Large free linear algebras of real and complex functions

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 $X$ 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 $G_\delta$ set $G$ is strongly $2^\mathfrak{c}$-algebrable if and only if $\mathbb{R}\setminus G$ is $\mathfrak{c}$-dense in itself.