Abstract
The aim of this paper is to introduce the concept of lambda operator of a fuzzy set in a fuzzy bitopological space. Then we study (i, j)-fuzzy Lembda Gamma- set and its properties. Moreover we define (i, j)-fuzzy Lembda-closed set, (i, j)-fuzzy Lembda Gamma-closed set and (i, j)-fuzzy generalized closed set in fuzzy bitopological space. The concepts (i, j)-fuzzy Lembda-closed set and (i, j)-fuzzy generalized closed set are independent to each other but jointly they gives the taui-fuzzy closed set. To this end as the application of (i, j)-fuzzy Lembda Gamma-closed set we shall study (i, j)-fuzzy Lembda Gamma continuity and (i, j)-fuzzy Lembda Gamma-generalized continuity and their properties.
Highlights
The purpose of our paper is to continue the research work in the similar direction but in different approach
First we recall some definitions from topology, fuzzy topology and fuzzy bitopological spaces
Let η, μ and μk be fuzzy subsets of a fuzzy bitopological space (X,τ i,τ j) for every k∈ Γ and xp be any point of X, the following properties holds: (i) η ≤ (i, j)-Λγ(η), (ii) if η ≤ μ (i, j)-Λγ(η) ≤(i, j)-Λγ(μ), (iii)(i, j)-Λγ((i, j)-Λγ(η))) =(i, j)-Λγ(η), (iv)if η ∈(i, j)FγO(X) η = (i, j)-Λγ(η), (v) (i, j)-Λγ (∨μk:k∈ Γ)= ∨{(i, j)-Λγ}, (vi) (i, j)-Λγ(∧μk:k∈ Γ)≤ ∧{(i, j)-Λγ(μk:k∈ Γ)} and (vii)(i, j)-Λγ(1X − η)=1X −(i, j)-Vγ (η)
Summary
The purpose of our paper is to continue the research work in the similar direction but in different approach. Let η, μ and μk be fuzzy subsets of a fuzzy bitopological space (X,τ i,τ j) for every k∈ Γ (an index set) and xp be any point of X, the following properties holds: (i) η ≤ (i, j)-Λγ(η), (ii) if η ≤ μ (i, j)-Λγ(η) ≤(i, j)-Λγ(μ), (iii)(i, j)-Λγ((i, j)-Λγ(η))) =(i, j)-Λγ(η), (iv)if η ∈(i, j)FγO(X) η = (i, j)-Λγ(η), (v) (i, j)-Λγ (∨μk:k∈ Γ)= ∨{(i, j)-Λγ (μk:k∈ Γ)}, (vi) (i, j)-Λγ(∧μk:k∈ Γ)≤ ∧{(i, j)-Λγ(μk:k∈ Γ)} and (vii)(i, j)-Λγ(1X − η)=1X −(i, j)-Vγ (η).
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