Commutative Finite State Automaton Group (FSAG) Having Cycles over the Binary Alphabet

The Cartesian composition A · B of a strongly connected automaton Group A and a cyclic commutative automaton Group B is defined. 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 and cyclic commutative automaton CCA.