{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:46GFULM4DPVBCIAIMR6XXGC3VE","short_pith_number":"pith:46GFULM4","schema_version":"1.0","canonical_sha256":"e78c5a2d9c1bea112008647d7b985ba92b55f3dcdddf6658541a3a0922d7ac41","source":{"kind":"arxiv","id":"1906.01266","version":1},"attestation_state":"computed","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)"},"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":"1906.01266","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-06-04T08:29:05Z","cross_cats_sorted":[],"title_canon_sha256":"fdff7ac5b7fcabd4a2d8671081cccf7756e98d72677947f0f4aa5707cccca57d","abstract_canon_sha256":"124d4aef53f762e31d567129968615f09e2e90bfef7534bb292925dcb582b848"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:17.258718Z","signature_b64":"TEa7hWoGUYCv7YJAQkEd0b1QNtMr+U+37QdsJlUd9s16TiIcdNWP16HxDPA+tnVQp6EaHJvPyMNFxcOf5I3hAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e78c5a2d9c1bea112008647d7b985ba92b55f3dcdddf6658541a3a0922d7ac41","last_reissued_at":"2026-05-17T23:44:17.258092Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:17.258092Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1906.01266","created_at":"2026-05-17T23:44:17.258188+00:00"},{"alias_kind":"arxiv_version","alias_value":"1906.01266v1","created_at":"2026-05-17T23:44:17.258188+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.01266","created_at":"2026-05-17T23:44:17.258188+00:00"},{"alias_kind":"pith_short_12","alias_value":"46GFULM4DPVB","created_at":"2026-05-18T12:33:10.108867+00:00"},{"alias_kind":"pith_short_16","alias_value":"46GFULM4DPVBCIAI","created_at":"2026-05-18T12:33:10.108867+00:00"},{"alias_kind":"pith_short_8","alias_value":"46GFULM4","created_at":"2026-05-18T12:33:10.108867+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/46GFULM4DPVBCIAIMR6XXGC3VE","json":"https://pith.science/pith/46GFULM4DPVBCIAIMR6XXGC3VE.json","graph_json":"https://pith.science/api/pith-number/46GFULM4DPVBCIAIMR6XXGC3VE/graph.json","events_json":"https://pith.science/api/pith-number/46GFULM4DPVBCIAIMR6XXGC3VE/events.json","paper":"https://pith.science/paper/46GFULM4"},"agent_actions":{"view_html":"https://pith.science/pith/46GFULM4DPVBCIAIMR6XXGC3VE","download_json":"https://pith.science/pith/46GFULM4DPVBCIAIMR6XXGC3VE.json","view_paper":"https://pith.science/paper/46GFULM4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1906.01266&json=true","fetch_graph":"https://pith.science/api/pith-number/46GFULM4DPVBCIAIMR6XXGC3VE/graph.json","fetch_events":"https://pith.science/api/pith-number/46GFULM4DPVBCIAIMR6XXGC3VE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/46GFULM4DPVBCIAIMR6XXGC3VE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/46GFULM4DPVBCIAIMR6XXGC3VE/action/storage_attestation","attest_author":"https://pith.science/pith/46GFULM4DPVBCIAIMR6XXGC3VE/action/author_attestation","sign_citation":"https://pith.science/pith/46GFULM4DPVBCIAIMR6XXGC3VE/action/citation_signature","submit_replication":"https://pith.science/pith/46GFULM4DPVBCIAIMR6XXGC3VE/action/replication_record"}},"created_at":"2026-05-17T23:44:17.258188+00:00","updated_at":"2026-05-17T23:44:17.258188+00:00"}