{"paper":{"title":"On the representation of finite distributive lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mark Siggers","submitted_at":"2014-11-28T14:25:12Z","abstract_excerpt":"A simple but elegant result of Rival states that every sublattice $L$ of a finite distributive lattice $\\mathcal{P}$ can be constructed from $\\mathcal{P}$ by removing a particular family $\\mathcal{I}_L$ of its irreducible intervals.\n  Applying this in the case that $\\mathcal{P}$ is a product of a finite set $\\mathcal{C}$ of chains, we get a one-to-one correspondence $L \\mapsto \\mathcal{D}_\\mathcal{P}(L)$ between the sublattices of $\\mathcal{P}$ and the preorders spanned by a canonical sublattice $\\mathcal{C}^\\infty$ of $\\mathcal{P}$.\n  We then show that $L$ is a tight sublattice of the product"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.0011","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"}