On skeleton automata

A skeleton automaton is an automaton in which the relation of mutual accessibility of states is the identity relation. We prove that automata that admit a regular enumeration of states are exactly skeleton automata. It is shown how for a given automaton one can construct an automaton with minimal number of states that has the same subautomata lattice, and is necessarily a skeleton automaton. A procedure is proposed to obtain a skeleton automaton from a given automaton by removal of minimal number of arcs in its transition diagram.