On Bounded Rational Trace Languages

In this paper, for a finitely generated monoid M, we tackle the following three questions: Is it possible to give a characterization of rational subsets of M which have polynomial growth?What is the structure of the counting function of rational sets which have polynomial growth?Is it true that every rational subset of M has either exponential growth or it has polynomial growth? Can one decide for a given rational set which of the two options holds? We give a positive answer to all the previous questions in the case that M is a direct product of free monoids. Some of the proved results also extend to trace monoid.