Free group of Hamel functions

TL;DR

This paper constructs a free group with continuum many generators from certain bijections of the real line that are also Hamel bases of the plane, revealing new composition properties of such functions.

Contribution

It introduces a novel free group structure among autobijections of  that are also Hamel bases of , and identifies new cases of function compositions involving these bases.

Findings

Constructed a free group with continuum many generators.

Identified new cases where a real function is a composition of two Hamel basis functions.

Included the identity function in the group.

Abstract

We construct a free group of continuum many generators among those autobijections of $\mathbb{R}$ which are also Hamel bases of $\mathbb{R}^2$, with identity function included. We also observe two new cases when a real function is a composition of two real functions which are Hamel bases of $\mathbb{R}^2$.