Fuzzy generalized bi–Γ–ideals of type (λ, θ) in ordered Γ–semigroups

Abstract

Subscribing to the Zadeh’s idea on fuzzy sets, many researchers strive to identify the key attributes of these sets for new finding in mathematics. In this perspective, we introduce a new concept of fuzzy generalized bi–Γ–ideal of an ordered Γ–semigroup G called a (λ, θ)–fuzzy generalized bi–Γ–ideal ofG. Fuzzy generalized bi–Γ–ideals of type (λ, θ) are the generalization of ordinary fuzzy generalized bi–Γ–ideals of an ordered Γ–semigroup G. A new classification of ordered Γ–semigroups in terms of a (λ, θ)–fuzzy generalized bi–Γ–ideal is given. Furthermore, we proved that U(μ, t) is a generalized bi–Γ–ideal if and only if the fuzzy subset μ is a (λ, θ)–fuzzy generalized bi–Γ–ideal of G for all t ∈(λ,θ]. Similarly, A is a generalized bi–Γ–ideal if and only if the characteristic function μA of A is a (λ, θ)–fuzzy generalized bi–Γ–ideal of G. Finally, the relationship between ordinary fuzzy generalized bi–Γ–ideal and (λ, θ)–fuzzy generalized bi–Γ–ideal is discussed