Isomorphisms on strongly connected group automaton

Abstract

Direct product of a permutation strongly connected automaton and a synchronizing strongly connected Aleshin Type automaton is also a strongly connected automaton. An automaton is called quasi-ideal  automaton if and only if all the following conditions are satisfied; (i) It is strongly connected (ii) the minimal ideal of its transition semi group is a right group (iii) the ranges of the idempotent elements of the minimal ideal of its transition group form a merging of a partition on its set of states. An automaton is isomorphic to the direct product of a permutation strongly connected automaton and synchronizing strongly connected Aleshin type automaton if and only if it is a quasi ideal automaton.