{"paper":{"title":"On the monoid of monotone injective partial selfmaps of $\\mathbb{N}^{2}_{\\leqslant}$ with cofinite domains and images","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Inna Pozdniakova, Oleg Gutik","submitted_at":"2016-02-21T22:40:59Z","abstract_excerpt":"Let $\\mathbb{N}^{2}_{\\leqslant}$ be the set $\\mathbb{N}^{2}$ with the partial order defined as the product of usual order $\\leq$ on the set of positive integers $\\mathbb{N}$. We study the semigroup $\\mathscr{P\\!O}\\!_{\\infty}(\\mathbb{N}^2_{\\leqslant})$ of monotone injective partial selfmaps of $\\mathbb{N}^{2}_{\\leqslant}$ having cofinite domain and image. We describe properties of elements of the semigroup $\\mathscr{P\\!O}\\!_{\\infty}(\\mathbb{N}^2_{\\leqslant})$ as monotone partial bijections of $\\mathbb{N}^{2}_{\\leqslant}$ and show that the group of units of $\\mathscr{P\\!O}\\!_{\\infty}(\\mathbb{N}^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.06593","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}