Skew two-sided bracoids

Abstract: Isabel Martin-Lyons and Paul J.Truman generalized 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. In this paper we study two-sided bracoids. In \cite{WR07} Rump showed that if a left brace $(B, \star ,\cdot )$ is a two-sided brace and the operation $\ast : B \times B \longrightarrow B$ is defined by $$a \ast b = a\cdot b \star \overline{a} \star \overline{b}$$ for all $a, b \in B$ then $(B, \star ,\ast )$ is a Jacobson radical ring. Lau showed that if $(B, \star ,\cdot )$ is a left brace and the operation is asssociative, then $B$ is a two-sided brace. We will prove bracoid versions of this results.