{"paper":{"title":"Ranks of $\\mathcal{F}$-limits of filter sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Adam Kwela, Ireneusz Rec{\\l}aw","submitted_at":"2014-10-02T14:24:26Z","abstract_excerpt":"We give an exact value of the rank of an $\\mathcal{F}$-Fubini sum of filters for the case where $\\mathcal{F}$ is a Borel filter of rank $1$. We also consider $\\mathcal{F}$-limits of filters $\\mathcal{F}_i$, which are of the form $\\lim_\\mathcal{F}\\mathcal{F}_i=\\left\\{A\\subset X: \\left\\{i\\in I: A\\in\\mathcal{F}_i\\right\\}\\in\\mathcal{F}\\right\\}$. We estimate the ranks of such filters; in particular we prove that they can fall to $1$ for $\\mathcal{F}$ as well as for $\\mathcal{F}_i$ of arbitrarily large ranks. At the end we prove some facts concerning filters of countable type and their ranks."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.0560","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"}