{"paper":{"title":"Large free sets in universal algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.LO","authors_text":"Artur Bartoszewicz, Szymon G{\\l}ab, Taras Banakh","submitted_at":"2012-09-28T08:01:09Z","abstract_excerpt":"We prove that for each universal algebra $(A,\\mathcal A)$ of cardinality $|A|\\ge 2$ and an infinite set $X$ of cardinality $|X|\\ge|\\mathcal A|$, the $X$-th power $(A^X,\\mathcal A^X)$ of the algebra $(A,\\mathcal A)$ contains a free subset $\\mathcal F\\subset A^X$ of cardinality $|\\mathcal F|=2^{|X|}$. This generalizes the classical Fichtenholtz-Kantorovitch-Hausdorff result on the existence of an independent family $\\mathcal I\\subset\\mathcal P(X)$ of cardinality $|\\mathcal I|=|\\mathcal P(X)|$ in the Boolean algebra $\\mathcal P(X)$ of subsets of an infinite set $X$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.6444","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"}