Semigroup algorithmic problems in metabelian groups
(2304.12893)Abstract
We consider semigroup algorithmic problems in finitely generated metabelian groups. Our paper focuses on three decision problems introduced by Choffrut and Karhum\"{a}ki (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.
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