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arXiv · 2205.15073

On bi-skew braces and brace blocks

L. Stefanello·S. Trappeniers
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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.

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L. Stefanello, S. Trappeniers. 2022-05-30. On bi-skew braces and brace blocks. https://doi.org/10.1016/j.jpaa.2022.107295

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