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：我们考虑特别艾芬集团中的半组算法问题 $\ mathsf{SA} (2,\ mathbb}}=\ mathbb}2\rtims\rtime\ mathsf{SL} (2,\ mathbb}$) $( 2,\ mathbb}$) 。 我们的论文侧重于Choffruty 和 Karhum\\"{a}ki(2005): 身份问题( 半组包含中性元素吗? ) 和组问题( 是一个半组? =mathb_ group $\ mathb}\rtimes\ groups f{SA} (2,\ mathbbbbb}\ mash\ math_r_moth_SL) 问题(3,\ mas\ mas_r_rxx) 问题。
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