  We study automaton structures, i.e. groups, monoids and semigroups generated
by an automaton, which, in this context, means a deterministic finite-state
letter-to-letter transducer. Instead of considering only complete automata, we
specifically investigate semigroups generated by partial automata. First, we
show that the class of semigroups generated by partial automata coincides with
the class of semigroups generated by complete automata if and only if the
latter class is closed under removing a previously adjoined zero, which is an
open problem in (complete) automaton semigroup theory stated by Cain. Then, we
show that no semidirect product (and, thus, also no direct product) of an
arbitrary semigroup with a (non-trivial) subsemigroup of the free monogenic
semigroup is an automaton semigroup. Finally, we concentrate on inverse
semigroups generated by invertible but partial automata, which we call
automaton-inverse semigroups, and show that any inverse automaton semigroup can
be generated by such an automaton (showing that automaton-inverse semigroups
and inverse automaton semigroups coincide).
