1) Is there a notion of (deterministic) finite state transducer (FST) that allows the possibility of producing an infinite stream of output symbols? In other words, one where the transduction relation is a subset of $\Sigma^* \times \Gamma^\omega$ (as opposed to $\Sigma^* \times \Gamma^*$)?
For example, the definition of an FST given on Wikipedia involves a transition relation $\delta \subseteq Q \times (\Sigma \cup \{\epsilon\}) \times (\Gamma \cup \{\epsilon\}) \times Q$, which might first seem to permit infinite streams of output. However, the transduction relation is defined inductively (basically, the reflexive-transitive closure of $\delta$), thereby ruling out infinitary outputs.
2) Relatedly, the definitions of FSTs that I've seen include a subset $F \subseteq Q$ of accepting states. Is there a name for the class of (deterministic) FSTs in which $F = Q$, i.e., in which all input strings are accepted and the emphasis is placed on the output?
Any references that you can point me to would be greatly appreciated!