On bi-skew braces and brace blocks
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.
Keywords
Cite
@article{arxiv.2205.15073,
title = {On bi-skew braces and brace blocks},
author = {L. Stefanello and S. Trappeniers},
journal= {arXiv preprint arXiv:2205.15073},
year = {2022}
}
Comments
Final version, published in Journal of Pure and Applied Algebra