The Identity Problem in the special affine group of $\mathbb{Z}^2
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 intr...
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| Published in: | Logical methods in computer science Vol. 21, Issue 2 |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Logical Methods in Computer Science e.V
09.06.2025
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| Subjects: | |
| ISSN: | 1860-5974, 1860-5974 |
| Online Access: | Get full text |
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| Summary: | 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 $\mathsf{SL}(3, \mathbb{Z})$. |
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| ISSN: | 1860-5974 1860-5974 |
| DOI: | 10.46298/lmcs-21(2:21)2025 |