Isoclinism of skew braces

TL;DR

This paper introduces the concept of isoclinism in skew braces, explores its properties, and applies it to the study of solutions to the Yang-Baxter equation, revealing invariants like right nilpotency.

Contribution

It defines isoclinism for skew braces and demonstrates its invariance properties, particularly right nilpotency, with applications to set-theoretic solutions of the Yang-Baxter equation.

Findings

Right nilpotency is an isoclinism invariant

Isoclinic solutions are characterized and studied

Multipermutation solutions under isoclinism are analyzed

Abstract

We define isoclinism of skew braces and present several applications. We study some properties of skew braces that are invariant under isoclinism. For example, we prove that right nilpotency is an isoclinism invariant. This result has application in the theory of set-theoretic solutions to the Yang-Baxter equation. We define isoclinic solutions and study multipermutation solutions under isoclinism.