{"paper":{"title":"Sets of Lengths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.GR","authors_text":"Alfred Geroldinger","submitted_at":"2015-09-24T18:15:03Z","abstract_excerpt":"Oftentimes the elements of a ring or semigroup $H$ can be written as finite products of irreducible elements, say $a=u_1 \\cdot \\ldots \\cdot u_k = v_1 \\cdot \\ldots \\cdot v_{\\ell}$, where the number of irreducible factors is distinct. The set $\\mathsf L (a) \\subset \\mathbb N$ of all possible factorization lengths of $a$ is called the set of lengths of $a$, and the full system $\\mathcal L (H) = \\{ \\mathsf L (a) \\mid a \\in H \\}$ is a well-studied means of describing the non-uniqueness of factorizations of $H$. We provide a friendly introduction, which is largely self-contained, to what is known ab"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07462","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"}