Semigroup algorithmic problems in metabelian groups

Abstract

We consider semigroup algorithmic problems in finitely generated metabelian groups. Our paper focuses on three decision problems introduced by Choffrut and Karhumäki (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 sub-semigroups of finitely generated metabelian groups. In particular, we establish a correspondence between polynomial semirings and sub-semigroups 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.

52 pages including appendices, 28 figures