{"paper":{"title":"Sets of lengths in atomic unit-cancellative finitely presented monoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.GR"],"primary_cat":"math.CO","authors_text":"Alfred Geroldinger, Emil Daniel Schwab","submitted_at":"2017-06-10T04:30:27Z","abstract_excerpt":"For an element $a$ of a monoid $H$, its set of lengths $\\mathsf L (a) \\subset \\mathbb N$ is the set of all positive integers $k$ for which there is a factorization $a=u_1 \\cdot \\ldots \\cdot u_k$ into $k$ atoms. We study the system $\\mathcal L (H) = \\{\\mathsf L (a) \\mid a \\in H \\}$ with a focus on the unions $\\mathcal U_k (H) \\subset \\mathbb N$ which are the unions of all sets of lengths containing a given $k \\in \\mathbb N$. The Structure Theorem for Unions -- stating that for all sufficiently large $k$, the sets $\\mathcal U_k (H)$ are almost arithmetical progressions with the same difference a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.03180","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"}