Skew Bracoids

Truman, Paul; Martin-Lyons, Isabel

Abstract

Skew braces are intensively studied owing to their wide ranging connections and applications. We generalize the definition of a skew brace to give a new algebraic object, which we term a \textit{skew bracoid}. Our construction involves two groups interacting in a manner analogous to the compatibility condition found in the definition of a skew brace. We formulate tools for characterizing and classifying skew bracoids, and study substructures, quotients, homomorphisms, and isomorphisms. As a first application, we prove that finite skew bracoids correspond with Hopf-Galois structures on finite separable extensions of fields, generalizing the existing connection between finite skew braces and Hopf-Galois structures on finite Galois extensions.

Citation

Truman, P., & Martin-Lyons, I. (2024). Skew Bracoids. Journal of Algebra, 638, 751-787. https://doi.org/10.1016/j.jalgebra.2023.10.005

• Journal Article Type: Article
• Acceptance Date: Oct 6, 2023
• Online Publication Date: Oct 12, 2023
• Publication Date: Jan 15, 2024
• Deposit Date: Oct 9, 2023
• Publicly Available Date: Oct 13, 2025
• Journal: Journal of Algebra
• Print ISSN: 0021-8693
• Electronic ISSN: 1090-266X
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 638
• Pages: 751-787
• DOI: https://doi.org/10.1016/j.jalgebra.2023.10.005
• Keywords: Skew left braces; Hopf-Galois structure; Hopf-Galois theory