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Two Open Problems on CA-Groupoids and Cancellativities of T2CA-Groupoids
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Cyclic associative groupoids (CA-groupoids) and Type-2 cyclic associative groupoids (T2CA-groupoids) are two types of non-associative groupoids which satisfy cyclic associative law and type-2 cyclic associative law, respectively. In this paper, we prove two theorems that weak cancellativity is cancellativity and right quasi-cancellativity is left quasi-cancellativity in a CA-groupoid, thus successfully solving two open problems. Moreover, the relationships among separativity, quasi-cancellativity and commutativity in a CA-groupoid are discussed. Finally, we study the various cancellativities of T2CA-groupoids such as power cancellativity, quasi-cancellativity and cancellativity. By determining the relationships between them, we can illuminate the structure of T2CA-groupoids.
Title: Two Open Problems on CA-Groupoids and Cancellativities of T2CA-Groupoids
Description:
Cyclic associative groupoids (CA-groupoids) and Type-2 cyclic associative groupoids (T2CA-groupoids) are two types of non-associative groupoids which satisfy cyclic associative law and type-2 cyclic associative law, respectively.
In this paper, we prove two theorems that weak cancellativity is cancellativity and right quasi-cancellativity is left quasi-cancellativity in a CA-groupoid, thus successfully solving two open problems.
Moreover, the relationships among separativity, quasi-cancellativity and commutativity in a CA-groupoid are discussed.
Finally, we study the various cancellativities of T2CA-groupoids such as power cancellativity, quasi-cancellativity and cancellativity.
By determining the relationships between them, we can illuminate the structure of T2CA-groupoids.
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