{"paper":{"title":"On the lattice of weak topologies on the bicyclic monoid with adjoined zero","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Oleg Gutik, Serhii Bardyla","submitted_at":"2019-08-13T10:15:46Z","abstract_excerpt":"A Hausdorff topology $\\tau$ on the bicyclic monoid with adjoined zero $\\mathcal{C}^0$ is called {\\em weak} if it is contained in the coarsest inverse semigroup topology on $\\mathcal{C}^0$. We show that the lattice $\\mathcal{W}$ of all weak shift-continuous topologies on $\\mathcal{C}^0$ is isomorphic to the lattice of all shift-invariant filters on $\\omega$ with an attached element $1$ endowed with the following partial order: $\\mathcal{F}\\leq \\mathcal{G}$ iff $\\mathcal{G}=1$ or $\\mathcal{F}\\subset \\mathcal{G}$. Also, we investigate cardinal characteristics of the lattice $\\mathcal{W}$. In part"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.04566","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.04566/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}