{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2005:7V465BY5X5HQEE56G5VVWFCDLL","short_pith_number":"pith:7V465BY5","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"},"canonical_sha256":"fd79ee871dbf4f0213be376b5b14435ac3db893550543be791b09a19d07e9e37","source":{"kind":"arxiv","id":"math/0501366","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0501366","created_at":"2026-05-18T01:08:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/0501366v1","created_at":"2026-05-18T01:08:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0501366","created_at":"2026-05-18T01:08:51Z"},{"alias_kind":"pith_short_12","alias_value":"7V465BY5X5HQ","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"7V465BY5X5HQEE56","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"7V465BY5","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2005:7V465BY5X5HQEE56G5VVWFCDLL","target":"record","payload":{"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"},"canonical_sha256":"fd79ee871dbf4f0213be376b5b14435ac3db893550543be791b09a19d07e9e37","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"},"source_kind":"arxiv","source_id":"math/0501366","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:08:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A0Do4K1BX3e5u+s2EGF/gdpFuIAF+Fq5Axl2Sy6QrUzapkgy88Vh5gbRdG7roXLjyCXYAfR7djPb5GFze9/OBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T03:18:05.380670Z"},"content_sha256":"371073396e80ed7bb6999e55bab0363bf744e0b1cde77fd490fbd68adc65db66","schema_version":"1.0","event_id":"sha256:371073396e80ed7bb6999e55bab0363bf744e0b1cde77fd490fbd68adc65db66"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2005:7V465BY5X5HQEE56G5VVWFCDLL","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:08:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u5iVBxGHdOSin6jbuj5Htb0nMrstz41mhMceZZRtk8OCCmBBLTAPsgc8HY/myentyrTbR9TOf4TRtDvGCZDVCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T03:18:05.381182Z"},"content_sha256":"5f01dc9ece266562d6ddcb4be38fe3a9d2d28658e4bfa22a54196de7b7ecdde9","schema_version":"1.0","event_id":"sha256:5f01dc9ece266562d6ddcb4be38fe3a9d2d28658e4bfa22a54196de7b7ecdde9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7V465BY5X5HQEE56G5VVWFCDLL/bundle.json","state_url":"https://pith.science/pith/7V465BY5X5HQEE56G5VVWFCDLL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7V465BY5X5HQEE56G5VVWFCDLL/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-27T03:18:05Z","links":{"resolver":"https://pith.science/pith/7V465BY5X5HQEE56G5VVWFCDLL","bundle":"https://pith.science/pith/7V465BY5X5HQEE56G5VVWFCDLL/bundle.json","state":"https://pith.science/pith/7V465BY5X5HQEE56G5VVWFCDLL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7V465BY5X5HQEE56G5VVWFCDLL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:7V465BY5X5HQEE56G5VVWFCDLL","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":"1ce75029196e63801cb26e05e9c6ebed7391d1415a0ed4cb5514a67ce4cb7813","cross_cats_sorted":[],"license":"","primary_cat":"math.GM","submitted_at":"2005-01-22T07:51:47Z","title_canon_sha256":"2b99c7b111ed76ac26f2eab605d419c13ae06f38b022698f7ca543bf93186b8c"},"schema_version":"1.0","source":{"id":"math/0501366","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0501366","created_at":"2026-05-18T01:08:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/0501366v1","created_at":"2026-05-18T01:08:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0501366","created_at":"2026-05-18T01:08:51Z"},{"alias_kind":"pith_short_12","alias_value":"7V465BY5X5HQ","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"7V465BY5X5HQEE56","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"7V465BY5","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:5f01dc9ece266562d6ddcb4be38fe3a9d2d28658e4bfa22a54196de7b7ecdde9","target":"graph","created_at":"2026-05-18T01:08:51Z","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":"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","authors_text":"Friedrich Wehrung (LMNO), George Gr\\\"atzer","cross_cats":[],"headline":"","license":"","primary_cat":"math.GM","submitted_at":"2005-01-22T07:51:47Z","title":"On the number of join-irreducibles in a congruence representation of a finite distributive lattice"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0501366","kind":"arxiv","version":1},"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":"sha256:371073396e80ed7bb6999e55bab0363bf744e0b1cde77fd490fbd68adc65db66","target":"record","created_at":"2026-05-18T01:08:51Z","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":"1ce75029196e63801cb26e05e9c6ebed7391d1415a0ed4cb5514a67ce4cb7813","cross_cats_sorted":[],"license":"","primary_cat":"math.GM","submitted_at":"2005-01-22T07:51:47Z","title_canon_sha256":"2b99c7b111ed76ac26f2eab605d419c13ae06f38b022698f7ca543bf93186b8c"},"schema_version":"1.0","source":{"id":"math/0501366","kind":"arxiv","version":1}},"canonical_sha256":"fd79ee871dbf4f0213be376b5b14435ac3db893550543be791b09a19d07e9e37","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fd79ee871dbf4f0213be376b5b14435ac3db893550543be791b09a19d07e9e37","first_computed_at":"2026-05-18T01:08:51.428889Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:08:51.428889Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Jt/6P/irom++m4pSxXcjj4bPMe8DRqGDQd/kiN7fQLmtDShsElr38lw+n01IpyCKe53Yyh+t9DrQmr2fR3ciDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:08:51.429431Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0501366","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:371073396e80ed7bb6999e55bab0363bf744e0b1cde77fd490fbd68adc65db66","sha256:5f01dc9ece266562d6ddcb4be38fe3a9d2d28658e4bfa22a54196de7b7ecdde9"],"state_sha256":"b05843771d5dbc023c2ee82d379044f19c08e7a06ddebff1bc93458078ad7038"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J7/lWFZ3M+n0Mu6c3vXxb7Q+SjqhwBtopp2j7yTPjhRk5p16G9TYLanKI8RMmJUphZluXpuzoVKCEfpLJeR0BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T03:18:05.384486Z","bundle_sha256":"f602afc3c63eac39c647dd3141fedec0fb89f870ae424a6fa681225f6c1a3c09"}}