of $$A^*$$.
Reviewer: H.Mitsch

##### MSC:
 20M05 Free semigroups, generators and relations, word problems 20M35 Semigroups in automata theory, linguistics, etc. 68Q70 Algebraic theory of languages and automata
##### Keywords:
free partially abelian monoid; finite alphabet; words; letters
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##### References:
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