Parikh's Theorem Does Not Hold for Multiplicities

We consider the question of whether the famous Parikh's theorem holds with multiplicities i.e., for formal power series instead of languages. We introduce two families of semilinear formal power series over a commutative product monoid. The strict hierarchy of algebraic, rational, recognizable and semi-linear formal power series is proved and in this way it is established that the Parikh's theorem does not hold with multiplicities.