  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 $(\mathbb{N},
+)$ arises as an automaton semigroup. (Previously, $(\mathbb{N},+)$ itself was
the unique example of a finitely generated residually finite semigroup that was
known not to arise as an automaton semigroup.)
