{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:NA73O54NC4VTSQY7WZ3C3X5K2I","short_pith_number":"pith:NA73O54N","canonical_record":{"source":{"id":"1109.5296","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-24T19:23:08Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"4ad238f41710047b9edcbbe2ab398d118be18b12c939317aa9ba5aa51b7011a4","abstract_canon_sha256":"11c129158674a913c8bff748e2ba8e25647c9409c96e4aae23b850704542a9c5"},"schema_version":"1.0"},"canonical_sha256":"683fb7778d172b39431fb6762ddfaad22c52be12e579470c780057754eb414de","source":{"kind":"arxiv","id":"1109.5296","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.5296","created_at":"2026-05-18T04:12:20Z"},{"alias_kind":"arxiv_version","alias_value":"1109.5296v1","created_at":"2026-05-18T04:12:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.5296","created_at":"2026-05-18T04:12:20Z"},{"alias_kind":"pith_short_12","alias_value":"NA73O54NC4VT","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"NA73O54NC4VTSQY7","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"NA73O54N","created_at":"2026-05-18T12:26:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:NA73O54NC4VTSQY7WZ3C3X5K2I","target":"record","payload":{"canonical_record":{"source":{"id":"1109.5296","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-24T19:23:08Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"4ad238f41710047b9edcbbe2ab398d118be18b12c939317aa9ba5aa51b7011a4","abstract_canon_sha256":"11c129158674a913c8bff748e2ba8e25647c9409c96e4aae23b850704542a9c5"},"schema_version":"1.0"},"canonical_sha256":"683fb7778d172b39431fb6762ddfaad22c52be12e579470c780057754eb414de","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:20.103094Z","signature_b64":"KRPowNe8ht1Y7/ET+E7GlZm3CGhK8IMBAWNVtZg55R0AjeX+jnRMuyfAjxiek2y+0tvZ3AtWo1V5FUexRvbYAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"683fb7778d172b39431fb6762ddfaad22c52be12e579470c780057754eb414de","last_reissued_at":"2026-05-18T04:12:20.102616Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:20.102616Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.5296","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":"2026-05-18T04:12:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dEKIJrw14e/OHlA8/yrQhz9wDhkvCfLqII8tedl56vHsmltzmqN9xP6cZo53jTeIvtqmPk8VqHUs7jxbcYsuDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T20:08:39.498451Z"},"content_sha256":"fe9f066b76a1cff69ccc6dadee46297cdcd40547577b9955f34095db9f54a174","schema_version":"1.0","event_id":"sha256:fe9f066b76a1cff69ccc6dadee46297cdcd40547577b9955f34095db9f54a174"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:NA73O54NC4VTSQY7WZ3C3X5K2I","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tamari Lattices and the symmetric Thompson monoid","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Patrick Dehornoy (LMNO)","submitted_at":"2011-09-24T19:23:08Z","abstract_excerpt":"We investigate the connection between Tamari lattices and the Thompson group F, summarized in the fact that F is a group of fractions for a certain monoid F+sym whose Cayley graph includes all Tamari lattices. Under this correspondence, the Tamari lattice operations are the counterparts of the least common multiple and greatest common divisor operations in F+sym. As an application, we show that, for every n, there exists a length l chain in the nth Tamari lattice whose endpoints are at distance at most 12l/n."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.5296","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":"2026-05-18T04:12:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6b2dQc2GYBU6hn9iedc5GtOVnmNcVaK+ZVZYWFUxncwGGKt7yyOULfcZ2KZvsAFi8PLbzxNSdouHzeeoe3gGDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T20:08:39.498793Z"},"content_sha256":"3809a65ff48af9d5c76cb9fed074c19dc96703f7a4a9a258bf5cf0c264946226","schema_version":"1.0","event_id":"sha256:3809a65ff48af9d5c76cb9fed074c19dc96703f7a4a9a258bf5cf0c264946226"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NA73O54NC4VTSQY7WZ3C3X5K2I/bundle.json","state_url":"https://pith.science/pith/NA73O54NC4VTSQY7WZ3C3X5K2I/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NA73O54NC4VTSQY7WZ3C3X5K2I/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":"2026-06-24T20:08:39Z","links":{"resolver":"https://pith.science/pith/NA73O54NC4VTSQY7WZ3C3X5K2I","bundle":"https://pith.science/pith/NA73O54NC4VTSQY7WZ3C3X5K2I/bundle.json","state":"https://pith.science/pith/NA73O54NC4VTSQY7WZ3C3X5K2I/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NA73O54NC4VTSQY7WZ3C3X5K2I/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:NA73O54NC4VTSQY7WZ3C3X5K2I","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":"11c129158674a913c8bff748e2ba8e25647c9409c96e4aae23b850704542a9c5","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-24T19:23:08Z","title_canon_sha256":"4ad238f41710047b9edcbbe2ab398d118be18b12c939317aa9ba5aa51b7011a4"},"schema_version":"1.0","source":{"id":"1109.5296","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.5296","created_at":"2026-05-18T04:12:20Z"},{"alias_kind":"arxiv_version","alias_value":"1109.5296v1","created_at":"2026-05-18T04:12:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.5296","created_at":"2026-05-18T04:12:20Z"},{"alias_kind":"pith_short_12","alias_value":"NA73O54NC4VT","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"NA73O54NC4VTSQY7","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"NA73O54N","created_at":"2026-05-18T12:26:37Z"}],"graph_snapshots":[{"event_id":"sha256:3809a65ff48af9d5c76cb9fed074c19dc96703f7a4a9a258bf5cf0c264946226","target":"graph","created_at":"2026-05-18T04:12:20Z","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 investigate the connection between Tamari lattices and the Thompson group F, summarized in the fact that F is a group of fractions for a certain monoid F+sym whose Cayley graph includes all Tamari lattices. Under this correspondence, the Tamari lattice operations are the counterparts of the least common multiple and greatest common divisor operations in F+sym. As an application, we show that, for every n, there exists a length l chain in the nth Tamari lattice whose endpoints are at distance at most 12l/n.","authors_text":"Patrick Dehornoy (LMNO)","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-24T19:23:08Z","title":"Tamari Lattices and the symmetric Thompson monoid"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.5296","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:fe9f066b76a1cff69ccc6dadee46297cdcd40547577b9955f34095db9f54a174","target":"record","created_at":"2026-05-18T04:12:20Z","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":"11c129158674a913c8bff748e2ba8e25647c9409c96e4aae23b850704542a9c5","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-24T19:23:08Z","title_canon_sha256":"4ad238f41710047b9edcbbe2ab398d118be18b12c939317aa9ba5aa51b7011a4"},"schema_version":"1.0","source":{"id":"1109.5296","kind":"arxiv","version":1}},"canonical_sha256":"683fb7778d172b39431fb6762ddfaad22c52be12e579470c780057754eb414de","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"683fb7778d172b39431fb6762ddfaad22c52be12e579470c780057754eb414de","first_computed_at":"2026-05-18T04:12:20.102616Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:12:20.102616Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KRPowNe8ht1Y7/ET+E7GlZm3CGhK8IMBAWNVtZg55R0AjeX+jnRMuyfAjxiek2y+0tvZ3AtWo1V5FUexRvbYAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:12:20.103094Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.5296","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fe9f066b76a1cff69ccc6dadee46297cdcd40547577b9955f34095db9f54a174","sha256:3809a65ff48af9d5c76cb9fed074c19dc96703f7a4a9a258bf5cf0c264946226"],"state_sha256":"fcc2da8080ab583a909d1997f028fd7667e6df88f37bbbc7f23315c841da2fa2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gM3g1AV94uWTWlQGk0Xrmi2Sm4Occqz5Ks6V6DPoG9DDilV/I03ybuuGtondOOKNd3Kvg8k6/sPnY/1m+4GXBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T20:08:39.500928Z","bundle_sha256":"ae670b01deb1e9db5d48d5f2f4ff87579b9c87de04349f9ab904adfe10f9e96f"}}