1. Introduction and Preliminaries
The fundamental concept of fuzzy sets was introduced by L.A. Zadeh [13] provided a natural foundation for building new branches. In 1968 C.L. Chang [3] introduced the concept of fuzzy topological spaces as a generalization of topological spaces.
The class of somewhat continuous mappings was first introduced by karl.R. Gentry and others in [5]. Later, the concept of "somewhat" in classical topology has been extented to fuzzy topological spaces. In fact, somewhat fuzzy continuous mappings and somewhat fuzzy semicontinuous mappings were introduced and studied by G. Thangaraj and G. Balasubramanian in [9] and [10] respectively. In 1989, A. Kandil [4] introduced the concept of fuzzy bitopological spaces. The product related spaces and the graph of a function were found in Azad [1]. The concept of somewhat pairwise fuzzy continuous mappings was introduced and devoloped by M.K. Uma and others in [11].
Meanwhile, the concept of fuzzy irresolute continuous mappings on a fuzzy topological space was introduced and studied by M.N. Mukherjee and S. P.Shina in [6] and fuzzy precontinuous mappings on a fuzzy topological space was introduced and studied by A.S. Bin Shahna in [2]. Also, fuzzy pre-irresolute continuous mappings on a fuzzy topological space were introduced and studiedby J.H. Park and B.H. Park in [7].
The concept of somewhat fuzzy pre-irresolute continuous mappings was introduced and studied by Young Bin Im and others in [12].
Recently, somewhat pairwise fuzzy precontinuous mappings on fuzzy bitopological spaces was introduced and studied by A. Swaminathan and others in [8].
In this paper, the concepts of somewhat pairwise fuzzy pre-irresolute continuous mappings and somewhat pairwise fuzzy irresolute preopen mappings on a fuzzy bitopological space are introduced and studied their properties.
Definition 1.1. A mapping f : (X, τ1, τ2) → (Y, η1, η2) is called pairwise fuzzy precontinuous [8] if f−1(ν) is a τ1-fuzzy preopen or τ2-fuzzy preopen set on (X, τ1, τ2) for any η1-fuzzy open or η2-fuzzy open set ν on (Y, η1, η2).
Definition 1.2. A mapping f : (X, τ1, τ2) → (Y, η1, η2) is called somewhat pairwise fuzzy precontinuous [8] if there exists a τ1-fuzzy preopen or τ2-fuzzy preopen set μ ≠ 0X on (X, τ1, τ2) such that μ ≤ f−1(ν) ≠ 0X for any η1-fuzzy open or η2-fuzzy open set ν on (Y, η1, η2) .
Definition 1.3. A mapping f : (X, τ1, τ2) → (Y, η1, η2) is called pairwise fuzzy preopen [8] if f(μ) is an η1-fuzzy preopen or η2-fuzzy preopen set on (Y, η1, η2) for any τ1-fuzzy open or τ2-fuzzy open set μ on (X, τ1, τ2).
Definition 1.4. A mapping f : (X, τ1, τ2) → (Y, η1, η2) is called somewhat pairwise fuzzy preopen [8] if there exists an η1-fuzzy preopen or η2-fuzzy preopen set ν ≠ 0Y on (Y, η1, η2) such that ν ≤ f(μ) ≠ 0Y for any τ1-fuzzy open or τ2-fuzzy open set μ on (X, τ1, τ2).
2. Somewhat pairwise fuzzy pre-irresolute continuous mappings
In this section, I introduce a somewhat pairwise fuzzy pre-irresolute continuous mapping which are stronger than a somewhat pairwise fuzzy precontinuous mapping. And we characterize a somewhat pairwise fuzzy pre-irresolute continuous mapping.
Definition 2.1. A mapping f : (X, τ1, τ2) → (Y, η1, η2) is called pairwise fuzzy pre-irresolute continuous if f−1(ν) is a τ1-fuzzy preopen or τ2-fuzzy preopen set on (X, τ1, τ2) for any η1-fuzzy preopen or η2-fuzzy preopen set ν on (Y, η1, η2).
Definition 2.2. A mapping f : (X, τ1, τ2) → (Y, η1, η2) is called somewhat pairwise fuzzy pre-irresolute continuous if there exists a τ1-fuzzy preopen or τ2-fuzzy preopen set μ ≠ 0X on (X, τ1, τ2) such that μ ≤ f−1(ν) ≠ 0X for any η1-fuzzy preopen or η2-fuzzy preopen set ν ≠ 0Y on (Y, η1, η2) .
From the definitions, it is clear that every pairwise fuzzy pre-irresolute continuous mapping is a somewhat pairwise fuzzy pre-irresolute continuous mapping. And every somewhat pairwise fuzzy pre-irresolute continuous mapping is a pairwise fuzzy precontinuous mapping. Also, every pairwise fuzzy precontinuous mapping is a somewhat pairwise fuzzy precontinuous mapping. But the converses are not true in general as the following examples show.
Example 2.3. Let λ1,λ2,λ3 be fuzzy sets on X = {a, b, c} and let σ1,σ2,σ3 be fuzzy sets on Y = {x, y, z}. Then , are defined as follows: Consider τ1 = {0X, 1X, λ1},, η1 = {0Y , 1Y , σ1},η2 = {0Y , 1Y , σ3}. Then (X, τ1, τ2) and (Y, η1, η2) are fuzzy bitopologies and f : (X, τ1, τ2) → (Y, η1, η2) defined by f(a) = y,f(b) = y,f(c) = y. Then we have f−1(σ1) = 0X, f−1(σ2) == 0X and λ1 < f−1(σ3) = λ2. Since λ1 is a τ1-fuzzy semiopen set on (X, τ1, τ2), f is somewhat pairwise fuzzy pre-irresolute continuous. But f−1(σ3) = λ2 is not a τ1-fuzzy preopen or τ2-fuzzy preopen set on (X, τ1, τ2). Hence f is not a pairwise fuzzy pre-irresolute continuous mapping.
Example 2.4. Let λ1,λ2,λ3 be fuzzy sets on X = {a, b, c} and let σ1,σ2,σ3 be fuzzy sets on Y = {x, y, z}. Then , are defined as follows: Consider τ1 = {0X, 1X, λ2},τ2 = {0X, 1X, λ3}, η1 = {0Y , 1Y , σ1},η2 = {0Y , 1Y , σ2}. Then (X, τ1, τ2) and (Y, η1, η2) are fuzzy bitopologies and f : (X, τ1, τ2) → (Y, η1, η2) defined by f(a) = y,f(b) = y,f(c) = y. Then we have f−1(σ1) = 0X and f−1(σ2) = λ2 are τ2-fuzzy preopen sets on (X, τ1, τ2), f is pairwise fuzzy precontinuous. But f−1(σ3) = λ1 of an η1-fuzzy preopen set σ3 on (Y, η1, η2) is not τ1-fuzzy preopen or τ2-fuzzy preopen on (X, τ1, τ2). Hence f is not a somewhat pairwise fuzzy pre-irresolute continuous mapping.
Example 2.5. Let λ1 and λ2 be fuzzy sets on X = {a, b, c} and let σ1 and σ2 be fuzzy sets on Y = {x, y, z}. Then . Consider τ1 = {0Y , 1Y , λ1},τ2 = {0Y , 1Y , λ2}, η1 = {0X, 1X, λ3},η2 = {0X, 1X, λ4}. Then (X, τ1, τ2) and (Y, η1, η2) are fuzzy bitopologies and consider an identity mapping f : (X, τ1, τ2) → (Y, η1, η2). Then we have λ1 < f−1(λ3) = λ3 and λ1 < f−1(λ4) = λ4.Since λ1 is τ1-fuzzy preopen set on (X, τ1, τ2), f is somewhat pairwise fuzzy precontinuous. But f−1(λ3) = λ3 and f−1(λ4) = λ4 are not τ1-fuzzy preopen or τ2-fuzzy preopen set on (X, τ1, τ2). Hence f is not a pairwise fuzzy precontinuous mapping.
Definition 2.6. A fuzzy set μ on a fuzzy bitopological space (X, τ1, τ2) is called pairwise predense fuzzy set if there exists no τ1-fuzzy preclosed or τ2-fuzzy preclosed set ν in (X, τ1, τ2) such that μ < ν < 1.
Theorem 2.7. Let f : (X, τ1, τ2) → (Y, η1, η2) be a mapping. Then the following are equivalent:
(1) f is somewhat pairwise fuzzy pre-irresolute continuous.
(2) If ν is an η1-fuzzy preclosed or η2-fuzzy preclosed set of (Y, η1, η2) such that f−1(ν) ≠ 1X, then there exists a τ1-fuzzy preclosed or τ2-fuzzy preclosed set μ ≠ 1X of (X, τ1, τ2) such that f−1(ν) ≤ μ.
(3) If μ is a pairwise predense fuzzy set on (X, τ1, τ2), then f(μ) is a pairwise predense fuzzy set on (Y, η1, η2).
Proof. (1) ⇒ (2): Let ν be an η1-fuzzy preclosed or η2-fuzzy preclosed set on (Y, η1, η2) such that f−1(ν) ≠ 1X. Then νc is an η1-fuzzy preopen or η2-fuzzy preopen set on (Y, η1, η2) and f−1(νc) = (f−1(ν))c ≠ 0X. Since f is somewhat pairwise fuzzy pre-irresolute continuous, there exists a τ1-fuzzy preopen or τ2-fuzzy preopen set λ ≠ 0X on (X, τ1, τ2) such that λ ≤ f−1(νc). Let μ = λc. Then μ ≠ 1X is a τ1-fuzzy preclosed or τ2-fuzzy preclosed set such that f−1(ν) = 1 − f−1(νc) ≤ 1 − λ = λc = μ.
(2) ⇒ (3): Let μ be a pairwise predense fuzzy set on (X, τ1, τ2) and suppose f(μ) is not pairwise predense fuzzy set on (Y, η1, η2). Then there exists an η1-fuzzy preclosed or η2-fuzzy preclosed set ν on (Y, η1, η2) such that f(μ) < ν < 1. Since ν < 1 and f−1(ν) ≠ 1X, there exists a τ1-fuzzy preclosed or τ2-fuzzy preclosed set δ ≠ 1X such that μ ≤ f−1(f(μ)) < f−1(ν) ≤ δ. This contradicts to the assumption that μ is a pairwise predense fuzzy set on (X, τ1, τ2). Hence f(μ) is a pairwise predense fuzzy set on (Y, η1, η2).
(3) ⇒ (1): Let ν ≠ 0Y be an η1-fuzzy preopen or η2-fuzzy preopen set on (Y, η1, η2) and let f−1(ν) ≠ 0X. Suppose that there exists no τ1-fuzzy preopen or τ2-fuzzy preopen set μ ≠ 0X on (X, τ1, τ2) such that μ ≤ f−1(ν). Then (f−1(ν))c is a τ1-fuzzy set or τ2-fuzzy set on (X, τ1, τ2) such that there is no τ1-fuzzy preclosed or τ2-fuzzy preclosed set δ on (X, τ1, τ2) with (f−1(ν))c < δ < 1. In fact, if there exists a τ1-fuzzy preopen or τ2-fuzzy preopen set δc such that δc ≤ f−1(ν), then it is a contradiction. So (f−1(ν))c is a pairwise predense fuzzy set on (X, τ1, τ2). Then f((f−1(ν))c) is a pairwise predense fuzzy set on (Y, η1, η2) . But f((f−1(ν))c) = f(f−1(νc)) ≠ νc < 1. This is a contradiction to the fact that f((f−1(ν))c) is pairwise predense fuzzy set on (Y, η1, η2) . Hence there exists a τ1-fuzzy preopen or τ2-fuzzy preopen set μ ≠ 0X on (X, τ1, τ2) such that μ ≤ f−1(ν). Consequently, f is somewhat pairwise fuzzy pre-irresolute continuous. □
Theorem 2.8. Let (X1, τ1, τ2), (X2, ω1, ω2), (Y1, η1, η2), (Y2, σ1, σ2) be fuzzy bitopological spaces. Let (X1, τ1, τ2) be product related to (X2, ω1, ω2) and let (Y1, η1, η2) be product related to (Y2, σ1, σ2). If f1 : (X1, τ1, τ2) →?Y1, η1, η2) and f2 : (X2, ω1, ω2) → (Y2, σ1, σ2) is a somewhat pairwise fuzzy pre-irresolute continuous mappings, then the product f1×f2 : (X1, τ1, τ2)×(X2, ω1, ω2) → (Y1, η1, η2)×(Y2, σ1, σ2) is also somewhat pairwise fuzzy pre-irresolute continuous.
Proof. Let be ηi-fuzzy preopen or σj-fuzzy preopen set on (Y1, η1, η2) × (Y2, σ1, σ2) where μi ≠ 0Y1 is ηi-fuzzy preopen set and νj ≠ 0Y2 is σj-fuzzy preopen set on (Y1, η1, η2) and (Y2, σ1, σ2) respectively. Then . Since f1 is somewhat pairwise fuzzy pre-irresolute continuous, there exists a τ1-fuzzy preopen or τ2-fuzzy preopen set δi ≠ 0X1 such that . And, since f2 is somewhat pairwise fuzzy pre-irresolute continuous, there exists a ω1-fuzzy preopen or ω2-fuzzy preopen set γj ≠ 0X2 such that . Now and δi × γj ≠ 0X1×X2 is a δi-fuzzy preopen or νj-fuzzy preopen set on (X1 × X2). Hence is a τi-fuzzy preopen or ωj-fuzzy preopen set on (X1, τ1, τ2) × (X2, ω1, ω2) such that . Therefore, f1 × f2 is somewhat pairwise fuzzy pre-irresolute continuous. □
Theorem 2.9. Let f : (X, τ1, τ2) → (Y, η1, η2) be a mapping. If the graph g : (X, τ1, τ2) → (X, τ1, τ2) × (Y, η1, η2) of f is a somewhat pairwise fuzzy pre-irresolute continuous mapping, then f is also somewhat pairwise fuzzy pre-irresolute continuous.
Proof. Let ν be an η1-fuzzy preopen or η2-fuzzy preopen set on (Y, η1, η2). Then f−1(ν) = 1 ∧ f−1(ν) = g−1(1 × ν). Since g is somewhat pairwise fuzzy pre-irresolute continuous and 1 × ν is a τi-fuzzy preopen or ηj-fuzzy preopen set on (X, τ1, τ2) × (Y, η1, η2), there exists a τ1-fuzzy preopen or τ2-fuzzy preopen set μ ≠ 0X on (X, τ1, τ2) such that μ ≤ g−1(1 × ν) = f−1(ν) ≠ 0X. Therefore, f is somewhat pairwise fuzzy pre-irresolute continuous. □
3. Somewhat pairwise fuzzy irresolute preopen mappings
In this section, I introduce a somewhat pairwise fuzzy irresolute preopen mapping which are stronger than a somewhat pairwise fuzzy preopen mapping. And we characterize a somewhat pairwise fuzzy irreolute preopen mapping.
Definition 3.1. A mapping f : (X, τ1, τ2) → (Y, η1, η2) is called pairwise fuzzy irresolute preopen if f(μ) is an η1-fuzzy preopen or η2-fuzzy preopen set on (Y, η1, η2) for any τ1-fuzzy preopen or τ2-fuzzy preopen set μ on (X, τ1, τ2).
Definition 3.2. A mapping f : (X, τ1, τ2) → (Y, η1, η2) is called somewhat pairwise fuzzy irresolute preopen if there exists an η1-fuzzy preopen or η2-fuzzy preopen set ν ≠ 0Y on (Y, η1, η2) such that ν ≤ f(μ) ≠ 0Y for any τ1-fuzzy preopen or τ2-fuzzy preopen set μ ≠ 0X on (X, τ1, τ2).
From the definitions, it is clear that every pairwise fuzzy irresolute preopen mapping is a somewhat pairwise fuzzy irresolute preopen mapping. And every somewhat pairwise fuzzy irresolute preopen mapping is a pairwise fuzzy preopen mapping. Also, every pairwise fuzzy preopen mapping is a somewhat pairwise fuzzy preopen mapping. But the converses are not true in general as the following examples show.
Example 3.3. Let λ1,λ2,λ3 be fuzzy sets on X = {a, b, c} and let σ1,σ2,σ3 be fuzzy sets on Y = {x, y, z}. Then , are defined as follows: Consider τ1 = {0X, 1X, λ1},τ2 = {0X, 1X, λ2}, η1 = {0Y , 1Y , σ1},. Then (X, τ1, τ2) and (Y, η1, η2) are fuzzy bitopologies and f : (X, τ1, τ2) → (Y, η1, η2) defined by f(a) = y, f(b) = y, f(c) = y. Then we have f(λ1) = σ1, σ1 < f(λ2) = σ2. Since f is somewhat pairwise fuzzy irresolute preopen mapping. But f(λ2) = σ2 is not η1-fuzzy preopen or η2-fuzzy preopen set on (Y, η1, η2). Hence f is not pairwise fuzzy irresolute preopen mapping.
Example 3.4. Let λ1,λ2,λ3 be fuzzy sets on X = {a, b, c} and let σ1,σ2,σ3 be fuzzy sets on Y = {x, y, z}. Then , are defined as follows: Consider τ1 = {0X, 1X, λ1},τ2 = {0X, 1X, λ2}, η1 = {0Y , 1Y , σ1},η2 = {0Y , 1Y , σ2}. Then (X, τ1, τ2) and (Y, η1, η2) are fuzzy bitopologies and f : (X, τ1, τ2) → (Y, η1, η2) defined by f(a) = y, f(b) = y, f(c) = y. Then we have f(λ1) = σ1, f(λ2) = σ2 and f(λ2) = 0Y are η2-fuzzy preopen set on (Y, η1, η2). Since f is pairwise fuzzy preopen mapping. But λ3 is a τ1-fuzzy preopen set on (X, τ1, τ2) and f(λ3) = 0Y . Hence f is not somewhat pairwise fuzzy irresolute preopen mapping.
Example 3.5. Let λ1,λ2,λ3,λ4,λ5 be fuzzy sets on I = [0, 1] with
Let τ1 = {0I , 1I , λ3} , τ2 = {0I , 1I , λ4}, η1 = {0I , 1I , λ1} and η2 = {0I , 1I , λ2}. Then (I, τ1, τ2) and (I, η1, η2) be fuzzy bitopologies on I. Consider an identity mapping f : (I, τ1, τ2) → (I, η1, η2) defined by f(x) = x, 0 ≤ x ≤ 1: We have λ2 < f(λ3) = λ3, λ2 < f(λ4) = λ4.Since λ2 is an η2-fuzzy preopen set on (I, η1, η2) ,f is somewhat pairwise fuzzy preopen. But f(λ3) = λ3 is not an η1-fuzzy preopen or η2-fuzzy preopen set on (I, τ1, τ2).Hence f is not a pairwise fuzzy preopen mapping.
Theorem 3.6. Let f : (X, τ1, τ2) → (Y, η1, η2) be a bijection. Then the following are equivalent:
(1) f is somewhat pairwise fuzzy irresolute preopen.
(2) If μ is a τ1-fuzzy preclosed or τ2-fuzzy preclosed set on (X, τ1, τ2) such that f(μ) ≠ 1Y , then there exists an η1-fuzzy preclosed or η2-fuzzy preclosed set ν ≠ 1Y on (Y, η1, η2) such that f(μ) < ν.
Proof. (1) ⇒ (2): Let μ be a τ1-fuzzy preclosed or τ2-fuzzy preclosed set on (X, τ1, τ2) such that f(μ) ≠ 1Y . Since f is bijective and μc is a τ1-fuzzy preopen or τ2-fuzzy preopen set on (X, τ1, τ2), f(μc) = (f(μ))c ≠ 0Y . And, since f is somewhat pairwise fuzzy irresolute preopen mapping, there exists an η1-fuzzy preopen or η2-fuzzy preopen set δ ≠ 0Y on (Y, η1, η2) such that δ < f(μc) = (f(μ))c. Consequently, f(μ) < δc = ν ≠ 1Y and ν is an η1-fuzzy preclosed or η2-fuzzy preclosed set on (Y, η1, η2).
(2) ⇒ (1): Let μ be a τ1-fuzzy preopen or τ2-fuzzy preopen set on (X, τ1, τ2) such that f(μ) ≠ 0Y . Then μc is a τ1-fuzzy preclosed or τ2-fuzzy preclosed set on (X, τ1, τ2) and f(μc) ≠ 1Y . Hence there exists an η1-fuzzy preclosed or η2-fuzzy preclosed set ν ≠ 1Y on (Y, η1, η2) such that f(μc) < ν. Since f is bijective, f(μc) = (f(μ))c < ν. Hence νc < f(μ) and νc ≠ 0X is an η1-fuzzy preopen or η2-fuzzy preopen set on (Y, η1, η2). Therefore, f is somewhat pairwise fuzzy irresolute preopen. □
Theorem 3.7. Let f : (X, τ1, τ2) → (Y, η1, η2) be a surjection. Then the following are equivalent:
(1) f is somewhat pairwise fuzzy irresolute preopen.
(2) If ν is a pairwise predense fuzzy set on (Y, η1, η2), then f−1(ν) is a pairwise predense fuzzy set on (X, τ1, τ2).
Proof. (1) ⇒ (2): Let ν be a pairwise predense fuzzy set on (Y, η1, η2). Suppose f−1(ν) is not pairwise predense fuzzy set on (X, τ1, τ2). Then there exists a τ1-fuzzy preclosed or τ2-fuzzy preclosed set μ on (X, τ1, τ2) such that f−1(ν) < μ < 1. Since f is somewhat pairwise fuzzy irreolute preopen and μc is a τ1-fuzzy preopen or τ2-fuzzy preopen set on (X, τ1, τ2), there exists an η1-fuzzy preopen or η2-fuzzy preopen set δ ≠ 0Y on (Y, η1, η2) such that δ ≤ f(Intμc) ≤ f(μc). Since f is surjective, δ ≤ f(μc) < f(f−1(νc)) = νc. Thus there exists an η1-fuzzy preclosed or η2-fuzzy preclosed set δc on (Y, η1, η2) such that ν < δc < 1. This is a contradiction. Hence f−1(ν) is pairwise predense fuzzy set on (X, τ1, τ2).
(2) ⇒ (1): Let μ be a τ1-fuzzy open or τ2-fuzzy open set on (X, τ1, τ2) and f(μ) ≠ 0Y . Suppose there exists no η1-fuzzy preopen or η2-fuzzy preopen set ν ≠ 0Y on (Y, η1, η2) such that ν ≤ f(μ). Then (f(μ))c is an η1-fuzzy set or η2-fuzzy set δ on (Y, η1, η2) such that there exists no η1-fuzzy preclosed or η2-fuzzy preclosed set δ on (Y, η1, η2) with (f(μ))c < δ < 1. This means that (f(μ))c is pairwise predense fuzzy set on (Y, η1, η2) . Thus f−1((f(μ))c) is pairwise predense fuzzy set on (X, τ1, τ2). But f−1((f(μ))c) = (f−1(f(μ)))c ≤ μc < 1. This is a contradiction to the fact that f−1(f(ν))c is pairwise predense fuzzy set on (X, τ1, τ2). Hence there exists an η1-fuzzy preopen or η1-fuzzy preopen set ν ≠ 0Y on (Y, η1, η2) such that ν ≤ f(μ). Therefore, f is somewhat pairwise fuzzy irresolute preopen. □
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