Free commutative skew braces

TL;DR

This paper constructs an explicit model of free commutative skew braces using rational functions, simplifying their comparison to that of free commutative groups, thus advancing algebraic understanding of these structures.

Contribution

It provides the first explicit construction of free commutative skew braces and characterizes their elements via rational functions, simplifying their analysis.

Findings

Explicit construction of free commutative skew braces.

Embedding into rational functions with a linear characterization.

Comparison of elements reduces to free commutative group comparison.

Abstract

The main result of this paper is an explicit construction of the free commutative skew brace -- that is, a skew brace whose circle group is commutative -- on an arbitrary generating set $X$. We embed this object into a set of rational functions and show that a simple linear equation characterizes the image of this embedding. As a consequence, comparing elements in this skew brace is no more difficult than comparing elements in the free commutative group generated by $X$.