Separability of rational subsets by recognizable subsets is decidable in $A^*\times\N^m$
Résumé
Given a direct product of monoids $M=A^*\times \N^m$ where $A$ is finite and $\N$ is the additive monoid of nonnegative integers, the following problem is recursively decidable: given two rational subsests of $M$, does there exist a recognizable subset which includes one of the subsets and excludes the other.