Critical classes for the \(\alpha _ 0\)-product
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.
scientific article; zbMATH DE number 4135406