{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2008:G4VN4SO5D6GLXTWGZSQOPQI6C7","short_pith_number":"pith:G4VN4SO5","schema_version":"1.0","canonical_sha256":"372ade49dd1f8cbbcec6cca0e7c11e17dad934f5e3bd7fdc5921524fe4026eac","source":{"kind":"arxiv","id":"0801.0315","version":1},"attestation_state":"computed","paper":{"title":"Uncountable families of prime z-ideals in C_0(R)","license":"","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Hung Le Pham","submitted_at":"2008-01-01T21:05:42Z","abstract_excerpt":"Denote by $\\continuum=2^{\\aleph_0}$ the cardinal of continuum. We construct an intriguing family $(P_\\alpha: \\alpha\\in\\continuum)$ of prime $z$-ideals in $\\C_0(\\reals)$ with the following properties:\n  If $f\\in P_{i_0}$ for some $i_0\\in\\continuum$, then $f\\in P_i$ for all but finitely many $i\\in \\continuum$;\n  $\\bigcap_{i\\neq i_0} P_i \\nsubset P_{i_0}$ for each $\\i_0\\in \\continuum$.\n  We also construct a well-ordered increasing chain, as well as a well-ordered decreasing chain, of order type $\\kappa$ of prime $z$-ideals in $\\C_0(\\reals)$ for any ordinal $\\kappa$ of cardinality $\\continuum$."},"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":"0801.0315","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.RA","submitted_at":"2008-01-01T21:05:42Z","cross_cats_sorted":[],"title_canon_sha256":"dd785fa981cae55e46f28b27fdb37da969bb12aa317b441efcc1d8cd724892b6","abstract_canon_sha256":"be2ad50ba1e48751da2ba7ec899be8cb140219f7bf57466026d5981b02a6be55"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:11.931527Z","signature_b64":"aIUEgRZj5Yn+gfBRM8hvEqAcNAkJa+eU9HRJqOTt+3qElJoHr41Jcrr8JohgZeVdAmkmcuJZFD9safDaStZ2CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"372ade49dd1f8cbbcec6cca0e7c11e17dad934f5e3bd7fdc5921524fe4026eac","last_reissued_at":"2026-05-18T02:58:11.930989Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:11.930989Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uncountable families of prime z-ideals in C_0(R)","license":"","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Hung Le Pham","submitted_at":"2008-01-01T21:05:42Z","abstract_excerpt":"Denote by $\\continuum=2^{\\aleph_0}$ the cardinal of continuum. We construct an intriguing family $(P_\\alpha: \\alpha\\in\\continuum)$ of prime $z$-ideals in $\\C_0(\\reals)$ with the following properties:\n  If $f\\in P_{i_0}$ for some $i_0\\in\\continuum$, then $f\\in P_i$ for all but finitely many $i\\in \\continuum$;\n  $\\bigcap_{i\\neq i_0} P_i \\nsubset P_{i_0}$ for each $\\i_0\\in \\continuum$.\n  We also construct a well-ordered increasing chain, as well as a well-ordered decreasing chain, of order type $\\kappa$ of prime $z$-ideals in $\\C_0(\\reals)$ for any ordinal $\\kappa$ of cardinality $\\continuum$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0801.0315","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":"0801.0315","created_at":"2026-05-18T02:58:11.931069+00:00"},{"alias_kind":"arxiv_version","alias_value":"0801.0315v1","created_at":"2026-05-18T02:58:11.931069+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0801.0315","created_at":"2026-05-18T02:58:11.931069+00:00"},{"alias_kind":"pith_short_12","alias_value":"G4VN4SO5D6GL","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_16","alias_value":"G4VN4SO5D6GLXTWG","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_8","alias_value":"G4VN4SO5","created_at":"2026-05-18T12:25:57.157939+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/G4VN4SO5D6GLXTWGZSQOPQI6C7","json":"https://pith.science/pith/G4VN4SO5D6GLXTWGZSQOPQI6C7.json","graph_json":"https://pith.science/api/pith-number/G4VN4SO5D6GLXTWGZSQOPQI6C7/graph.json","events_json":"https://pith.science/api/pith-number/G4VN4SO5D6GLXTWGZSQOPQI6C7/events.json","paper":"https://pith.science/paper/G4VN4SO5"},"agent_actions":{"view_html":"https://pith.science/pith/G4VN4SO5D6GLXTWGZSQOPQI6C7","download_json":"https://pith.science/pith/G4VN4SO5D6GLXTWGZSQOPQI6C7.json","view_paper":"https://pith.science/paper/G4VN4SO5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0801.0315&json=true","fetch_graph":"https://pith.science/api/pith-number/G4VN4SO5D6GLXTWGZSQOPQI6C7/graph.json","fetch_events":"https://pith.science/api/pith-number/G4VN4SO5D6GLXTWGZSQOPQI6C7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/G4VN4SO5D6GLXTWGZSQOPQI6C7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/G4VN4SO5D6GLXTWGZSQOPQI6C7/action/storage_attestation","attest_author":"https://pith.science/pith/G4VN4SO5D6GLXTWGZSQOPQI6C7/action/author_attestation","sign_citation":"https://pith.science/pith/G4VN4SO5D6GLXTWGZSQOPQI6C7/action/citation_signature","submit_replication":"https://pith.science/pith/G4VN4SO5D6GLXTWGZSQOPQI6C7/action/replication_record"}},"created_at":"2026-05-18T02:58:11.931069+00:00","updated_at":"2026-05-18T02:58:11.931069+00:00"}