{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:F34RMDAQV66MUCQIW3NVIO2UNP","short_pith_number":"pith:F34RMDAQ","canonical_record":{"source":{"id":"1801.05282","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-01-16T15:13:51Z","cross_cats_sorted":[],"title_canon_sha256":"be2cf5616ffe7bb0ed8292f80ccf0aa00dc194b877d26867e74509b2ee003ef7","abstract_canon_sha256":"b81718669d087d8884ec06dd5b3482ee9a9de32183d7c9b1f7b639b549e2b8c0"},"schema_version":"1.0"},"canonical_sha256":"2ef9160c10afbcca0a08b6db543b546bcae5faa50dfa6c6da2f6a8368b814faa","source":{"kind":"arxiv","id":"1801.05282","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.05282","created_at":"2026-05-18T00:25:33Z"},{"alias_kind":"arxiv_version","alias_value":"1801.05282v2","created_at":"2026-05-18T00:25:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.05282","created_at":"2026-05-18T00:25:33Z"},{"alias_kind":"pith_short_12","alias_value":"F34RMDAQV66M","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_16","alias_value":"F34RMDAQV66MUCQI","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_8","alias_value":"F34RMDAQ","created_at":"2026-05-18T12:32:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:F34RMDAQV66MUCQIW3NVIO2UNP","target":"record","payload":{"canonical_record":{"source":{"id":"1801.05282","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-01-16T15:13:51Z","cross_cats_sorted":[],"title_canon_sha256":"be2cf5616ffe7bb0ed8292f80ccf0aa00dc194b877d26867e74509b2ee003ef7","abstract_canon_sha256":"b81718669d087d8884ec06dd5b3482ee9a9de32183d7c9b1f7b639b549e2b8c0"},"schema_version":"1.0"},"canonical_sha256":"2ef9160c10afbcca0a08b6db543b546bcae5faa50dfa6c6da2f6a8368b814faa","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:25:33.312269Z","signature_b64":"T7mfvrl9uo7ppvvdmLMoz5w763kusnYryslWVdmK/hZ/yBQC7FaLtQDCSRt0dzwrURf6Ybluuv5I0BLy2e6GCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ef9160c10afbcca0a08b6db543b546bcae5faa50dfa6c6da2f6a8368b814faa","last_reissued_at":"2026-05-18T00:25:33.311573Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:25:33.311573Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1801.05282","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:25:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z3woUsxMG8h2D/2DkK85eOldOMDDhqe6E6T5HHoWFa8bMSZdpLJtRUpN/MawyZjcNyPr0n6vxZUvgvfp8cylCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T04:58:50.104170Z"},"content_sha256":"5dbcab3c553c0a6312e55a813d3f7c309d2341d75f827599baa89dbdb64dd03c","schema_version":"1.0","event_id":"sha256:5dbcab3c553c0a6312e55a813d3f7c309d2341d75f827599baa89dbdb64dd03c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:F34RMDAQV66MUCQIW3NVIO2UNP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some Extremal Values of the Number of Congruences of a Finite Lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Claudia Mure\\c{s}an, J\\' ulia Kulin","submitted_at":"2018-01-16T15:13:51Z","abstract_excerpt":"We study the smallest, as well as the largest numbers of congruences of lattices of an arbitrary finite cardinality $n$. Continuing the work of Freese and Cz\\' edli, we prove that the third, fourth and fifth largest numbers of congruences of an $n$--element lattice are: $5\\cdot 2^{n-5}$ if $n\\geq 5$, respectively $2^{n-3}$ and $7\\cdot 2^{n-6}$ if $n\\geq 6$. We also determine the structures of the $n$--element lattices having $5\\cdot 2^{n-5}$, respectively $2^{n-3}$ congruences, along with the structures of their congruence lattices."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.05282","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:25:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TIjjr8Ai6qahzoq5NFv01VfhDYeiCadKsuR23qd4zz2opf3NhMzJgt1Sca3Ze3f+giqx/vq1LUKDGINVtNUYCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T04:58:50.104823Z"},"content_sha256":"2e7259eb7ccc680de4e9862eddb901007fab126620714992e25b60400c2059cd","schema_version":"1.0","event_id":"sha256:2e7259eb7ccc680de4e9862eddb901007fab126620714992e25b60400c2059cd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F34RMDAQV66MUCQIW3NVIO2UNP/bundle.json","state_url":"https://pith.science/pith/F34RMDAQV66MUCQIW3NVIO2UNP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F34RMDAQV66MUCQIW3NVIO2UNP/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":"2026-06-11T04:58:50Z","links":{"resolver":"https://pith.science/pith/F34RMDAQV66MUCQIW3NVIO2UNP","bundle":"https://pith.science/pith/F34RMDAQV66MUCQIW3NVIO2UNP/bundle.json","state":"https://pith.science/pith/F34RMDAQV66MUCQIW3NVIO2UNP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F34RMDAQV66MUCQIW3NVIO2UNP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:F34RMDAQV66MUCQIW3NVIO2UNP","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":"b81718669d087d8884ec06dd5b3482ee9a9de32183d7c9b1f7b639b549e2b8c0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-01-16T15:13:51Z","title_canon_sha256":"be2cf5616ffe7bb0ed8292f80ccf0aa00dc194b877d26867e74509b2ee003ef7"},"schema_version":"1.0","source":{"id":"1801.05282","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.05282","created_at":"2026-05-18T00:25:33Z"},{"alias_kind":"arxiv_version","alias_value":"1801.05282v2","created_at":"2026-05-18T00:25:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.05282","created_at":"2026-05-18T00:25:33Z"},{"alias_kind":"pith_short_12","alias_value":"F34RMDAQV66M","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_16","alias_value":"F34RMDAQV66MUCQI","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_8","alias_value":"F34RMDAQ","created_at":"2026-05-18T12:32:22Z"}],"graph_snapshots":[{"event_id":"sha256:2e7259eb7ccc680de4e9862eddb901007fab126620714992e25b60400c2059cd","target":"graph","created_at":"2026-05-18T00:25:33Z","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 study the smallest, as well as the largest numbers of congruences of lattices of an arbitrary finite cardinality $n$. Continuing the work of Freese and Cz\\' edli, we prove that the third, fourth and fifth largest numbers of congruences of an $n$--element lattice are: $5\\cdot 2^{n-5}$ if $n\\geq 5$, respectively $2^{n-3}$ and $7\\cdot 2^{n-6}$ if $n\\geq 6$. We also determine the structures of the $n$--element lattices having $5\\cdot 2^{n-5}$, respectively $2^{n-3}$ congruences, along with the structures of their congruence lattices.","authors_text":"Claudia Mure\\c{s}an, J\\' ulia Kulin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-01-16T15:13:51Z","title":"Some Extremal Values of the Number of Congruences of a Finite Lattice"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.05282","kind":"arxiv","version":2},"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":"sha256:5dbcab3c553c0a6312e55a813d3f7c309d2341d75f827599baa89dbdb64dd03c","target":"record","created_at":"2026-05-18T00:25:33Z","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":"b81718669d087d8884ec06dd5b3482ee9a9de32183d7c9b1f7b639b549e2b8c0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-01-16T15:13:51Z","title_canon_sha256":"be2cf5616ffe7bb0ed8292f80ccf0aa00dc194b877d26867e74509b2ee003ef7"},"schema_version":"1.0","source":{"id":"1801.05282","kind":"arxiv","version":2}},"canonical_sha256":"2ef9160c10afbcca0a08b6db543b546bcae5faa50dfa6c6da2f6a8368b814faa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2ef9160c10afbcca0a08b6db543b546bcae5faa50dfa6c6da2f6a8368b814faa","first_computed_at":"2026-05-18T00:25:33.311573Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:25:33.311573Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"T7mfvrl9uo7ppvvdmLMoz5w763kusnYryslWVdmK/hZ/yBQC7FaLtQDCSRt0dzwrURf6Ybluuv5I0BLy2e6GCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:25:33.312269Z","signed_message":"canonical_sha256_bytes"},"source_id":"1801.05282","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5dbcab3c553c0a6312e55a813d3f7c309d2341d75f827599baa89dbdb64dd03c","sha256:2e7259eb7ccc680de4e9862eddb901007fab126620714992e25b60400c2059cd"],"state_sha256":"3339a1ab54a85d307dc813c7dce077167451103f426c6e6d9599e3f9f92d6b45"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nhYWrywGDY6LuwOwxxPeZLdQxRehfgVlmFU/vYvmtR0jlzBYLraoVK+s0m+YO4iHYifsLtTx34v7QaybKMypBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T04:58:50.108168Z","bundle_sha256":"fae8959db40064ff453cb291ea56df6c29a77fed8652c05778bd2331728cf54d"}}