{"paper":{"title":"Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT","math.RA"],"primary_cat":"math.AC","authors_text":"Alfred Geroldinger, Florian Kainrath, Salvatore Tringali, Yushuang Fan","submitted_at":"2016-12-09T18:16:32Z","abstract_excerpt":"Let $H$ be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every $k \\in \\mathbb N$, let $\\mathscr U_k (H)$ denote the set of all $\\ell \\in \\mathbb N$ with the property that there are atoms $u_1, \\ldots, u_k, v_1, \\ldots, v_{\\ell}$ such that $u_1 \\cdot \\ldots \\cdot u_k = v_1 \\cdot \\ldots \\cdot v_{\\ell}$ (thus, $\\mathscr U_k (H)$ is the union of all sets of lengths containing $k$).\n  The Structure Theorem for Unions states that, for all sufficiently large $k$, the sets $\\mathscr U_k (H)$ are almost arithme"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.03116","kind":"arxiv","version":3},"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"}