Fuzzy generalized bi-ideals of \(\Gamma\)-semigroups. (Q1932793)

Let \(S\) and \(\Gamma\) be two non-empty sets. Then \(S\) is called a \(\Gamma\)-semigroup if there exists a mapping \(S\times\Gamma\times S\to S\) (images to be denoted by \(a\alpha b\)) that satisfies 1) \(x\gamma y\in S\), 2) \((x\beta y)\gamma z=x\beta(y\gamma z)\), \(\forall x,y,z\in S\), \(\forall\beta,\gamma\in\Gamma\). In this paper the authors introduce the notion of generalized bi-ideal and fuzzy generalized bi-ideal of a \(\Gamma\)-semigroup, and characterize regular \(\Gamma\)-semigroups in terms of fuzzy generalized bi-ideals.