{"paper":{"title":"Chains, Antichains, and Complements in Infinite Partition Lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Jakob Grue Simonsen, James Emil Avery, Jean-Yves Moyen, Pavel Ruzicka","submitted_at":"2015-01-21T20:19:23Z","abstract_excerpt":"We consider the partition lattice $\\Pi_\\kappa$ on any set of transfinite cardinality $\\kappa$ and properties of $\\Pi_\\kappa$ whose analogues do not hold for finite cardinalities. Assuming the Axiom of Choice we prove: (I) the cardinality of any maximal well-ordered chain is always exactly $\\kappa$; (II) there are maximal chains in $\\Pi_\\kappa$ of cardinality $> \\kappa$; (III) if, for every cardinal $\\lambda < \\kappa$, we have $2^{\\lambda} < 2^\\kappa$, there exists a maximal chain of cardinality $< 2^{\\kappa}$ (but $\\ge \\kappa$) in $\\Pi_{2^\\kappa}$; (IV) every non-trivial maximal antichain in $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.05284","kind":"arxiv","version":4},"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"}