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Given any element $a$ of $\\mathcal{S}$, let ${\\rm D}_a(\\mathcal{S})$ be the largest length of the irreducible sequence such that the sum of all terms from the sequence is equal to $a$. In case that any ascending chain of principal ideals starting from the ideal $(a)$ terminates in $\\mathcal{S}$, we found the sufficient and necessary conditions of ${\\rm D}_a(\\mathcal{S})$ being f"},"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":"1504.06818","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-26T13:16:53Z","cross_cats_sorted":["math.AC","math.GR","math.NT"],"title_canon_sha256":"8de43031be4d8abfedcc1a8eed71d0d4651129456488d51a308d2fe365d86ab8","abstract_canon_sha256":"7439ef1660d7e71ddd095489e45d858298abfa6223efa5072ede7cfa47ac5e6e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:39:54.806917Z","signature_b64":"8aa4yba0KZyTFGiHxO5LvqMgC7l/tk0YKR7WAe+FCDkMnRV7McAlJukfDqej7V3/IDpF2nV9evFE0X9n0fTbDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"05a6b442a04852dead9c3fa341565d4904b0c4684ec2ab8269abfbd232b339ce","last_reissued_at":"2026-05-18T01:39:54.806386Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:39:54.806386Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Additively irreducible sequences in commutative semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.GR","math.NT"],"primary_cat":"math.CO","authors_text":"Guoqing Wang","submitted_at":"2015-04-26T13:16:53Z","abstract_excerpt":"Let $\\mathcal{S}$ be a commutative semigroup, and let $T$ be a sequence of terms from the semigroup $\\mathcal{S}$. 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