{"paper":{"title":"On the semigroup $\\textbf{ID}_{\\infty}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Anatolii Savchuk, Oleg Gutik","submitted_at":"2019-04-14T07:33:09Z","abstract_excerpt":"We study the semigroup $\\textbf{{ID}}_{\\infty}$ of all partial isometries of the set of integers $\\mathbb{Z}$. It is proved that the quotient semigroup $\\textbf{{ID}}_{\\infty}/\\mathfrak{C}_{\\textsf{mg}}$, where $\\mathfrak{C}_{\\textsf{mg}}$ is the minimum group congruence, is isomorphic to the group ${\\textsf{Iso}}(\\mathbb{Z})$ of all isometries of $\\mathbb{Z}$, $\\textbf{{ID}}_{\\infty}$ is an $F$-inverse semigroup, and $\\textbf{{ID}}_{\\infty}$ is isomorphic to the semidirect product ${\\textsf{Iso}}(\\mathbb{Z})\\ltimes_\\mathfrak{h}\\mathscr{P}_{\\!\\infty}(\\mathbb{Z})$ of the free semilattice with u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.06644","kind":"arxiv","version":1},"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"}