The Left-Discrimination Sequence of an Automaton

Left-discrimination of a semigroup is defined and shown to be a sufficient condition that a semigroup be isomorphic to the input semigroup of its semigroup automaton, a necessary condition if the semigroup is finite. The left-discrimination sequence of an automaton is defined as a sequence of semigroups beginning with the input semigroup of the automaton, each member being the input semigroup of the semigroup automaton of its predecessor. It is related directly to a particular monotonically decreasing sequence of subautomata of the original automaton. This sequence is shown to be preserved by homomorphisms and is extended and used in an algorithm for determining the homomorphisms on one finite automaton to another.