{"paper":{"title":"Closure operators, frames, and neatest representations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Rob Egrot","submitted_at":"2017-02-08T02:51:57Z","abstract_excerpt":"Given a poset $P$ and a standard closure operator $\\Gamma:\\wp(P)\\to\\wp(P)$ we give a necessary and sufficient condition for the lattice of $\\Gamma$-closed sets of $\\wp(P)$ to be a frame in terms of the recursive construction of the $\\Gamma$-closure of sets. We use this condition to show that given a set $\\mathcal{U}$ of distinguished joins from $P$, the lattice of $\\mathcal{U}$-ideals of $P$ fails to be a frame if and only if it fails to be $\\sigma$-distributive, with $\\sigma$ depending on the cardinalities of sets in $\\mathcal{U}$. From this we deduce that if a poset has the property that whe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02257","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"}