{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:YCLNV34OZSSSAPXW6DT7HKPHL7","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c6a2ffae071b60e58f6e617f689386e903fa7522b70cf5ff14a19a8f061fd9a1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-01-21T20:19:23Z","title_canon_sha256":"271780c7e1b72935f18d3e2caad07c51e036991fabd0b1c6c95f876b1382aeb9"},"schema_version":"1.0","source":{"id":"1501.05284","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.05284","created_at":"2026-05-18T00:50:43Z"},{"alias_kind":"arxiv_version","alias_value":"1501.05284v4","created_at":"2026-05-18T00:50:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.05284","created_at":"2026-05-18T00:50:43Z"},{"alias_kind":"pith_short_12","alias_value":"YCLNV34OZSSS","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"YCLNV34OZSSSAPXW","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"YCLNV34O","created_at":"2026-05-18T12:29:50Z"}],"graph_snapshots":[{"event_id":"sha256:f98e4af5e081ea789d7b35cf160732a66893101c293ebb9c1d04b8d3f9ce5102","target":"graph","created_at":"2026-05-18T00:50:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 $","authors_text":"Jakob Grue Simonsen, James Emil Avery, Jean-Yves Moyen, Pavel Ruzicka","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-01-21T20:19:23Z","title":"Chains, Antichains, and Complements in Infinite Partition Lattices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.05284","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:05b2cc7645a7f727216b147bc8b81da23ef7a37fcae49200ee5be84c67b72456","target":"record","created_at":"2026-05-18T00:50:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c6a2ffae071b60e58f6e617f689386e903fa7522b70cf5ff14a19a8f061fd9a1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-01-21T20:19:23Z","title_canon_sha256":"271780c7e1b72935f18d3e2caad07c51e036991fabd0b1c6c95f876b1382aeb9"},"schema_version":"1.0","source":{"id":"1501.05284","kind":"arxiv","version":4}},"canonical_sha256":"c096daef8ecca5203ef6f0e7f3a9e75fc680df5398e6725505b0354b29f07d64","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c096daef8ecca5203ef6f0e7f3a9e75fc680df5398e6725505b0354b29f07d64","first_computed_at":"2026-05-18T00:50:43.202622Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:50:43.202622Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Qg1Yx6Ti6qcDnh8KPJHxoD75ldNgv40prlfMrlu2+hEQeXpgxIG8FiNWEE48LJQFmmR42fXtlxwEUm+BSW+8CA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:50:43.203294Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.05284","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:05b2cc7645a7f727216b147bc8b81da23ef7a37fcae49200ee5be84c67b72456","sha256:f98e4af5e081ea789d7b35cf160732a66893101c293ebb9c1d04b8d3f9ce5102"],"state_sha256":"509cd122cec470b086d3ef12d135bedb160e7840f9c4fd05016dcfe11ed2f9bf"}