On bi-skew braces and brace blocks

TL;DR

This paper provides a comprehensive analysis of bi-skew braces, exploring their properties, connections to the Yang--Baxter equation, and introducing brace blocks with new examples and characterizations.

Contribution

It offers a systematic study of bi-skew braces, addresses Byott's conjecture for them, and characterizes brace blocks with new insights and examples.

Findings

Bi-skew braces are connected to set-theoretic solutions of the Yang--Baxter equation.

The paper characterizes brace blocks and shows how existing constructions fit into this framework.

New examples of brace blocks are provided, expanding the known landscape.

Abstract

L. N. Childs defined a bi-skew brace to be a skew brace such that if we swap the role of the two operations, then we find again a skew brace. In this paper, we give a systematic analysis of bi-skew braces. We study nilpotency and solubility, and connections between bi-skew braces and set-theoretic solutions of the Yang--Baxter equation. Further, we deal with Byott's conjecture in the case of bi-skew braces, and we use bi-skew braces as a tool to solve a classification problem proposed by L. Vendramin. In the final part, we investigate brace blocks, defined by A. Koch to be families of group operations on a given set such that any two of them yield a bi-skew brace. We provide a characterisation of brace blocks, illustrate how all known constructions in literature follow in a natural way from our characterisation, and give several new examples.