{"paper":{"title":"Categories of frame-completions and join-specifications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Rob Egrot","submitted_at":"2018-06-02T14:18:11Z","abstract_excerpt":"Given a poset $P$, a join-specification $\\mathcal U$ for $P$ is a set of subsets of $P$ whose joins are all defined. The set $\\mathcal I_{\\mathcal U}$ of downsets closed under joins of sets in $\\mathcal U$ forms a complete lattice, and is, in a sense, the free $\\mathcal U$-join preserving join-completion of $P$. The main aim of this paper is to address two questions. First, given a join-specification $\\mathcal U$, when is $\\mathcal I_{\\mathcal U}$ a frame? And second, given a poset $P$, what is the structure of its set of frame-generating join-specifications?\n  To answer the first question we "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.00642","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"}