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 $\mathsf{SL}(3, \mathbb{Z})$.

翻译：本文研究特殊仿射群 $\mathsf{SA}(2, \mathbb{Z}) = \mathbb{Z}^2 \rtimes \mathsf{SL}(2, \mathbb{Z})$（即保持定向的格 $\mathbb{Z}^2$ 仿射变换群）中的半群算法问题。我们聚焦于 Choffrut 与 Karhumäki (2005) 提出的两个判定问题：对于 $\mathsf{SA}(2, \mathbb{Z})$ 的有限生成子半群，其恒等问题（半群是否包含单位元？）与群问题（半群是否为群？）。我们证明这两个问题都是可判定的且为 NP 完全问题。由于 $\mathsf{SL}(2, \mathbb{Z}) \leq \mathsf{SA}(2, \mathbb{Z}) \leq \mathsf{SL}(3, \mathbb{Z})$，我们的结果推广了 Bell、Hirvensalo 与 Potapov (2017) 关于这两个问题在 $\mathsf{SL}(2, \mathbb{Z})$ 中 NP 完全性的结论，并为 $\mathsf{SL}(3, \mathbb{Z})$ 中的未解问题迈出了第一步。