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We consider the system $\\mathcal L (H) = \\{ \\mathsf L (a) \\mid a \\in H \\}$ of all sets of lengths, and we characterize (in terms of the class group $G$) when $\\mathcal L (H)$ is additively closed under set addition."},"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":"1407.1967","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-07-08T06:30:05Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"7e33197c55d87fc06f0da9a8ee54f5629c76d0e14372a7c57852ec4cd1e8bc17","abstract_canon_sha256":"34f7e860e0665c929197b78678e0d2e7aa1d573386e479eed89b7b319d483030"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:03:52.799192Z","signature_b64":"MfEdNAACXqsAOg8UpeP/ntGZ+x2JbozY7M1I38rdg2LlfkUWpJGY3snDL8QIsdKSAIlSA4hOJ+8z/l1rGgLDAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4d8b5d845d0a4ed727e0c4b0969a22d9124af870721a3f3149459c0e9b98bf93","last_reissued_at":"2026-05-18T02:03:52.798373Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:03:52.798373Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The system of sets of lengths in Krull monoids under set addition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Alfred Geroldinger (IM), Wolfgang Schmid (LAGA)","submitted_at":"2014-07-08T06:30:05Z","abstract_excerpt":"Let $H$ be a Krull monoid with class group $G$ and suppose that each class contains a prime divisor. 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We consider the system $\\mathcal L (H) = \\{ \\mathsf L (a) \\mid a \\in H \\}$ of all sets of lengths, and we characterize (in terms of the class group $G$) when $\\mathcal L (H)$ is additively closed under set addition."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.1967","kind":"arxiv","version":3},"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":"1407.1967","created_at":"2026-05-18T02:03:52.798504+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.1967v3","created_at":"2026-05-18T02:03:52.798504+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.1967","created_at":"2026-05-18T02:03:52.798504+00:00"},{"alias_kind":"pith_short_12","alias_value":"JWFV3BC5BJHN","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_16","alias_value":"JWFV3BC5BJHNOJ7A","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_8","alias_value":"JWFV3BC5","created_at":"2026-05-18T12:28:35.611951+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/JWFV3BC5BJHNOJ7AYSYJNGRC3E","json":"https://pith.science/pith/JWFV3BC5BJHNOJ7AYSYJNGRC3E.json","graph_json":"https://pith.science/api/pith-number/JWFV3BC5BJHNOJ7AYSYJNGRC3E/graph.json","events_json":"https://pith.science/api/pith-number/JWFV3BC5BJHNOJ7AYSYJNGRC3E/events.json","paper":"https://pith.science/paper/JWFV3BC5"},"agent_actions":{"view_html":"https://pith.science/pith/JWFV3BC5BJHNOJ7AYSYJNGRC3E","download_json":"https://pith.science/pith/JWFV3BC5BJHNOJ7AYSYJNGRC3E.json","view_paper":"https://pith.science/paper/JWFV3BC5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.1967&json=true","fetch_graph":"https://pith.science/api/pith-number/JWFV3BC5BJHNOJ7AYSYJNGRC3E/graph.json","fetch_events":"https://pith.science/api/pith-number/JWFV3BC5BJHNOJ7AYSYJNGRC3E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JWFV3BC5BJHNOJ7AYSYJNGRC3E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JWFV3BC5BJHNOJ7AYSYJNGRC3E/action/storage_attestation","attest_author":"https://pith.science/pith/JWFV3BC5BJHNOJ7AYSYJNGRC3E/action/author_attestation","sign_citation":"https://pith.science/pith/JWFV3BC5BJHNOJ7AYSYJNGRC3E/action/citation_signature","submit_replication":"https://pith.science/pith/JWFV3BC5BJHNOJ7AYSYJNGRC3E/action/replication_record"}},"created_at":"2026-05-18T02:03:52.798504+00:00","updated_at":"2026-05-18T02:03:52.798504+00:00"}