{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:U7KGTTLOKEX5A6VINIPM44YK5A","short_pith_number":"pith:U7KGTTLO","schema_version":"1.0","canonical_sha256":"a7d469cd6e512fd07aa86a1ece730ae82abe4a7724de48f9c0ef7818834a6f7e","source":{"kind":"arxiv","id":"1603.06356","version":2},"attestation_state":"computed","paper":{"title":"Minimal relations and catenary degrees in Krull monoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Alfred Geroldinger, Yushuang Fan","submitted_at":"2016-03-21T09:05:09Z","abstract_excerpt":"Let $H$ be a Krull monoid with class group $G$. Then $H$ is factorial if and only if $G$ is trivial. Sets of lengths and sets of catenary degrees are well studied invariants describing the arithmetic of $H$ in the non-factorial case. In this note we focus on the set $Ca (H)$ of catenary degrees of $H$ and on the set $\\mathcal R (H)$ of distances in minimal relations. We show that every finite nonempty subset of $\\mathbb N_{\\ge 2}$ can be realized as the set of catenary degrees of a Krull monoid with finite class group. This answers Problem 4.1 of {arXiv:1506.07587}. Suppose in addition that ev"},"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":"1603.06356","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-03-21T09:05:09Z","cross_cats_sorted":[],"title_canon_sha256":"60e86052210c73baa53ff0938bf9adb0ee3ee2dd00f1d9301a144e8965589db6","abstract_canon_sha256":"5167f60333f0a114aa1be81f2f38ee6c9f44cd659b4e14d1728aa9afbdfff342"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:06.556487Z","signature_b64":"ji1w/1vFbNoJsnpUGTM5kae0ioowtIQwwoW57Mn3y5CALnTt5XarXiqCIT/cKLdfxO28C4o/zp8QAbaLktuODw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7d469cd6e512fd07aa86a1ece730ae82abe4a7724de48f9c0ef7818834a6f7e","last_reissued_at":"2026-05-18T01:09:06.556014Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:06.556014Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimal relations and catenary degrees in Krull monoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Alfred Geroldinger, Yushuang Fan","submitted_at":"2016-03-21T09:05:09Z","abstract_excerpt":"Let $H$ be a Krull monoid with class group $G$. Then $H$ is factorial if and only if $G$ is trivial. Sets of lengths and sets of catenary degrees are well studied invariants describing the arithmetic of $H$ in the non-factorial case. In this note we focus on the set $Ca (H)$ of catenary degrees of $H$ and on the set $\\mathcal R (H)$ of distances in minimal relations. We show that every finite nonempty subset of $\\mathbb N_{\\ge 2}$ can be realized as the set of catenary degrees of a Krull monoid with finite class group. This answers Problem 4.1 of {arXiv:1506.07587}. Suppose in addition that ev"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06356","kind":"arxiv","version":2},"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":"1603.06356","created_at":"2026-05-18T01:09:06.556094+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.06356v2","created_at":"2026-05-18T01:09:06.556094+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.06356","created_at":"2026-05-18T01:09:06.556094+00:00"},{"alias_kind":"pith_short_12","alias_value":"U7KGTTLOKEX5","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_16","alias_value":"U7KGTTLOKEX5A6VI","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_8","alias_value":"U7KGTTLO","created_at":"2026-05-18T12:30:46.583412+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/U7KGTTLOKEX5A6VINIPM44YK5A","json":"https://pith.science/pith/U7KGTTLOKEX5A6VINIPM44YK5A.json","graph_json":"https://pith.science/api/pith-number/U7KGTTLOKEX5A6VINIPM44YK5A/graph.json","events_json":"https://pith.science/api/pith-number/U7KGTTLOKEX5A6VINIPM44YK5A/events.json","paper":"https://pith.science/paper/U7KGTTLO"},"agent_actions":{"view_html":"https://pith.science/pith/U7KGTTLOKEX5A6VINIPM44YK5A","download_json":"https://pith.science/pith/U7KGTTLOKEX5A6VINIPM44YK5A.json","view_paper":"https://pith.science/paper/U7KGTTLO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.06356&json=true","fetch_graph":"https://pith.science/api/pith-number/U7KGTTLOKEX5A6VINIPM44YK5A/graph.json","fetch_events":"https://pith.science/api/pith-number/U7KGTTLOKEX5A6VINIPM44YK5A/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/U7KGTTLOKEX5A6VINIPM44YK5A/action/timestamp_anchor","attest_storage":"https://pith.science/pith/U7KGTTLOKEX5A6VINIPM44YK5A/action/storage_attestation","attest_author":"https://pith.science/pith/U7KGTTLOKEX5A6VINIPM44YK5A/action/author_attestation","sign_citation":"https://pith.science/pith/U7KGTTLOKEX5A6VINIPM44YK5A/action/citation_signature","submit_replication":"https://pith.science/pith/U7KGTTLOKEX5A6VINIPM44YK5A/action/replication_record"}},"created_at":"2026-05-18T01:09:06.556094+00:00","updated_at":"2026-05-18T01:09:06.556094+00:00"}