Large free linear algebras of real and complex functions

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

Strongc-algebrability of strong Sierpiński–Zygmund, smooth nowhere analytic and other sets of functions

“…The construction of the function from the assertion of Proposition 3.3 was presented for example in [9]. By Theorem 1.5 we can obtain the following result.…”

“…Moreover, in [9], the authors obtained strong c-algebrability of C G , for some sets G and asked a question about strong c-algebrability of C R\Q . In this paper we fully answer for the question about strong c-algebrability of C G in the general case (i.e.…”

Subsets of some families of real functions and their algebrability

“…Bartoszewicz, M. Bienias, and S. Głąb showed in [6,Theorem 8] that the family of nowhere continuous Darboux functions is 2 c -algebrable. This result was improved up to strongly 2 c -algebrability by A. Bartoszewicz, S. Głąb, and A. Paszkiewicz in [9,Theorem 3.3]. Hence D is strongly 2 c -algebrable.…”

“…Recall that the mentioned earlier nowhere continuous Darboux functions (which are contained in D \ (Q ∪Ś)) are strongly 2 c -algebrable [9,Theorem 3.3]. In the same paper (Theorem 10) it is shown that also family of everywhere discontinuous functions, that have finitely many values, is 2 c -algebrable.…”

Large free sets in powers of universal algebras

“…A special case of this theorem was proved in [4]. The proof of this theorem uses the Fichtenholz-Kantorovitch-Hausdorff Theorem, which in turn, can be considered as a special case of Theorem 1.1.…”