{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:53:29Z","timestamp":1753894409546,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T00:00:00Z","timestamp":1749427200000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T00:00:00Z","timestamp":1749427200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T00:00:00Z","timestamp":1749427200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,3,29]]},"abstract":"<jats:p>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\\&amp;quot;{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 (2017) on the NP-completeness of both problems in $\\mathsf{SL}(2, \\mathbb{Z})$, and contributes a first step towards the open problems in $\\mathsf{SL}(3, \\mathbb{Z})$.<\/jats:p>","DOI":"10.46298\/lmcs-21(2:21)2025","type":"journal-article","created":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T08:25:07Z","timestamp":1749457507000},"source":"Crossref","is-referenced-by-count":0,"title":["The Identity Problem in the special affine group of $\\mathbb{Z}^2$"],"prefix":"10.46298","volume":"Volume 21, Issue 2","author":[{"given":"Ruiwen","family":"Dong","sequence":"first","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2025,6,9]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/2301.09502v8","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/2301.09502v8","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T11:43:08Z","timestamp":1751024588000},"score":1,"resource":{"primary":{"URL":"http:\/\/lmcs.episciences.org\/13602"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,9]]},"references-count":0,"URL":"https:\/\/doi.org\/10.46298\/lmcs-21(2:21)2025","relation":{"has-preprint":[{"id-type":"arxiv","id":"2301.09502v6","asserted-by":"subject"},{"id-type":"arxiv","id":"2301.09502v5","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"2301.09502","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.2301.09502","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"type":"electronic","value":"1860-5974"}],"subject":[],"published":{"date-parts":[[2025,6,9]]},"article-number":"13602"}}