斜双边bracoid
Skew two-sided bracoids
AI总结：
本文在斜bracoid的现有研究基础上，针对双边bracoid展开研究，将Rump、Lau关于双边brace的相关经典结论推广至bracoid情形，丰富了该类代数结构的理论成果。
摘要：
Isabel Martin-Lyons与Paul J.Truman推广了斜brace（斜 brace，一种代数结构）的定义，提出了一种被称为skew bracoid（斜bracoid）的新型代数对象。他们的构造涉及两个群，其相互作用方式与斜brace定义中的相容性条件相类似。二人构建了用于刻画与分类斜bracoid的工具，并研究了斜bracoid的子结构与商结构。本文研究双边bracoid。在文献\textit{\cite{WR07}}中，Rump证明：若左brace $(B, \star ,\cdot )$ 是双边brace，且对所有 $a, b \in B$，运算 $\ast : B \times B \longrightarrow B$ 定义为 $a \ast b = a\cdot b \star \overline{a} \star \overline{b}$，则 $(B, \star ,\n\ast )$ 是一个Jacobson根环。Lau证明：若 $(B, \star ,\cdot )$ 是左brace且该运算满足结合律，则 $B$ 是双边brace。本文将证明上述结果的bracoid版本。
英文摘要：
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.
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