{"paper":{"title":"On the monoid of monotone injective partial selfmaps of $\\mathbb{N}^{2}_{\\leqslant}$ with cofinite domains and images, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Inna Pozdniakova, Oleg Gutik","submitted_at":"2017-01-27T11:23:00Z","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 the natural partial order on $\\mathscr{P\\!O}\\!_{\\infty}(\\mathbb{N}^2_{\\leqslant})$ and show that it coincides with the natural partial order which is induced from symmetric inverse monoid $\\mathscr{I}_{\\mathbb{N}\\times\\mathbb{N}}$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.08015","kind":"arxiv","version":2},"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"}