  The projected language of a general deterministic automaton with $n$ states
is recognizable by a deterministic automaton with $2^{n-1} + 2^{n-m} - 1$
states, where $m$ denotes the number of states incident to unobservable
non-loop transitions, and this bound is best possible. Here, we derive the
tight bound $2^{n - \lceil \frac{m}{2} \rceil} - 1$ for permutation automata.
For a state-partition automaton with $n$ states (also called automata with the
observer property) the projected language is recognizable with $n$ states. Up
to now, these, and finite languages projected onto unary languages, were the
only classes of automata known to possess this property. We show that this is
also true for commutative automata and we find commutative automata that are
not state-partition automata.
