{"paper":{"title":"Cauchy-Davenport type theorems for semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.GR","authors_text":"Salvatore Tringali","submitted_at":"2013-07-31T17:31:04Z","abstract_excerpt":"Let $\\mathbb{A} = (A, +)$ be a (possibly non-commutative) semigroup. For $Z \\subseteq A$ we define $Z^\\times := Z \\cap \\mathbb A^\\times$, where $\\mathbb A^\\times$ is the set of the units of $\\mathbb{A}$, and $$\\gamma(Z) := \\sup_{z_0 \\in Z^\\times} \\inf_{z_0 \\ne z \\in Z} {\\rm ord}(z - z_0).$$ The paper investigates some properties of $\\gamma(\\cdot)$ and shows the following extension of the Cauchy-Davenport theorem: If $\\mathbb A$ is cancellative and $X, Y \\subseteq A$, then $$|X+Y| \\ge \\min(\\gamma(X+Y),|X| + |Y| - 1).$$ This implies a generalization of Kemperman's inequality for torsion-free gro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.8396","kind":"arxiv","version":2},"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"}