Large free linear algebras of real and complex functions

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.

💡 Research Summary

The paper investigates the existence of large free linear algebras inside the spaces of all real‑valued or complex‑valued functions on a set X, and applies this structural result to several “pathological’’ families of functions. The main set‑theoretic hypothesis is that the cardinal κ = |X| satisfies κ^ω = κ (for example any infinite cardinal of uncountable cofinality, in particular κ = 𝔠). Under this assumption the authors prove that the algebra ℝ^X (or ℂ^X) contains a free linear algebra with 2^κ algebraically independent generators. A free algebra here means that no non‑trivial polynomial relation with coefficients in ℝ (or ℂ) holds among the generators; consequently any non‑zero polynomial in the generators yields a new function that still belongs to the ambient space.

The construction proceeds by a transfinite recursion of length κ. At each stage a new function f_α is defined so that for every finite set of previously constructed functions {f_β1,…,f_βn} and every non‑zero polynomial P in n variables, the function P(f_β1,…,f_βn) does not coincide with any earlier function on a large set of points. The cardinal arithmetic κ^ω = κ guarantees that at each step there are enough points left to separate the new function from all previously generated polynomial combinations, thereby preserving algebraic independence. The authors also verify that the set of all finite polynomial expressions in the generators is closed under pointwise addition and multiplication, giving a genuine sub‑algebra of ℝ^X (or ℂ^X).

Having established the existence of such a massive free algebra, the paper turns to concrete algebrability problems. The first application concerns the family PEFS of “perfectly everywhere surjective’’ functions f : ℂ → ℂ, i.e. functions that map every non‑empty open set onto the whole complex plane. Earlier work showed that PEFS is 𝔠‑algebrable (it contains a sub‑algebra generated by 𝔠 many functions). By embedding a free algebra of size 2^𝔠 into ℂ^ℂ and carefully selecting its generators inside PEFS, the authors prove that PEFS is actually strongly 2^𝔠‑algebrable. Strong algebrability means that the generated algebra is not only large but also free: any non‑zero polynomial in the generators remains a perfectly everywhere surjective function. The proof exploits the fact that surjectivity is preserved under pointwise algebraic operations when the functions are constructed to be “wild’’ on a dense family of disjoint open sets.

The second major application deals with the set C_G of real‑valued functions whose set of continuity points equals a prescribed G_δ set G ⊂ ℝ. Classical results (due to Baire, Zahorski, etc.) describe the possible continuity sets of real functions, but the algebrability of C_G has been largely unexplored. The authors show that C_G is strongly 2^𝔠‑algebrable if and only if the complement ℝ \ G is “𝔠‑dense in itself”, meaning that every non‑empty open interval intersecting ℝ \ G contains 𝔠 many points of ℝ \ G. Under this condition they construct a free algebra of size 2^𝔠 whose generators are functions that are discontinuous precisely on a carefully chosen dense subset of ℝ \ G and continuous elsewhere; polynomial combinations of these generators inherit the same continuity set G. Conversely, if ℝ \ G fails to be 𝔠‑dense in itself (for instance if ℝ \ G is countable or has isolated points), any algebra of functions contained in C_G must be at most 𝔠‑generated, and strong 2^𝔠‑algebrability is impossible. This dichotomy provides a precise set‑theoretic criterion for the maximal algebraic size of C_G.

Beyond these two examples, the authors present a general theorem: for any set X with |X| = κ and κ^ω = κ, the space ℝ^X (or ℂ^X) contains a free algebra with 2^κ generators. This result unifies and extends many earlier scattered algebrability statements in the literature, showing that the phenomenon is not accidental but rooted in basic cardinal arithmetic.

In summary, the paper makes three significant contributions: (1) it establishes a robust set‑theoretic method for building free algebras of maximal possible cardinality inside function spaces; (2) it applies this method to prove strong 2^𝔠‑algebrability of the class of perfectly everywhere surjective complex functions; (3) it characterizes precisely when the class of functions with a given G_δ continuity set is strongly 2^𝔠‑algebrable, linking the answer to the density properties of the complement of G. These results deepen our understanding of the algebraic richness hidden in spaces of highly irregular functions and open new avenues for exploring large algebraic structures in analysis and topology.