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This paper studies the class of automaton semigroups from two perspectives:closure under constructions, and examples of semigroups that are not automatonsemigroups. We prove that (semigroup) free products of finite semigroups alwaysarise as automaton semigroups, and that the class of automaton monoids isclosed under forming wreath products with finite monoids. We also considerclosure under certain kinds of Rees matrix constructions, strong semilattices,and small extensions. Finally, we prove that no subsemigroup of $(\mathbb{N},+)$ arises as an automaton semigroup. (Previously, $(\mathbb{N},+)$ itself wasthe unique example of a finitely generated residually finite semigroup that wasknown not to arise as an automaton semigroup.)

Automaton semigroups: new construction results and examples of non-automaton semigroups