Representations of commutative asynchronous automata

Ito (1976, 1978) [14,17] provided representations of strongly connected automata by group-matrix type automata. This shows the close connection between strongly connected automata with their automorphism groups. In this paper we deal with commutative asynchronous automata. In particular, we introduce and study normal commutative asynchronous automata and cyclic commutative asynchronous automata. Some properties on endomorphism monoids of these automata are given. Also, the representations of normal commutative asynchronous automata and cyclic commutative asynchronous automata are provided by S-automata and regular S-automata, respectively. The cartesian composition A∘B of a strongly connected automaton A and a cyclic commutative asynchronous automaton B is studied. It is shown that the endomorphism monoid E(A∘B) of automaton A∘B is a Clifford monoid. Finally, a representation of A∘B is provided by regular Clifford monoid matrix-type automaton. This generalizes and extends the representations of strongly connected automata given by Ito (1976) [14].