| Περιγραφή: |
The skew bracoid is a novel algebraic object consisting of two groups connected by a transitive action of one on the other and satisfying a compatibility relation. This generalises the skew brace, as defined by Guarnieri and Vendramin, which instead has two group structures on a single underlying set, whose interaction is governed by an analogous compatibility relation. This thesis introduces the skew bracoid, giving numerous examples, making key characterisations, and defining notions of substructure and homomorphism. We discuss three nested sub-classes of skew bracoids, the first specifies that the acting group admits a certain exact factorisation, the next that this is a semi-direct product, and the last additionally specifies the relationship between the two groups. Broadening the connection between skew braces and Hopf-Galois structures on Galois extensions of fields, we show that skew bracoids are deeply connected to Hopf-Galois structures on separable extensions of fields. We give preliminary results suggesting the use of skew bracoids in addressing questions concerning the Hopf-Galois correspondence, and generalise a product construction of Crespo, Rio and Vela from the Galois setting to the merely separable. Finally, we connect a large family of skew bracoids to solutions of the set-theoretic Yang-Baxter equation, showing that a single skew bracoid can yield distinct solutions by our procedure. |