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On noncounting regular classes. (English) Zbl 0780.68084
Summary: Let \(A^*\) be the free monoid of base \(A\) and \(n\) a fixed positive integer. For any word \(w\in A^*\) we consider the set \([w]_ n\) of all the words which are equivalent to \(w\) modulus the congruence \(\theta_ n\) generated by the relation \(x^ n\sim x^{n+1}\), where \(x\) is any word of \(A^*\). The main result of the paper is that if \(n>4\) then for any word \(w\in A^*\) the congruence class \([w]_ n\) is a regular language. We also prove that the word problem for the quotient monoid \(M_ n=A^*/\theta_ n\) is recursively solvable.

MSC:
68Q45 Formal languages and automata
20M35 Semigroups in automata theory, linguistics, etc.
20M05 Free semigroups, generators and relations, word problems
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[1] Book, R.V., Thue systems and the church-rosser property: replacement systems, specification of formal languages, and presentations of monoids, (), 1-38
[2] Brzozowski, J.; Culik, K.; Gabrielian, A., Classification of noncounting events, J. comput. system sci., 5, 41-53, (1971) · Zbl 0241.94050
[3] Brzozowski, J., Open problems about regular languages, (), 23-45
[4] Ehrenfeucht, A.; Rozenberg, G., On regularity of languages generated by copying systems, Discrete appl. math., 8, 313-317, (1984) · Zbl 0549.68075
[5] Eilenberg, S.; Eilenberg, S., Automata, languages and machines, Automata, languages and machines, vol B, (1976), Academic Press · Zbl 0359.94067
[6] Fine, N.J.; Wilf, M.S., Uniqueness theorem for periodic functions, Proc. amer. math. soc., 16, 109-114, (1965) · Zbl 0131.30203
[7] Green, J.A.; Rees, D., On semigroups in which xr=x, Proc. Cambridge philos. soc., 48, 35-40, (1952) · Zbl 0046.01903
[8] Hotzel, E., On finiteness conditions in semigroups, J. algebra, 60, 352-370, (1979) · Zbl 0421.20032
[9] Lallement, G., Semigroups and combinatorial applications, (1979), John Wiley New York · Zbl 0421.20025
[10] Lothaire, M., Combinatorics on words, (1983), Addison-Wesley Reading, MA · Zbl 0514.20045
[11] de Luca, A.; Varricchio, S., A finiteness condition for semigroups generalizing a theorem of hotzel, J. algebra, 136, 60-72, (1991) · Zbl 0741.20038
[12] Theoret. Comput. Sci. to appear
[13] McNaughton, R.; Papert, S., Counter-free automata, (1971), The M.I.T. Press Cambridge, MA · Zbl 0232.94024
[14] Simon, I., Notes on noncounting languages of order, 2, (1970), manuscript
[15] Thue, A., Über die gegenseitige lage gleicher teile gewisser zeichenreihen, Videnskabsselskabets skrifter, 1, (1912), Mat.-Nat. K.1., Christiania · JFM 44.0462.01
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