I'm interested in the (very restricted) subset of regular languages for which there is an automaton having the following property: for every letter $a$ of the alphabet, the automaton has at most one transition labeled with $a$. (Of course, the automaton is then necessarily a partial deterministic automaton.) Is there a name for this class of regular languages?
These languages are quite limited but not entirely trivial either, for instance you could have $(ab)^*$, $ab^*|cd^*$, or $(ab)^*a$ (which I guess cannot be written as a regular expression with only one occurrence of each letter, so my class is more general than imposing the analogous requirement on regular expressions). These automata are clearly local automata, so this is a subclass of the local languages. These automata are also aperiodic, so these languages are star-free. Of course, for any alphabet, there are only finitely many languages of this kind over the alphabet.
I thought this class was pretty natural, but looking around, I could not find anything about it. Does it have a name, or has it been studied somehow?