Algebraic structures of automata

To determine the structure of an automaton A, we define the layers of A, which form a poset. We show that for any finite poset P there exists an automaton whose poset of layers is isomorphic to P. All subautomata of an automaton A are characterized by the layers of A, and they form a Ps-type upper semilattice. Conversely, for any Ps-type upper semilattice L there exists an automaton whose upper semilattice of subautomata is isomorphic to L. Among the classes of automata, we consider the class of single bottom automata, and we provide the composition of a single loop automaton and a strongly connected automaton together with its automorphism group.