Skew bracoids containing a skew brace

TL;DR

This paper explores a specific family of skew bracoids, establishing a bijective correspondence with left cancellative semibraces, and demonstrates their applications in constructing solutions to the set-theoretic Yang-Baxter equation.

Contribution

It introduces a new correspondence between skew bracoids and semibraces, enabling the study of Yang-Baxter solutions through skew bracoids.

Findings

Establishes a bijection between skew bracoids and left cancellative semibraces.

Provides numerous examples of skew bracoids satisfying the new hypothesis.

Shows how skew bracoids can be used to generate solutions to the Yang-Baxter equation.

Abstract

Skew bracoids have been shown to have applications in Hopf-Galois theory. We show that a certain family of skew bracoids correspond bijectively with left cancellative semibraces. A consequence of this correspondence is that skew bracoids in this family can be used to obtain and study solutions of the set-theoretic Yang--Baxter equation; we study this process and the resulting solutions. We give numerous examples of skew bracoids satisfying our hypothesis, drawing upon a variety of constructions in the literature.