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Then the following conditions are equivalent: (i) There exists a finite lattice L such that (J(L),EL) is isomorphic to the quasi-ordered set (P,E). (ii) There are not exactly two elements x in P such that p E x, for any p in P. For a finite lattice L, let je(L) = |J(L)|-|J(Con L)|, where Con L is the congruence"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0501366","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GM","submitted_at":"2005-01-22T07:51:47Z","cross_cats_sorted":[],"title_canon_sha256":"2b99c7b111ed76ac26f2eab605d419c13ae06f38b022698f7ca543bf93186b8c","abstract_canon_sha256":"1ce75029196e63801cb26e05e9c6ebed7391d1415a0ed4cb5514a67ce4cb7813"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:51.429431Z","signature_b64":"Jt/6P/irom++m4pSxXcjj4bPMe8DRqGDQd/kiN7fQLmtDShsElr38lw+n01IpyCKe53Yyh+t9DrQmr2fR3ciDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd79ee871dbf4f0213be376b5b14435ac3db893550543be791b09a19d07e9e37","last_reissued_at":"2026-05-18T01:08:51.428889Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:51.428889Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the number of join-irreducibles in a congruence representation of a finite distributive lattice","license":"","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Friedrich Wehrung (LMNO), George Gr\\\"atzer","submitted_at":"2005-01-22T07:51:47Z","abstract_excerpt":"For a finite lattice L, let EL denote the reflexive and transitive closure of the join-dependency relation on L, defined on the set J(L) of all join-irreducible elements of L. We characterize the relations of the form EL, as follows: Theorem. Let E be a quasi-ordering on a finite set P. Then the following conditions are equivalent: (i) There exists a finite lattice L such that (J(L),EL) is isomorphic to the quasi-ordered set (P,E). (ii) There are not exactly two elements x in P such that p E x, for any p in P. For a finite lattice L, let je(L) = |J(L)|-|J(Con L)|, where Con L is the congruence"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0501366","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0501366","created_at":"2026-05-18T01:08:51.428968+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0501366v1","created_at":"2026-05-18T01:08:51.428968+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0501366","created_at":"2026-05-18T01:08:51.428968+00:00"},{"alias_kind":"pith_short_12","alias_value":"7V465BY5X5HQ","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_16","alias_value":"7V465BY5X5HQEE56","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_8","alias_value":"7V465BY5","created_at":"2026-05-18T12:25:53.335082+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7V465BY5X5HQEE56G5VVWFCDLL","json":"https://pith.science/pith/7V465BY5X5HQEE56G5VVWFCDLL.json","graph_json":"https://pith.science/api/pith-number/7V465BY5X5HQEE56G5VVWFCDLL/graph.json","events_json":"https://pith.science/api/pith-number/7V465BY5X5HQEE56G5VVWFCDLL/events.json","paper":"https://pith.science/paper/7V465BY5"},"agent_actions":{"view_html":"https://pith.science/pith/7V465BY5X5HQEE56G5VVWFCDLL","download_json":"https://pith.science/pith/7V465BY5X5HQEE56G5VVWFCDLL.json","view_paper":"https://pith.science/paper/7V465BY5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0501366&json=true","fetch_graph":"https://pith.science/api/pith-number/7V465BY5X5HQEE56G5VVWFCDLL/graph.json","fetch_events":"https://pith.science/api/pith-number/7V465BY5X5HQEE56G5VVWFCDLL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7V465BY5X5HQEE56G5VVWFCDLL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7V465BY5X5HQEE56G5VVWFCDLL/action/storage_attestation","attest_author":"https://pith.science/pith/7V465BY5X5HQEE56G5VVWFCDLL/action/author_attestation","sign_citation":"https://pith.science/pith/7V465BY5X5HQEE56G5VVWFCDLL/action/citation_signature","submit_replication":"https://pith.science/pith/7V465BY5X5HQEE56G5VVWFCDLL/action/replication_record"}},"created_at":"2026-05-18T01:08:51.428968+00:00","updated_at":"2026-05-18T01:08:51.428968+00:00"}