{"paper":{"title":"Union of sets of lengths of numerical semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"A. Vigneron-Tenorio, D. Mar\\'in-Arag\\'on, J.I. Garc\\'ia-Garc\\'ia","submitted_at":"2019-06-04T08:29:05Z","abstract_excerpt":"Let $S=\\langle a_1,\\ldots,a_p\\rangle$ be a numerical semigroup, $s\\in S$ and ${\\sf z}(s)$ its set of factorizations. The set of length is denoted by ${\\mathcal L}(s)=\\{{\\tt L}(x_1,\\dots,x_p)\\mid (x_1,\\dots,x_p)\\in{\\sf Z}(s)\\}$ where ${\\tt L}(x_1,\\dots,x_p)=x_1+\\ldots+x_p$. From these definitions, the following sets can be defined ${\\textsf W}(n)=\\{s\\in S\\mid \\exists x\\in{\\sf z}(s) \\textrm{ such that } {\\tt{L}}(x)=n\\}$, $\\nu(n)=\\cup_{s\\in {\\textsf W}(n)} {\\mathcal L}(s)=\\{l_1<l_2<\\ldots< l_r\\}$ and $\\Delta\\nu(n)=\\{l_2-l_1,\\ldots,l_r-l_{r-1}\\}$. In this paper, we prove that the set $\\Delta\\nu(S)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.01266","kind":"arxiv","version":1},"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"}