Skew two-sided bracoids

Abstract

AbstractIsabel Martin-Lyons and Paul J.Truman generalised the definition of a skew brace to give a new algebraic object, which they termed a skew bracoid. Their construction involves two groups interacting in a manner analogous to the compatibility condition found in the definition of a skew brace. They formulated tools for characterizing and classifying skew bracoids, and studied substructures and quotients of skew bracoids. As an application, they proved 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. In this paper we study two-sided bracoids. Rump showed that if a left brace $$(B, \star ,\cdot )$$ ( B , ⋆ , · ) is a two-sided brace and the operation $$*: B \times B \longrightarrow B$$ ∗ : B × B ⟶ B is defined by $$a *b = a\cdot b \star \overline{a} \star \overline{b}$$ a ∗ b = a · b ⋆ a ¯ ⋆ b ¯ for all $$a, b \in B$$ a , b ∈ B then $$(B, \star ,*)$$ ( B , ⋆ , ∗ ) is a Jacobson radical ring. Lau showed that if $$(B, \star ,\cdot )$$ ( B , ⋆ , · ) is a left brace and the operation is asssociative, then $$(B, \star ,\cdot )$$ ( B , ⋆ , · ) is a two-sided brace. We will prove bracoid versions of this results.