Addendum/Corrigendum to "On solubility of skew left braces and solutions of the Yang-Baxter equation"

TL;DR

This paper revises a previous proof regarding the solubility of solutions to the Yang-Baxter equation by introducing i-homomorphisms, strengthening the connection between solution solubility and skew brace structure.

Contribution

It introduces i-homomorphisms of solutions and redefines solubility, ensuring the validity of Theorem C and improving previous results.

Findings

Theorem C remains valid with the new definition.

Indecomposable solutions are characterized by i-simplicity.

Soluble solutions have soluble structure skew braces.

Abstract

In our previous work: Adv. Math. 455 (2024), no. 109880, solubility of solutions was introduced as an extension of solubility of skew braces in the classification context of non-degenerate solutions of the Yang-Baxter equation. One of our main results (Theorem C) proved that a skew brace is soluble if, and only if, its associated solution is soluble. A minor step depending on the definition of homomorphism of solutions was overlooked. In this work, proof of Theorem C is repaired by means of a new class of homomorphisms of solutions: i-homomorphisms of solutions. The importance of this new class is twofold: indecomposable solutions are characterised by means of i-simplicity of solutions, and i-kernels of i-homomorphisms generate ideals in structure skew braces of solutions. Hence, solubility of solutions is redefined as an opposite class of indecomposable solutions. The results obtained with this definition improve our previous outcomes: every soluble solution is proved to have a soluble structure skew brace, and consequently, Theorem C still holds. Several results stemming from this new analysis are outlined.