{"paper":{"title":"Large free linear algebras of real and complex functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.RA","authors_text":"Adam Paszkiewicz, Artur Bartoszewicz, Szymon G\\l\\cab","submitted_at":"2012-12-18T12:30:47Z","abstract_excerpt":"Let $X$ be a set of cardinality $\\kappa$ such that $\\kappa^\\omega=\\kappa$. We prove that the linear algebra $\\mathbb{R}^X$ (or $\\mathbb{C}^X$) contains a free linear algebra with $2^\\kappa$ generators. Using this, we prove several algebrability results for spaces $\\mathbb{C}^\\mathbb{C}$ and $\\mathbb{R}^\\mathbb{R}$. In particular, we show that the set of all perfectly everywhere surjective functions $f:\\mathbb{C}\\to\\mathbb{C}$ is strongly $2^\\mathfrak{c}$-algebrable. We also show that the set of all functions $f:\\mathbb{R}\\to\\mathbb{R}$ whose sets of continuity points equals some fixed $G_\\delt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.4329","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"}