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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
##### Keywords:
free monoid; regular language; word problem; quotient monoid
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##### References:
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