The Identity Problem in the special affine group of $\mathbb{Z}^2$

TL;DR

This paper proves that the Identity and Group problems for finitely generated sub-semigroups of the special affine group of rac12;2 are decidable and NP-complete, extending previous results in related matrix groups.

Contribution

It establishes the decidability and NP-completeness of key semigroup problems in rac12;2, advancing understanding of algorithmic problems in affine and matrix groups.

Findings

Both the Identity and Group problems are decidable in rac12;2.

These problems are NP-complete for finitely generated sub-semigroups.

Results extend previous NP-completeness findings from rac12;2 to the special affine group.

Abstract

We consider semigroup algorithmic problems in the Special Affine group $\mathsf{SA}(2, \mathbb{Z}) = \mathbb{Z}^2 \rtimes \mathsf{SL}(2, \mathbb{Z})$, which is the group of affine transformations of the lattice $\mathbb{Z}^2$ that preserve orientation. Our paper focuses on two decision problems introduced by Choffrut and Karhum\"{a}ki (2005): the Identity Problem (does a semigroup contain a neutral element?) and the Group Problem (is a semigroup a group?) for finitely generated sub-semigroups of $\mathsf{SA}(2, \mathbb{Z})$. We show that both problems are decidable and NP-complete. Since $\mathsf{SL}(2, \mathbb{Z}) \leq \mathsf{SA}(2, \mathbb{Z}) \leq \mathsf{SL}(3, \mathbb{Z})$, our result extends that of Bell, Hirvensalo and Potapov (2017) on the NP-completeness of both problems in $\mathsf{SL}(2, \mathbb{Z})$, and contributes a first step towards the open problems in $\mathsf{SL}(3, \mathbb{Z})$.