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：本文研究特殊仿射群$\mathsf{SA}(2, \mathbb{Z}) = \mathbb{Z}^2 \rtimes \mathsf{SL}(2, \mathbb{Z})$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。我们聚焦于Choffrut与Karhumäki（2005）提出的两个判定问题：有限生成子半群的恒等问题（半群是否包含单位元？）与群问题（半群是否构成群？）。我们证明这两个问题均可判定且为NP完全问题。由于$\mathsf{SL}(2, \mathbb{Z}) \leq \mathsf{SA}(2, \mathbb{Z}) \leq \mathsf{SL}(3, \mathbb{Z})$，本结果不仅扩展了Bell、Hirvensalo和Potapov（SODA 2017）关于$\mathsf{SL}(2, \mathbb{Z})$中两问题NP完全性的结论，更向解决$\mathsf{SL}(3, \mathbb{Z})$中的开放问题迈出了第一步。