{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:UM3ARBABN2BEXG5WVYLH4HDSHA","short_pith_number":"pith:UM3ARBAB","canonical_record":{"source":{"id":"1004.0007","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-03-31T20:12:29Z","cross_cats_sorted":[],"title_canon_sha256":"4f59108e72db57760e3c649d7a009d157646df329726b5bae0193af9aa67d636","abstract_canon_sha256":"cd6fb1feb4eb11154d1b68bbe9e34169ed20ff35ef1a7007394ccf4eba2033ee"},"schema_version":"1.0"},"canonical_sha256":"a3360884016e824b9bb6ae167e1c72380c9f17bc8a069e2e5ffcb1153233c5de","source":{"kind":"arxiv","id":"1004.0007","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.0007","created_at":"2026-05-18T03:44:34Z"},{"alias_kind":"arxiv_version","alias_value":"1004.0007v1","created_at":"2026-05-18T03:44:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.0007","created_at":"2026-05-18T03:44:34Z"},{"alias_kind":"pith_short_12","alias_value":"UM3ARBABN2BE","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"UM3ARBABN2BEXG5W","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"UM3ARBAB","created_at":"2026-05-18T12:26:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:UM3ARBABN2BEXG5WVYLH4HDSHA","target":"record","payload":{"canonical_record":{"source":{"id":"1004.0007","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-03-31T20:12:29Z","cross_cats_sorted":[],"title_canon_sha256":"4f59108e72db57760e3c649d7a009d157646df329726b5bae0193af9aa67d636","abstract_canon_sha256":"cd6fb1feb4eb11154d1b68bbe9e34169ed20ff35ef1a7007394ccf4eba2033ee"},"schema_version":"1.0"},"canonical_sha256":"a3360884016e824b9bb6ae167e1c72380c9f17bc8a069e2e5ffcb1153233c5de","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:44:34.145378Z","signature_b64":"X3rglLqnwwaI7Wy4T9se6C3iK+qg/MKrXW9++t48Rn4pq7mdRwM9czxtCcaSnIeyn+JitCgCBt9xJcSoUlmUAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a3360884016e824b9bb6ae167e1c72380c9f17bc8a069e2e5ffcb1153233c5de","last_reissued_at":"2026-05-18T03:44:34.144529Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:44:34.144529Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1004.0007","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":"2026-05-18T03:44:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IPSmaj+srKefN0peDmxl2wTO0XU4yaTEzY+6G5FTa+25P0tYxYXP/uzR99gqHc5MvZBWurtAP6ukqnpGFWecAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T09:21:48.252913Z"},"content_sha256":"68f0020bfaccac62d7f37c99d0d344946fc21a311ecdc826cfbfcecd26b7a64e","schema_version":"1.0","event_id":"sha256:68f0020bfaccac62d7f37c99d0d344946fc21a311ecdc826cfbfcecd26b7a64e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:UM3ARBABN2BEXG5WVYLH4HDSHA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Axioms for unary semigroups via division operations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Joao Araujo, Michael Kinyon","submitted_at":"2010-03-31T20:12:29Z","abstract_excerpt":"When a semigroup has a unary operation, it is possible to define two binary operations, namely, left and right division. In addition it is well known that groups can be defined in terms of those two divisions. The aim of this paper is to extend those results to other classes of unary semigroups. In the first part of the paper we provide characterizations for several classes of unary semigroups, including (a special class of) E-inversive, regular, completely regular, inverse, Clifford, etc., in terms of left and right division. In the second part we solve a problem that was posed elsewhere. The"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.0007","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":"2026-05-18T03:44:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZUEi+G3AZH6IOWO5rsvn3GVieQOFANeUqXJcGCieEDTglf94DhExMh5Msgz3kxBaYJRKl55vllW3PmpkiN+KAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T09:21:48.253267Z"},"content_sha256":"3d8458ec312937ef557452a6da2e03f6624f87fddac4ac55175b1ab5ca2abb8c","schema_version":"1.0","event_id":"sha256:3d8458ec312937ef557452a6da2e03f6624f87fddac4ac55175b1ab5ca2abb8c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UM3ARBABN2BEXG5WVYLH4HDSHA/bundle.json","state_url":"https://pith.science/pith/UM3ARBABN2BEXG5WVYLH4HDSHA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UM3ARBABN2BEXG5WVYLH4HDSHA/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":"2026-05-31T09:21:48Z","links":{"resolver":"https://pith.science/pith/UM3ARBABN2BEXG5WVYLH4HDSHA","bundle":"https://pith.science/pith/UM3ARBABN2BEXG5WVYLH4HDSHA/bundle.json","state":"https://pith.science/pith/UM3ARBABN2BEXG5WVYLH4HDSHA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UM3ARBABN2BEXG5WVYLH4HDSHA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:UM3ARBABN2BEXG5WVYLH4HDSHA","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":"cd6fb1feb4eb11154d1b68bbe9e34169ed20ff35ef1a7007394ccf4eba2033ee","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-03-31T20:12:29Z","title_canon_sha256":"4f59108e72db57760e3c649d7a009d157646df329726b5bae0193af9aa67d636"},"schema_version":"1.0","source":{"id":"1004.0007","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.0007","created_at":"2026-05-18T03:44:34Z"},{"alias_kind":"arxiv_version","alias_value":"1004.0007v1","created_at":"2026-05-18T03:44:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.0007","created_at":"2026-05-18T03:44:34Z"},{"alias_kind":"pith_short_12","alias_value":"UM3ARBABN2BE","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"UM3ARBABN2BEXG5W","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"UM3ARBAB","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:3d8458ec312937ef557452a6da2e03f6624f87fddac4ac55175b1ab5ca2abb8c","target":"graph","created_at":"2026-05-18T03:44:34Z","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":"When a semigroup has a unary operation, it is possible to define two binary operations, namely, left and right division. In addition it is well known that groups can be defined in terms of those two divisions. The aim of this paper is to extend those results to other classes of unary semigroups. In the first part of the paper we provide characterizations for several classes of unary semigroups, including (a special class of) E-inversive, regular, completely regular, inverse, Clifford, etc., in terms of left and right division. In the second part we solve a problem that was posed elsewhere. The","authors_text":"Joao Araujo, Michael Kinyon","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-03-31T20:12:29Z","title":"Axioms for unary semigroups via division operations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.0007","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":"sha256:68f0020bfaccac62d7f37c99d0d344946fc21a311ecdc826cfbfcecd26b7a64e","target":"record","created_at":"2026-05-18T03:44:34Z","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":"cd6fb1feb4eb11154d1b68bbe9e34169ed20ff35ef1a7007394ccf4eba2033ee","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-03-31T20:12:29Z","title_canon_sha256":"4f59108e72db57760e3c649d7a009d157646df329726b5bae0193af9aa67d636"},"schema_version":"1.0","source":{"id":"1004.0007","kind":"arxiv","version":1}},"canonical_sha256":"a3360884016e824b9bb6ae167e1c72380c9f17bc8a069e2e5ffcb1153233c5de","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a3360884016e824b9bb6ae167e1c72380c9f17bc8a069e2e5ffcb1153233c5de","first_computed_at":"2026-05-18T03:44:34.144529Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:44:34.144529Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"X3rglLqnwwaI7Wy4T9se6C3iK+qg/MKrXW9++t48Rn4pq7mdRwM9czxtCcaSnIeyn+JitCgCBt9xJcSoUlmUAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:44:34.145378Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.0007","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:68f0020bfaccac62d7f37c99d0d344946fc21a311ecdc826cfbfcecd26b7a64e","sha256:3d8458ec312937ef557452a6da2e03f6624f87fddac4ac55175b1ab5ca2abb8c"],"state_sha256":"c0f7f9d9eb08b640cfbd58d9e5d907636b89c77cd934192a03d1cd55846b39cc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TwL07ITSR+WcRO1KZ7HR/yWiJoxdSc/gLHW3Q4/68rvgPYZG1Nc+CMs8e+jaONo0azAjY3rBqEHO3Hz3bNgdAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T09:21:48.255451Z","bundle_sha256":"27c2183d641b9f67c6f918e22b07db33e7a6095bfbf82c477041a29e3542302a"}}