{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:W5DOV5NCFPEOQLHBOVL2XFBTF3","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":"fd6aa7d687842d51b471b23193c503a903357f170ce80f85b59a5eeb88a1953b","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-09-10T15:43:06Z","title_canon_sha256":"db68cdfc4dd0f8d14055d026f6f3399ee42b9c6ef6e7a41f30947942e136c11b"},"schema_version":"1.0","source":{"id":"1209.2042","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.2042","created_at":"2026-05-18T03:18:08Z"},{"alias_kind":"arxiv_version","alias_value":"1209.2042v2","created_at":"2026-05-18T03:18:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.2042","created_at":"2026-05-18T03:18:08Z"},{"alias_kind":"pith_short_12","alias_value":"W5DOV5NCFPEO","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"W5DOV5NCFPEOQLHB","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"W5DOV5NC","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:78527f0192dc562a5a73623616df1352988cb1994ebfde50ae38de0433e51dfc","target":"graph","created_at":"2026-05-18T03:18:08Z","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":"Let $S$ be an algebraic semigroup (not necessarily linear) defined over a field $F$. We show that there exists a positive integer $n$ such that $x^n$ belongs to a subgroup of $S(F)$ for any $x \\in S(F)$. In particular, the semigroup $S(F)$ is strongly {\\pi}-regular.","authors_text":"Lex E. Renner, Michel Brion","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-09-10T15:43:06Z","title":"Algebraic Semigroups are Strongly {\\pi}-regular"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.2042","kind":"arxiv","version":2},"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":"sha256:3fb0feb070f99c3be7471b43e886f49110c3e7053c27f36c0e8b1783e316c925","target":"record","created_at":"2026-05-18T03:18:08Z","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":"fd6aa7d687842d51b471b23193c503a903357f170ce80f85b59a5eeb88a1953b","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-09-10T15:43:06Z","title_canon_sha256":"db68cdfc4dd0f8d14055d026f6f3399ee42b9c6ef6e7a41f30947942e136c11b"},"schema_version":"1.0","source":{"id":"1209.2042","kind":"arxiv","version":2}},"canonical_sha256":"b746eaf5a22bc8e82ce17557ab94332ec7dc12d94253972b889701bbe03266f1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b746eaf5a22bc8e82ce17557ab94332ec7dc12d94253972b889701bbe03266f1","first_computed_at":"2026-05-18T03:18:08.261567Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:18:08.261567Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vVWz//j8AuZk8Dh39+ae5v0XTIgL80mF3dodc5PoE6FXoV02WVihdBALNbisAztte9e9iKNMl2d2O96FhkZ2Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:18:08.262066Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.2042","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3fb0feb070f99c3be7471b43e886f49110c3e7053c27f36c0e8b1783e316c925","sha256:78527f0192dc562a5a73623616df1352988cb1994ebfde50ae38de0433e51dfc"],"state_sha256":"591ce27fc53a2b182cba275a252a83673a8f8cb888702bbfe86b21378f08ddc0"}