  The aim of this paper is to investigate whether the class of automaton
semigroups is closed under certain semigroup constructions. We prove that the
free product of two automaton semigroups that contain left identities is again
an automaton semigroup. We also show that the class of automaton semigroups is
closed under the combined operation of 'free product followed by adjoining an
identity'. We present an example of a free product of finite semigroups that we
conjecture is not an automaton semigroup. Turning to wreath products, we
consider two slight generalizations of the concept of an automaton semigroup,
and show that a wreath product of an automaton monoid and a finite monoid
arises as a generalized automaton semigroup in both senses. We also suggest a
potential counterexample that would show that a wreath product of an automaton
monoid and a finite monoid is not a necessarily an automaton monoid in the
usual sense.
