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 (SODA 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：我们考虑特殊仿射群 $\mathsf{SA}(2, \mathbb{Z}) = \mathbb{Z}^2 \rtimes \mathsf{SL}(2, \mathbb{Z})$ 中的半群算法问题，该群是保持定向的格 $\mathbb{Z}^2$ 的仿射变换群。本文聚焦于 Choffrut 和 Karhumäki (2005) 提出的两个判定问题：对于 $\mathsf{SA}(2, \mathbb{Z})$ 中有限生成的子半群，研究恒等元问题（半群是否包含单位元？）和群问题（半群是否为一个群？）。我们证明这两个问题都是可判定的，且属于 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 和 Potapov (SODA 2017) 关于 $\mathsf{SL}(2, \mathbb{Z})$ 中两个问题 NP 完全性的结论，并为 $\mathsf{SL}(3, \mathbb{Z})$ 中的公开问题迈出了第一步。