The Identity Problem in the special affine group of ℤ^2

01/23/2023

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by   Ruiwen Dong, et al.

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We consider semigroup algorithmic problems in the Special Affine group 𝖲𝖠(2, ℤ) = ℤ^2 ⋊𝖲𝖫(2, ℤ), which is the group of affine transformations of the lattice 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 𝖲𝖠(2, ℤ). We show that both problems are decidable and NP-complete. Since 𝖲𝖫(2, ℤ) ≤𝖲𝖠(2, ℤ) ≤𝖲𝖫(3, ℤ), our result extends that of Bell, Hirvensalo and Potapov (SODA 2017) on the NP-completeness of both problems in 𝖲𝖫(2, ℤ), and contributes a first step towards the open problems in 𝖲𝖫(3, ℤ).