Automaton semigroups: New constructions results and examples of non-automaton semigroups

Abstract

This paper studies the class of automaton semigroups from two perspectives: closure under constructions, and examples of semigroups that are not automaton semigroups. We prove that (semigroup) free products of finite semigroups always arise as automaton semigroups, and that the class of automaton monoids is closed under forming wreath products with finite monoids. We also consider closure under certain kinds of Rees matrix constructions, strong semilattices, and small extensions. Finally, we prove that no subsemigroup of (N,+) arises as an automaton semigroup. (Previously, (N,+) itself was the unique example of a semigroup having the ‘general’ properties of automaton semigroups (such as residual finiteness, solvable word problem, etc.) but that was known not to arise as an automaton semigroup.)