Induced permutation automata and coverings of strongly connected automata

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Let A = ( S, Σ, N) be a strongly connected automaton and e be a minimal idempotent of characteristic semigroup B( A). The unique (up to isomorphism) subgroup eB( A) e is a very important group in automaton automorphism theory. For example, the automorphism group A( A) of A is a homomorphic image of a subgroup of eB( A) e [7, 8]. If H is a subgroup of A( A), then the automorphism group of the factor automaton A H is also homomorphic to a subgroup of eB( A) e. In this paper we show another property of eB( A) e. First, we introduce the induced permutation automaton whose characteristic semigroup is isomorphic to eB( A) e, and the generalized factor automaton. Using these two automata, we construct a cascade product covering of A, if eB( A) e is not {id} (identity permutation group). This is an example of an effective admissible subset system covering [3], as well as a generalization of the result of Krohn et al. [6]which gave a decomposition of A, if the automorphism group of A is not {id}.