{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:F2WL5X2ATCJMDFXNIFQURSPP5I","short_pith_number":"pith:F2WL5X2A","canonical_record":{"source":{"id":"1409.1044","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-09-03T11:43:33Z","cross_cats_sorted":[],"title_canon_sha256":"85bb79b7e2ab9dc90974c999d40b68ce99287364541bf8a8c4a442e9247ac0a5","abstract_canon_sha256":"e20c6d2d0da44fdb2fa15f3d7849955daaf1a117ed668b86ecbd338bc9fbaeb9"},"schema_version":"1.0"},"canonical_sha256":"2eacbedf409892c196ed416148c9efea08d93a2d81b499371f387ad5813f4379","source":{"kind":"arxiv","id":"1409.1044","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.1044","created_at":"2026-05-18T02:43:44Z"},{"alias_kind":"arxiv_version","alias_value":"1409.1044v1","created_at":"2026-05-18T02:43:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.1044","created_at":"2026-05-18T02:43:44Z"},{"alias_kind":"pith_short_12","alias_value":"F2WL5X2ATCJM","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"F2WL5X2ATCJMDFXN","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"F2WL5X2A","created_at":"2026-05-18T12:28:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:F2WL5X2ATCJMDFXNIFQURSPP5I","target":"record","payload":{"canonical_record":{"source":{"id":"1409.1044","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-09-03T11:43:33Z","cross_cats_sorted":[],"title_canon_sha256":"85bb79b7e2ab9dc90974c999d40b68ce99287364541bf8a8c4a442e9247ac0a5","abstract_canon_sha256":"e20c6d2d0da44fdb2fa15f3d7849955daaf1a117ed668b86ecbd338bc9fbaeb9"},"schema_version":"1.0"},"canonical_sha256":"2eacbedf409892c196ed416148c9efea08d93a2d81b499371f387ad5813f4379","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:44.486451Z","signature_b64":"NRT1FrZk7lHTcEza2UqIif2YdjUhhA5UKdpnsHqZLVnWTOjy8UMZEqfeeFm5aXFgk0ZPVXcCrSZhmk43LI9vAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2eacbedf409892c196ed416148c9efea08d93a2d81b499371f387ad5813f4379","last_reissued_at":"2026-05-18T02:43:44.485978Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:44.485978Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1409.1044","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":"2026-05-18T02:43:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HAMAfo7kBq1Pa7VVQPKI5fqJTRhoJyrX67wfOeYUh+YGfeWQoowDwEmtQoeFpjY4M5i/1Zgz3L2wXDEGoaXtDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T20:46:29.566436Z"},"content_sha256":"8fe225de898254ea088f8a8d8b5766555541807d2170ea9e2ac9e8e6bd118e4b","schema_version":"1.0","event_id":"sha256:8fe225de898254ea088f8a8d8b5766555541807d2170ea9e2ac9e8e6bd118e4b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:F2WL5X2ATCJMDFXNIFQURSPP5I","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ends of Semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"J. D. Mitchell, N. Ruskuc, R. Gray, S. Craik, V. Kilibarda","submitted_at":"2014-09-03T11:43:33Z","abstract_excerpt":"We define the notion of the partial order of ends of the Cayley graph of a semigroup. We prove that the structure of the ends of a semigroup is invariant under change of finite generating set and at the same time is inherited by subsemigroups and extensions of finite Rees index. We prove an analogue of Hopf's Theorem, stating that a group has 1, 2 or infinitely many ends, for left cancellative semigroups and that the cardinality of the set of ends is invariant in subsemigroups and extension of finite Green index in left cancellative semigroups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.1044","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":"2026-05-18T02:43:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yQ/YUxH6P58Lz3zcSwaqRxuLYj6KD76UGvv5mXPWifCursCdx73ngsGzcm6jRaCWneBYjAkvw5CU+UyL4kSxCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T20:46:29.566786Z"},"content_sha256":"7500724811144611d636aa4fb10ca818ec36d1a8d15809f9bc30d95133ea9638","schema_version":"1.0","event_id":"sha256:7500724811144611d636aa4fb10ca818ec36d1a8d15809f9bc30d95133ea9638"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F2WL5X2ATCJMDFXNIFQURSPP5I/bundle.json","state_url":"https://pith.science/pith/F2WL5X2ATCJMDFXNIFQURSPP5I/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F2WL5X2ATCJMDFXNIFQURSPP5I/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":"2026-06-02T20:46:29Z","links":{"resolver":"https://pith.science/pith/F2WL5X2ATCJMDFXNIFQURSPP5I","bundle":"https://pith.science/pith/F2WL5X2ATCJMDFXNIFQURSPP5I/bundle.json","state":"https://pith.science/pith/F2WL5X2ATCJMDFXNIFQURSPP5I/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F2WL5X2ATCJMDFXNIFQURSPP5I/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:F2WL5X2ATCJMDFXNIFQURSPP5I","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":"e20c6d2d0da44fdb2fa15f3d7849955daaf1a117ed668b86ecbd338bc9fbaeb9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-09-03T11:43:33Z","title_canon_sha256":"85bb79b7e2ab9dc90974c999d40b68ce99287364541bf8a8c4a442e9247ac0a5"},"schema_version":"1.0","source":{"id":"1409.1044","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.1044","created_at":"2026-05-18T02:43:44Z"},{"alias_kind":"arxiv_version","alias_value":"1409.1044v1","created_at":"2026-05-18T02:43:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.1044","created_at":"2026-05-18T02:43:44Z"},{"alias_kind":"pith_short_12","alias_value":"F2WL5X2ATCJM","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"F2WL5X2ATCJMDFXN","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"F2WL5X2A","created_at":"2026-05-18T12:28:28Z"}],"graph_snapshots":[{"event_id":"sha256:7500724811144611d636aa4fb10ca818ec36d1a8d15809f9bc30d95133ea9638","target":"graph","created_at":"2026-05-18T02:43:44Z","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 define the notion of the partial order of ends of the Cayley graph of a semigroup. We prove that the structure of the ends of a semigroup is invariant under change of finite generating set and at the same time is inherited by subsemigroups and extensions of finite Rees index. We prove an analogue of Hopf's Theorem, stating that a group has 1, 2 or infinitely many ends, for left cancellative semigroups and that the cardinality of the set of ends is invariant in subsemigroups and extension of finite Green index in left cancellative semigroups.","authors_text":"J. D. Mitchell, N. Ruskuc, R. Gray, S. Craik, V. Kilibarda","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-09-03T11:43:33Z","title":"Ends of Semigroups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.1044","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":"sha256:8fe225de898254ea088f8a8d8b5766555541807d2170ea9e2ac9e8e6bd118e4b","target":"record","created_at":"2026-05-18T02:43:44Z","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":"e20c6d2d0da44fdb2fa15f3d7849955daaf1a117ed668b86ecbd338bc9fbaeb9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-09-03T11:43:33Z","title_canon_sha256":"85bb79b7e2ab9dc90974c999d40b68ce99287364541bf8a8c4a442e9247ac0a5"},"schema_version":"1.0","source":{"id":"1409.1044","kind":"arxiv","version":1}},"canonical_sha256":"2eacbedf409892c196ed416148c9efea08d93a2d81b499371f387ad5813f4379","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2eacbedf409892c196ed416148c9efea08d93a2d81b499371f387ad5813f4379","first_computed_at":"2026-05-18T02:43:44.485978Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:43:44.485978Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NRT1FrZk7lHTcEza2UqIif2YdjUhhA5UKdpnsHqZLVnWTOjy8UMZEqfeeFm5aXFgk0ZPVXcCrSZhmk43LI9vAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:43:44.486451Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.1044","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8fe225de898254ea088f8a8d8b5766555541807d2170ea9e2ac9e8e6bd118e4b","sha256:7500724811144611d636aa4fb10ca818ec36d1a8d15809f9bc30d95133ea9638"],"state_sha256":"b94d4c74c8c3fc2b0655f58bb7cfb2c921128d112998b051b68a3bc52738b577"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"joLHtQMXlvvVHyQdZ9kEmkKhrSKaEgQDUlwgoM770j6GfJV06hqAO4utJc4U+yRYcsHkohMWd0DA5Uk3nOjVCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T20:46:29.568802Z","bundle_sha256":"a9425e051f9d99b33252bf6d42626e380d2475fc54bba0270ce91662fa82dc41"}}