Almost classical skew bracoids

TL;DR

This paper studies special subclasses of skew bracoids called almost a brace and almost classical, exploring their structure and applications in Hopf-Galois theory and solutions to the Yang-Baxter equation.

Contribution

It introduces a new perspective on skew bracoids, providing constructions, recovering known results, and analyzing solutions related to the Hopf-Galois correspondence.

Findings

New construction based on induced Hopf-Galois structures

Recovered Greither and Pareigis' result on Hopf-Galois correspondence

Analyzed solutions arising from skew bracoids, including multiple solutions from one skew bracoid

Abstract

We investigate two sub-classes of skew bracoids, the first consists of those we term almost a brace, meaning the multiplicative group decomposes as a certain semi-direct product, and then those that are almost classical, which additionally specifies the relationship between the multiplicative group and the additive. Skew bracoids with these properties have applications in Hopf-Galois theory, in particular for questions concerning the Hopf-Galois correspondence, and can also yield solutions to the set-theoretic Yang-Baxter equation. We use this skew bracoid perspective to give a new construction building on the induced Hopf-Galois structures of Crespo, Rio and Vela, recover a result of Greither and Pareigis on the Hopf-Galois correspondence, and examine the solutions that arise from skew bracoids, in particular where more than one solution may be drawn from a single skew bracoid.