  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.
