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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1706.03180","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-10T04:30:27Z","cross_cats_sorted":["math.AC","math.GR"],"title_canon_sha256":"cc45fb479f634a3de77a6af42ab9cbe480f1dfa6aeb140a0b811dfaaaccd4935","abstract_canon_sha256":"2f384795cf4ab20d409cb9a3519d251f5985c9bd35baab6eefc0ecc789f37b86"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:38.257049Z","signature_b64":"6q1AThns/G4gsDCVa0srbh3nuvExnLUFR9BJ6gEzEYFba/fF/sC6RQNt2mf99UeDW6xym7AO+xCvvfTtGGObCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e4a560bd0052cbd29de197a8d16e2ef50805add16e6edbb70838868c227d96f","last_reissued_at":"2026-05-18T00:42:38.256413Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:38.256413Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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