Polynomials, fragments of temporal logic and the variety over traces

We show that some language-theoretic and logical characterizations of recognizable word languages whose syntactic monoid is in the variety also hold over traces. To this end we give algebraic characterizations for the language operations of generating the polynomial closure and generating the unambiguous polynomial closure over traces.We also show that there exist natural fragments of local temporal logic that describe this class of languages corresponding to . All characterizations are known to hold for words.