Large free linear algebras of real and complex functions
Artur Bartoszewicz, Szymon G\l\cab, Adam Paszkiewicz

TL;DR
This paper demonstrates the existence of large free linear algebras within spaces of real and complex functions, revealing significant algebrability properties of certain function sets.
Contribution
It establishes the presence of free linear algebras with maximal generators in function spaces and explores their algebrability properties.
Findings
Existence of free linear algebras with 2^κ generators in ℝ^X and ℂ^X.
Strong 2^𝔠-algebrability of perfectly everywhere surjective functions.
Characterization of functions with a fixed G_δ set of continuity points as strongly 2^𝔠-algebrable.
Abstract
Let be a set of cardinality such that . We prove that the linear algebra (or ) contains a free linear algebra with generators. Using this, we prove several algebrability results for spaces and . In particular, we show that the set of all perfectly everywhere surjective functions is strongly -algebrable. We also show that the set of all functions whose sets of continuity points equals some fixed set is strongly -algebrable if and only if is -dense in itself.
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