Semigroup Algorithmic Problems in Metabelian Groups

We consider semigroup algorithmic problems in finitely generated metabelian groups. Our paper focuses on three decision problems introduced by Choffrut and Karhum'aki (2005): the Identity Problem (does a semigroup contain a neutral element?), the Group Problem (is a semigroup a group?) and the Inverse Problem (does a semigroup contain the inverse of a generator?). We show that all three problems are decidable for finitely generated subsemigroups of finitely generated metabelian groups. In particular, we establish a correspondence between polynomial semirings and subsemigroups of metabelian groups using an interaction of graph theory, convex polytopes, algebraic geometry and number theory. Since the Semigroup Membership problem (does a semigroup contain a given element?) is known to be undecidable in finitely generated metabelian groups, our result completes the decidability characterization of semigroup algorithmic problems in metabelian groups.