{"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"}