The Identity Problem in $\mathbb{Z} \wr \mathbb{Z}$ is decidable

TL;DR

This paper proves that the Identity and Group Problems are decidable for finitely generated sub-semigroups of the wreath product   , advancing understanding of algorithmic problems in metabelian groups.

Contribution

It establishes the decidability of the Identity and Group Problems in   , a significant step beyond previous undecidability results.

Findings

Both the Identity and Group Problems are decidable in   .

The results complement known undecidability of the Semigroup Membership Problem.

Advances understanding of algorithmic problems in metabelian groups.

Abstract

We consider semigroup algorithmic problems in the wreath product $\mathbb{Z} \wr \mathbb{Z}$. Our paper focuses on two decision problems introduced by Choffrut and Karhum\"{a}ki (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 $\mathbb{Z} \wr \mathbb{Z}$. 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 $\mathbb{Z} \wr \mathbb{Z}$ shown by Lohrey, Steinberg and Zetzsche (ICALP 2013), and contributes an important step towards solving semigroup algorithmic problems in general metabelian groups.