  We improve on earlier results on the closure under free products of the class
of automaton semigroups. We consider partial automata and show that the free
product of two self-similar semigroups (or automaton semigroups) is
self-similar (an automaton semigroup) if there is a homomorphism from one of
the base semigroups to the other. The construction used is computable and
yields further consequences. One of them is that we can adjoin a free generator
to any self-similar semigroup (or automaton semigroup) and preserve the
property of self-similarity (or being an automaton semigroup).
  The existence of a homomorphism between two semigroups is a very lax
requirement; in particular, it is satisfied if one of the semigroups contains
an idempotent. To explore the limits of this requirement, we show that no
simple or $0$-simple idempotent-free semigroup is a finitely generated
self-similar semigroup (or an automaton semigroup). Furthermore, we give an
example of a pair of residually finite semigroups without a homomorphism from
one to the other.
