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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 $\mathbb{Z}^2$ thatpreserve orientation. Our paper focuses on two decision problems introduced byChoffrut and Karhum\"{a}ki (2005): the Identity Problem (does a semigroupcontain a neutral element?) and the Group Problem (is a semigroup a group?) forfinitely generated sub-semigroups of $\mathsf{SA}(2, \mathbb{Z})$. We show thatboth 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 resultextends that of Bell, Hirvensalo and Potapov (SODA 2017) on the NP-completenessof both problems in $\mathsf{SL}(2, \mathbb{Z})$, and contributes a first steptowards the open problems in $\mathsf{SL}(3, \mathbb{Z})$.

The Identity Problem in the special affine group of $\mathbb{Z}^2$