On Fuzzy-Γ -ideals of Γ-Abel-Grassmann's Groupoids

Abstract

In this study, we have introduced the notion of Γ-fuzzification in Γ-AG-groupoids which is in fact the generalization of fuzzy AG-groupoids. We have studied several properties of an intra-regular Γ-AG **-groupoids in terms of fuzzy Γ-left (right, two-sided, quasi, interior, generalized bi-, bi-) ideals. We have proved that all fuzzy Γ-ideals coincide in intra-regular Γ-AG **-groupoids. We have also shown that the set of fuzzy Γ-two-sided ideals of an intra-regular Γ-AG **-groupoid forms a semilattice structure.

On Fuzzy Primary Γ-Ideals in Γ-Left Almost Semigroups

“…As is well known (Khan et al, 2013). Fuzzy subset a of S is called a fuzzy sub Γ-left almost semigroup (fuzzy sub Γ-left almost semigroup) of S if:…”

“…(a(xγy)≥min{a(x),a(y)}), for all x,y in S, γ∈Γ and is called a fuzzy left (right) Γ-ideal of S if a(xγy)≥a(y) (a(xγy)≥a(x)) for all γ∈Γ, x,y∈S if a is both fuzzy right and left Γ-ideal of S, then a is called a fuzzy Γ-ideal of S (Khan et al, 2013). It is easy to see that a is a fuzzy Γ-ideal of S if and only if a(xγy)≥max{a(x),a(y)} for all x,y∈S,γ∈Γ and any fuzzy right (left) Γ-ideal of S is a fuzzy Γ-left almost subsemigroup of S. Equivalently, We can prove easily that A is a (right, left) Γ-ideal of S if and only if the function ƒ A of A is a fuzzy (right, left) Γ-ideal of S (Shah et al, 2014).…”

“…Khan et al (2013) Let S be a Γ-left almost semigroup with left identity. If ƒ,g,h and k are fuzzy subsets of S, then:…”