{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:TPEY3DPKVP56L3RINCDB57DZFZ","short_pith_number":"pith:TPEY3DPK","canonical_record":{"source":{"id":"1111.7191","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-11-28T04:07:34Z","cross_cats_sorted":["hep-th","math-ph","math.MP"],"title_canon_sha256":"f3d5d8c29cea57a3a4315f8fecaa899ef0b71878880daac6633d18a2e206679a","abstract_canon_sha256":"50f6e7fb0193a4277409b506906dbe3ce014f688e997e12f79d8baf1fe5eb47a"},"schema_version":"1.0"},"canonical_sha256":"9bc98d8deaabfbe5ee2868861efc792e55452813b22e1fd66587fc3aebcf3233","source":{"kind":"arxiv","id":"1111.7191","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.7191","created_at":"2026-05-18T04:07:18Z"},{"alias_kind":"arxiv_version","alias_value":"1111.7191v1","created_at":"2026-05-18T04:07:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.7191","created_at":"2026-05-18T04:07:18Z"},{"alias_kind":"pith_short_12","alias_value":"TPEY3DPKVP56","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"TPEY3DPKVP56L3RI","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"TPEY3DPK","created_at":"2026-05-18T12:26:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:TPEY3DPKVP56L3RINCDB57DZFZ","target":"record","payload":{"canonical_record":{"source":{"id":"1111.7191","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-11-28T04:07:34Z","cross_cats_sorted":["hep-th","math-ph","math.MP"],"title_canon_sha256":"f3d5d8c29cea57a3a4315f8fecaa899ef0b71878880daac6633d18a2e206679a","abstract_canon_sha256":"50f6e7fb0193a4277409b506906dbe3ce014f688e997e12f79d8baf1fe5eb47a"},"schema_version":"1.0"},"canonical_sha256":"9bc98d8deaabfbe5ee2868861efc792e55452813b22e1fd66587fc3aebcf3233","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:07:18.527411Z","signature_b64":"Nv5tPTFIxW8PEsLCwpMIFRoaMrxZY0DW0TWZ3FcqblKbkNcjSPX/qiW4+EC7GZy2gxdFhuTOkgESvKYDCAL+Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9bc98d8deaabfbe5ee2868861efc792e55452813b22e1fd66587fc3aebcf3233","last_reissued_at":"2026-05-18T04:07:18.526869Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:07:18.526869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.7191","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":"2026-05-18T04:07:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vfgk7shhsCsoaw2EXaBSDB2Psfhtochjpb+vWf94zUA5Lv04mQB7wwi9Hc4+GwfX4sKk2ssA5KZx8CZVqPMiDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T18:36:15.249130Z"},"content_sha256":"3e3fac8033ba933b9292d706547c86775e9af6380062b90e78941520b0ddfb08","schema_version":"1.0","event_id":"sha256:3e3fac8033ba933b9292d706547c86775e9af6380062b90e78941520b0ddfb08"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:TPEY3DPKVP56L3RINCDB57DZFZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"N-ary Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math-ph","math.MP"],"primary_cat":"math.GR","authors_text":"Alexander M. Gal'mak (Department of Mathematics, Belarus), Mogilev, Mogilev State University","submitted_at":"2011-11-28T04:07:34Z","abstract_excerpt":"The book \"N-ary Groups\" (in Russian) consists of two Parts. It is intended on the one hand as an initial introduction to the theory of n-ary groups, and on the other hand it contains the published results by the author on this subject. At present, the theory of n-ary groups developing but slowly from group theory. Nonetheless, ternary and n-ary structures have recently been applied to modern models of elementary particle physics. One of the author's goals in this book is to draw the attention of mathematicians and theoretical physicists to the theory of n-ary groups, to some of its distinguish"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.7191","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":"2026-05-18T04:07:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L5XQB9vlL2Hv10vqulnryBXQcqDZj78LH+UdbZQRmV26Gch9KuBauOr4w4lWNF1jj0WTT6zE4s06eU9F9/lNDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T18:36:15.249503Z"},"content_sha256":"77620a0ac76a2ea5f3e130244f5ed0ec46d95335ed6dfd3b7ab610b26fa0c200","schema_version":"1.0","event_id":"sha256:77620a0ac76a2ea5f3e130244f5ed0ec46d95335ed6dfd3b7ab610b26fa0c200"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TPEY3DPKVP56L3RINCDB57DZFZ/bundle.json","state_url":"https://pith.science/pith/TPEY3DPKVP56L3RINCDB57DZFZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TPEY3DPKVP56L3RINCDB57DZFZ/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":"2026-06-03T18:36:15Z","links":{"resolver":"https://pith.science/pith/TPEY3DPKVP56L3RINCDB57DZFZ","bundle":"https://pith.science/pith/TPEY3DPKVP56L3RINCDB57DZFZ/bundle.json","state":"https://pith.science/pith/TPEY3DPKVP56L3RINCDB57DZFZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TPEY3DPKVP56L3RINCDB57DZFZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:TPEY3DPKVP56L3RINCDB57DZFZ","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":"50f6e7fb0193a4277409b506906dbe3ce014f688e997e12f79d8baf1fe5eb47a","cross_cats_sorted":["hep-th","math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-11-28T04:07:34Z","title_canon_sha256":"f3d5d8c29cea57a3a4315f8fecaa899ef0b71878880daac6633d18a2e206679a"},"schema_version":"1.0","source":{"id":"1111.7191","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.7191","created_at":"2026-05-18T04:07:18Z"},{"alias_kind":"arxiv_version","alias_value":"1111.7191v1","created_at":"2026-05-18T04:07:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.7191","created_at":"2026-05-18T04:07:18Z"},{"alias_kind":"pith_short_12","alias_value":"TPEY3DPKVP56","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"TPEY3DPKVP56L3RI","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"TPEY3DPK","created_at":"2026-05-18T12:26:42Z"}],"graph_snapshots":[{"event_id":"sha256:77620a0ac76a2ea5f3e130244f5ed0ec46d95335ed6dfd3b7ab610b26fa0c200","target":"graph","created_at":"2026-05-18T04:07:18Z","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":"The book \"N-ary Groups\" (in Russian) consists of two Parts. It is intended on the one hand as an initial introduction to the theory of n-ary groups, and on the other hand it contains the published results by the author on this subject. At present, the theory of n-ary groups developing but slowly from group theory. Nonetheless, ternary and n-ary structures have recently been applied to modern models of elementary particle physics. One of the author's goals in this book is to draw the attention of mathematicians and theoretical physicists to the theory of n-ary groups, to some of its distinguish","authors_text":"Alexander M. Gal'mak (Department of Mathematics, Belarus), Mogilev, Mogilev State University","cross_cats":["hep-th","math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-11-28T04:07:34Z","title":"N-ary Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.7191","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":"sha256:3e3fac8033ba933b9292d706547c86775e9af6380062b90e78941520b0ddfb08","target":"record","created_at":"2026-05-18T04:07:18Z","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":"50f6e7fb0193a4277409b506906dbe3ce014f688e997e12f79d8baf1fe5eb47a","cross_cats_sorted":["hep-th","math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-11-28T04:07:34Z","title_canon_sha256":"f3d5d8c29cea57a3a4315f8fecaa899ef0b71878880daac6633d18a2e206679a"},"schema_version":"1.0","source":{"id":"1111.7191","kind":"arxiv","version":1}},"canonical_sha256":"9bc98d8deaabfbe5ee2868861efc792e55452813b22e1fd66587fc3aebcf3233","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9bc98d8deaabfbe5ee2868861efc792e55452813b22e1fd66587fc3aebcf3233","first_computed_at":"2026-05-18T04:07:18.526869Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:07:18.526869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Nv5tPTFIxW8PEsLCwpMIFRoaMrxZY0DW0TWZ3FcqblKbkNcjSPX/qiW4+EC7GZy2gxdFhuTOkgESvKYDCAL+Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T04:07:18.527411Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.7191","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e3fac8033ba933b9292d706547c86775e9af6380062b90e78941520b0ddfb08","sha256:77620a0ac76a2ea5f3e130244f5ed0ec46d95335ed6dfd3b7ab610b26fa0c200"],"state_sha256":"aa08f9b4c7d435a44b4bd373a569e3f54fcc5ca98d9c79ebd77ecb9a57ad7cfd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3dJGHBSN+vjHxnpAvpbQ+NrfQ+Yu1levNn1XHG0C3MOzeZB1sphbpXb2+MDqSR1J7kH4JukEv5mm5iQgPlxmAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T18:36:15.251444Z","bundle_sha256":"2a320f08c0fae3d6928d6761c92daa3ef742bab7b7fa64ec1331f3c95741226b"}}