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Critical classes for the \(\alpha _ 0\)-product. (English) Zbl 0693.68033

A class \({\mathcal K}\) of automata is precomplete if it satisfies: (i) there exists an \(A\in {\mathcal K}\) such that the characteristic semigroup of the two-state identity reset automaton is isomorphic to a subsemigroup of the characteristic semigroup S(A) of A; (ii) for every simple group G there is an \(A\in {\mathcal K}\) such that \(G| S(A)\). A class \({\mathcal K}_ 0\) of automata is critical if for every precomplete class \({\mathcal K}\), the inclusion \({\mathcal K}_ 0\subseteq HSP_{\alpha_ 0}({\mathcal K})\) implies that \({\mathcal K}\) is complete for the \(\alpha_ 0\)-product. The paper gives a full description of the critical classes of automata. The notions of weakly precomplete and strongly critical classes are also considered.
Reviewer: G.Gauthier

MSC:

68Q70 Algebraic theory of languages and automata
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References:

[1] (Arbib, M. A., Algebraic Theory of Machines, Languages and Semigroups (1968), Academic Press: Academic Press New York) · Zbl 0181.01501
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