The Identity Problem in ℤ ≀ ℤ Is Decidable

We consider semigroup algorithmic problems in the wreath product ℤ ≀ ℤ. Our paper focuses on two decision problems introduced by Choffrut and Karhumäki (2005): the Identity Problem (does a semigroup contain the neutral element?) and the Group Problem (is a semigroup a group?) for finitely generated sub-semigroups of ℤ ≀ ℤ. We show that both problems are decidable. Our result complements the undecidability of the Semigroup Membership Problem (does a semigroup contain a given element?) in ℤ ≀ ℤ shown by Lohrey, Steinberg and Zetzsche (ICALP 2013), and contributes an important step towards solving semigroup algorithmic problems in general metabelian groups.
LIPIcs, Vol. 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), pages 124:1-124:20