{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:JUPRQKN7W6B6C5BWTOIPU6GRH3","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":"ed10756c4d920d93b5955f35c235aaf1d2d4adad95711dc9e5527a6317b11d5b","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-15T21:42:36Z","title_canon_sha256":"6d6991a4d7cf37d3abf29044273a166b410fb613cf993d2afa5e3bb4ae3d8250"},"schema_version":"1.0","source":{"id":"1210.4203","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.4203","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"arxiv_version","alias_value":"1210.4203v5","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.4203","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"pith_short_12","alias_value":"JUPRQKN7W6B6","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JUPRQKN7W6B6C5BW","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JUPRQKN7","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:5992fc2fed6941ceaebbf45268597b4a85054e351dfc89e2e1e17c4fcee43ecd","target":"graph","created_at":"2026-05-18T02:28:21Z","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 generalize the Davenport transform and use it to prove that, for a (possibly non-commutative) cancellative semigroup $\\mathbb A = (A, +)$ and non-empty subsets $X,Y$ of $A$ such that the subsemigroup generated by $Y$ is commutative, we have $|X + Y| \\ge \\min(\\omega(Y), |X| + |Y| - 1)$, where $\\omega(Y) := \\sup_{y_0 \\in Y \\cap \\mathbb A^{\\times}} \\inf_{y \\in Y \\setminus \\{y_0\\}} |<y - y_0>|$. This carries over the Cauchy-Davenport theorem to the broader setting of semigroups, and it implies, in particular, an extension of I. Chowla's and S.S. Pillai's theorems for cyclic groups and a notable","authors_text":"Salvatore Tringali","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-15T21:42:36Z","title":"A Cauchy-Davenport theorem for semigroups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.4203","kind":"arxiv","version":5},"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:a296009c63067258c31671c6bb7990ab985abdb273c668a041502a006114452d","target":"record","created_at":"2026-05-18T02:28:21Z","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":"ed10756c4d920d93b5955f35c235aaf1d2d4adad95711dc9e5527a6317b11d5b","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-15T21:42:36Z","title_canon_sha256":"6d6991a4d7cf37d3abf29044273a166b410fb613cf993d2afa5e3bb4ae3d8250"},"schema_version":"1.0","source":{"id":"1210.4203","kind":"arxiv","version":5}},"canonical_sha256":"4d1f1829bfb783e174369b90fa78d13eca4e76a682e9a60c5b3021d5f5cca3eb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4d1f1829bfb783e174369b90fa78d13eca4e76a682e9a60c5b3021d5f5cca3eb","first_computed_at":"2026-05-18T02:28:21.165452Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:28:21.165452Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"t3+Ajz20kWzN053I/TqBPAjRJ3dODkM+/YiVlcI3dEOrnqI/6fHUxbMtWN+cacRyYosbuJyGUjli51j5fxXZBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:28:21.166136Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.4203","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a296009c63067258c31671c6bb7990ab985abdb273c668a041502a006114452d","sha256:5992fc2fed6941ceaebbf45268597b4a85054e351dfc89e2e1e17c4fcee43ecd"],"state_sha256":"a1fda747b59b6e7d2eb9866e32612f63b2cb867ca10a076f0149dcbe4cfa63c4"}