QUASI-CANCELLATIVE CYCLIC-ASSOCIATIVE $Gamma$-AG-GROUPOIDS

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Thanawat Saenyakul
Pairote Yiarayong

Abstract

In this paper, we extend the concept of gif.latex?\Gamma-semigroup to quasi-cancellative and cyclic-associative AG-groupoids and investigate various of its properties. We introduce the concept of quasi-cancellative cyclic-associative gif.latex?\Gamma-AG-groupoids. Moreover, we prove that the following conditions are equivalent:


  1. gif.latex?\left&space;(&space;x\alpha&space;y&space;\right&space;)\beta&space;a=&space;\left&space;(&space;x\alpha&space;y&space;\right&space;)\beta&space;b for all gif.latex?a,&space;b,&space;x,&space;y\in&space;S and  gif.latex?\alpha&space;,&space;\beta&space;\in&space;\Gamma.

  2. gif.latex?a\beta&space;\left&space;(&space;y\alpha&space;x&space;\right&space;)=b\beta&space;\left&space;(&space;y\alpha&space;x&space;\right&space;) for all  gif.latex?a,&space;b,&space;x,&space;y\in&space;S and  gif.latex?\alpha&space;,&space;\beta&space;\in&space;\Gamma.

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Research Article

References

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