一句话：证明 Z² 的特殊仿射群中有限生成半群的恒等元问题与群问题可判定且 NP 完全。
论文研究特殊仿射群 SA(2,Z)=Z² ⋊ SL(2,Z) 中有限生成子半群的算法问题，聚焦 Choffrut 与 Karhumäki 提出的恒等元问题（半群是否含单位元）和群问题（半群是否为群）。作者证明这两个判定问题都是可判定的，并且是 NP 完全的。由于 SL(2,Z) ≤ SA(2,Z) ≤ SL(3,Z)，该结果扩展了 Bell、Hirvensalo 与 Potapov（SODA 2017）关于 SL(2,Z) 中两个问题 NP 完全性的结论，并向 SL(3,Z) 中相关公开问题迈出第一步。
We consider semigroup algorithmic problems in the Special Affine group ${\text{SA}}(2,{\mathbb{Z}}) = {{\mathbb{Z}}^2} \rtimes {\text{SL}}(2,{\mathbb{Z}})$, 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 Karhumäki (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 ${\text{SA}}(2,{\mathbb{Z}})$. We show that both problems are decidable and NP-complete. Since ${\text{SL}}(2,{\mathbb{Z}}) \leq {\text{SA}}(2,{\mathbb{Z}}) \leq {\text{SL}}(3,{\mathbb{Z}})$, our result extends that of Bell, Hirvensalo and Potapov (SODA 2017) on the NP-completeness of both problems in ${\text{SL}}(2,{\mathbb{Z}})$, and contributes a first step towards the open problems in ${\text{SL}}(3,{\mathbb{Z}})$.