Symmetric skew braces and brace systems

TL;DR

This paper studies symmetric skew braces, particularly those with $ ext{lambda}$-homomorphic properties, introduces a construction method for such braces, and explores their applications on infinite sets.

Contribution

It develops a new construction method for symmetric skew braces with $ ext{lambda}$-homomorphic properties, unifying many existing constructions and extending them to infinite sets.

Findings

Most known symmetric skew brace constructions fit within the new framework.

A method for constructing $ ext{lambda}$-homomorphic symmetric skew braces on infinite sets.

Theoretical foundation for generating a broad class of symmetric skew braces.

Abstract

For a skew left brace $(G, \cdot, \circ)$, the map $\lambda : (G, \circ) \to \Aut \,(G, \cdot),~~a \mapsto \lambda_a,$ where $\lambda_a(b) = a^{-1} \cdot (a \circ b)$ for all $a, b \in G$, is a group homomorphism. Then $\lambda$ can also be viewed as a map from $(G, \cdot)$ to $\Aut \, (G, \cdot)$, which, in general, may not be a homomorphism. A skew left brace will be called $\lambda$-anti-homomorphic ($\lambda$-homomorphic) if $\lambda : (G, \cdot) \to \Aut \, (G, \cdot)$ is an anti-homomorphism (a homomorphism). We mainly study such skew left braces. We device a method for constructing a class of binary operations on a given set so that the set with any two such operations constitute a $\lambda$-homomorphic symmetric skew brace. Most of the constructions of symmetric skew braces dealt with in the literature fall in the framework of our construction. We then carry out various such constructions on specific infinite sets.