{"paper":{"title":"On the system of sets of lengths and the elasticity of submonoids of a finite-rank free commutative monoid","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Felix Gotti","submitted_at":"2018-06-29T05:52:40Z","abstract_excerpt":"Let $H$ be an atomic monoid. For $x \\in H$, let $\\mathsf{L}(x)$ denote the set of all possible lengths of factorizations of $x$ into irreducibles. The system of sets of lengths of $H$ is the set $\\mathcal{L}(H) = \\{\\mathsf{L}(x) \\mid x \\in H\\}$. On the other hand, the elasticity of $x$, denoted by $\\rho(x)$, is the quotient $\\sup \\mathsf{L}(x)/\\inf \\mathsf{L}(x)$ and the elasticity of $H$ is the supremum of the set $\\{\\rho(x) \\mid x \\in H\\}$. The system of sets of lengths and the elasticity of $H$ both measure how far is $H$ from being half-factorial, i.e., $|\\mathsf{L}(x)| = 1$ for each $x \\i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.11273","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"}