{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:KDVICVYEP7VV5446HLH3ODB5SV","short_pith_number":"pith:KDVICVYE","canonical_record":{"source":{"id":"1301.6023","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-25T12:05:50Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"7cdbf8f889badc5f520941bb8f405016944a7264780ed2ab057faaf7197165f2","abstract_canon_sha256":"398ed891e9ef292827d267cd8de2785dbdf06b705d854cd41c3f5f1f60f2bd25"},"schema_version":"1.0"},"canonical_sha256":"50ea8157047feb5ef39e3acfb70c3d956db3005f1839569bcbfacb9ea4c505e6","source":{"kind":"arxiv","id":"1301.6023","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1301.6023","created_at":"2026-05-18T02:54:12Z"},{"alias_kind":"arxiv_version","alias_value":"1301.6023v1","created_at":"2026-05-18T02:54:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.6023","created_at":"2026-05-18T02:54:12Z"},{"alias_kind":"pith_short_12","alias_value":"KDVICVYEP7VV","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"KDVICVYEP7VV5446","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"KDVICVYE","created_at":"2026-05-18T12:27:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:KDVICVYEP7VV5446HLH3ODB5SV","target":"record","payload":{"canonical_record":{"source":{"id":"1301.6023","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-25T12:05:50Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"7cdbf8f889badc5f520941bb8f405016944a7264780ed2ab057faaf7197165f2","abstract_canon_sha256":"398ed891e9ef292827d267cd8de2785dbdf06b705d854cd41c3f5f1f60f2bd25"},"schema_version":"1.0"},"canonical_sha256":"50ea8157047feb5ef39e3acfb70c3d956db3005f1839569bcbfacb9ea4c505e6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:12.832385Z","signature_b64":"SmFGZZzqjrlItuji3Nsxv8tqdGN2waJAkXklyc9OBRti9zxm9gwhNyWamgWV8DEfSNUZeXmZjC9RAKTXkjmLCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50ea8157047feb5ef39e3acfb70c3d956db3005f1839569bcbfacb9ea4c505e6","last_reissued_at":"2026-05-18T02:54:12.831847Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:12.831847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1301.6023","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":"2026-05-18T02:54:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ymQ+rJ6dVcjdENPtlOrSUBUePCo7JBCRzGuf6fmzlrfnsexPwJoK6h42xdsW+91IL/077G2mQbbSb7muuW8ABw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T16:28:50.520906Z"},"content_sha256":"6fbdedca8825ea3ac5d285970206ea5ba972b13e6bd379864bb9c4e4ced2a6d4","schema_version":"1.0","event_id":"sha256:6fbdedca8825ea3ac5d285970206ea5ba972b13e6bd379864bb9c4e4ced2a6d4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:KDVICVYEP7VV5446HLH3ODB5SV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The semigroups of order 9 and their automorphism groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.CO","authors_text":"Andreas Distler, Tom Kelsey","submitted_at":"2013-01-25T12:05:50Z","abstract_excerpt":"We report the number of semigroups with 9 elements up to isomorphism or anti-isomorphism to be 52,989,400,714,478 and up to isomorphism to be 105,978,177,936,292. We obtained these results by combining computer search with recently published formulae for the number of nilpotent semigroups of degree 3. We further provide a complete account of the automorphism groups of the semigroups with at most 9 elements. We use this information to deduce that there are 148,195,347,518,186 distinct associative binary operations on an 8-element set and 38,447,365,355,811,944,462 on a 9-element set."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.6023","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":"2026-05-18T02:54:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K41FlyR9jTnPe0raoFymYJYfrk4iDPlOPlqNbOmyw51NnnZWQf7HD79uVdbME3j5nd2IuDdOk8N381F4EvSEDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T16:28:50.521522Z"},"content_sha256":"9ffa471740afc1c244173287c9b9bf95bbd0ca3982abf77ff8dc3af146b23413","schema_version":"1.0","event_id":"sha256:9ffa471740afc1c244173287c9b9bf95bbd0ca3982abf77ff8dc3af146b23413"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KDVICVYEP7VV5446HLH3ODB5SV/bundle.json","state_url":"https://pith.science/pith/KDVICVYEP7VV5446HLH3ODB5SV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KDVICVYEP7VV5446HLH3ODB5SV/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":"2026-05-31T16:28:50Z","links":{"resolver":"https://pith.science/pith/KDVICVYEP7VV5446HLH3ODB5SV","bundle":"https://pith.science/pith/KDVICVYEP7VV5446HLH3ODB5SV/bundle.json","state":"https://pith.science/pith/KDVICVYEP7VV5446HLH3ODB5SV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KDVICVYEP7VV5446HLH3ODB5SV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:KDVICVYEP7VV5446HLH3ODB5SV","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":"398ed891e9ef292827d267cd8de2785dbdf06b705d854cd41c3f5f1f60f2bd25","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-25T12:05:50Z","title_canon_sha256":"7cdbf8f889badc5f520941bb8f405016944a7264780ed2ab057faaf7197165f2"},"schema_version":"1.0","source":{"id":"1301.6023","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1301.6023","created_at":"2026-05-18T02:54:12Z"},{"alias_kind":"arxiv_version","alias_value":"1301.6023v1","created_at":"2026-05-18T02:54:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.6023","created_at":"2026-05-18T02:54:12Z"},{"alias_kind":"pith_short_12","alias_value":"KDVICVYEP7VV","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"KDVICVYEP7VV5446","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"KDVICVYE","created_at":"2026-05-18T12:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:9ffa471740afc1c244173287c9b9bf95bbd0ca3982abf77ff8dc3af146b23413","target":"graph","created_at":"2026-05-18T02:54:12Z","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 report the number of semigroups with 9 elements up to isomorphism or anti-isomorphism to be 52,989,400,714,478 and up to isomorphism to be 105,978,177,936,292. We obtained these results by combining computer search with recently published formulae for the number of nilpotent semigroups of degree 3. We further provide a complete account of the automorphism groups of the semigroups with at most 9 elements. We use this information to deduce that there are 148,195,347,518,186 distinct associative binary operations on an 8-element set and 38,447,365,355,811,944,462 on a 9-element set.","authors_text":"Andreas Distler, Tom Kelsey","cross_cats":["math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-25T12:05:50Z","title":"The semigroups of order 9 and their automorphism groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.6023","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":"sha256:6fbdedca8825ea3ac5d285970206ea5ba972b13e6bd379864bb9c4e4ced2a6d4","target":"record","created_at":"2026-05-18T02:54:12Z","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":"398ed891e9ef292827d267cd8de2785dbdf06b705d854cd41c3f5f1f60f2bd25","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-25T12:05:50Z","title_canon_sha256":"7cdbf8f889badc5f520941bb8f405016944a7264780ed2ab057faaf7197165f2"},"schema_version":"1.0","source":{"id":"1301.6023","kind":"arxiv","version":1}},"canonical_sha256":"50ea8157047feb5ef39e3acfb70c3d956db3005f1839569bcbfacb9ea4c505e6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"50ea8157047feb5ef39e3acfb70c3d956db3005f1839569bcbfacb9ea4c505e6","first_computed_at":"2026-05-18T02:54:12.831847Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:54:12.831847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SmFGZZzqjrlItuji3Nsxv8tqdGN2waJAkXklyc9OBRti9zxm9gwhNyWamgWV8DEfSNUZeXmZjC9RAKTXkjmLCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:54:12.832385Z","signed_message":"canonical_sha256_bytes"},"source_id":"1301.6023","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6fbdedca8825ea3ac5d285970206ea5ba972b13e6bd379864bb9c4e4ced2a6d4","sha256:9ffa471740afc1c244173287c9b9bf95bbd0ca3982abf77ff8dc3af146b23413"],"state_sha256":"722b1929594f50e0dbd035a3c715f99e8442022cba53a54ae863911c2d15fefe"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FcNnZLRIUfqkpV04O1Ux0IdhErwmAQnlf6LsC+xegBj6q3j2cP3FCRINDhEWilf61hBN5m3gZ+E3QeCR3F0dBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T16:28:50.524860Z","bundle_sha256":"cc3929c6eaf0e28020d7f26c1b40a529084a5ca2f4b49ca8d6f706f58e6b1083"}}