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.
Cyclic associativity can be regarded as a kind of variation symmetry, and cyclic associative groupoid (CA-groupoid) is a generalization of commutative semigroup. In this paper, the various cancellation properties of CA-groupoids, including cancellation, quasi-cancellation and power cancellation, are studied. The relationships among cancellative CA-groupoids, quasi-cancellative CA-groupoids and power cancellative CA-groupoids are found out. Moreover, the concept of variant CA-groupoid is proposed firstly, some examples are presented. It is shown that the structure of variant CA-groupoid is very interesting, and the construction methods and decomposition theorem of variant CA-groupoids are established.
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