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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.09623 |
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| _version_ | 1866909169923129344 |
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| author | Malinowska, Izabela Agata |
| author_facet | Malinowska, Izabela Agata |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09623 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Skew two-sided bracoids Malinowska, Izabela Agata Rings and Algebras Group Theory 16T25, 81R50 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. |
| title | Skew two-sided bracoids |
| topic | Rings and Algebras Group Theory 16T25, 81R50 |
| url | https://arxiv.org/abs/2404.09623 |