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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

20M05 Free semigroups, generators and relations, word problems
20M35 Semigroups in automata theory, linguistics, etc.
68Q70 Algebraic theory of languages and automata
Full Text: DOI
[1] Cartier, P.; Foata, D., Prolèmes combinatoires de commutation et réarrangements, () · Zbl 0186.30101
[2] Clerbout, M.; Latteux, M., Partial commutations and faithful rational transductions, Publication de l’équipe lilloise d’informatique théorique no IT 54-83, (1983) · Zbl 0548.68073
[3] R. Cori and D. Perrin, Sur la reconnaissabilité dans les monoides partiellement commutatifs libres, RAIRO Inform. Théoret., to appear.
[4] Eilenberg, S., ()
[5] Flé, M.P.; Roucairol, G., On serializability of iterated transactions, ACM sigact-sigops, 194-200, (1982) · Zbl 0492.68019
[6] Lothaire, M., Combinatorics on words, (1983), Addison-Wesley Reading, MA · Zbl 0514.20045
[7] Pin, J.-E., Variétés de langages formels, (1984), Masson Paris · Zbl 0636.68093
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