A new representation of α-openness, α-continuity, α-irresoluteness, and α-compactness in L-fuzzy pretopological spaces

A. Ghareeb; H. S. Al-Saadi; O. H. Khalil

doi:10.1515/math-2019-0047

Article Open Access

A new representation of α-openness, α-continuity, α-irresoluteness, and α-compactness in L-fuzzy pretopological spaces

A. Ghareeb , H. S. Al-Saadi and O. H. Khalil

Published/Copyright: June 22, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

This paper presents a new representation of α-openness, α-continuity, α-irresoluteness, and α-compactness based on L-fuzzy α-open operators introduced by Nannan and Ruiying [1] and implication operation. The proposed representation extends the properties of α-openness, α-continuity, α-irresoluteness, and α-compactness to the setting of L-fuzzy pretopological spaces based on graded concepts. Moreover, we introduce and establish the relationships among the new concepts.

Keywords: L-fuzzy pretopology; L-fuzzy α-open operator; L-fuzzy α-openness degree; L-fuzzy α-continuty degree; L-fuzzy α-irresolutness degree; L-fuzzy α-compactness degree

PACS: 03E72; 54A40; 54C20

1 Introduction

Continuity is an important concept in topology, which has developed extensively with the emergence of fuzzy mathematics. In [2, 3], Šostak considered the degrees to which a mapping is continuous, open, and closed between two (L, M)-fuzzy topological spaces (including the fuzzifying case) for the first time. Subsequently, the degrees of continuity, openness, and closeness of mappings between L-fuzzifying topological spaces were discussed in detail by Pang [4]. Later on, Liang and Shi [5] clarified the relationship among these degrees and the degree of compactness and connectedness in the case of (L, M)-fuzzy setting.

Recently, Shi [6] measured preopenness and semiopenness of L-subset by introducing the concepts of L-fuzzy preopen operators and L-fuzzy semiopen operators, respectively. In [7], Shi and Li used L-fuzzy semiopen operators to introduce and characterize the semicompactness. Later on in [8] the degree of preconnectedness was introduced with the help of L-fuzzy preopen operators. In addition, he used Shi’s operators to define new operators such as L-fuzzy semipreopen operators [9] and L-fuzzy F-open operators [10]. These operators have proved to be of great importance in studying the characteristics of many concepts of L-fuzzy topology (see [11, 12, 13, 14]).

In [1], Nannan and Ruiying introduced L-fuzzy α-open operators in L-fuzzy topological spaces and used it to study L-fuzzy α-compactness. Moreover, the concept of open cover and a-fuzzy α-compact are given and its related properties are discussed. Also, the relationship between L-fuzzy α-compactness and fuzzy α-compactness are discussed.

This paper first discusses some important properties of L-fuzzy α-open operators. It then introduces α-openness, α-continuity, α-irresoluteness, and α-compactness degree based on the implication operation and L-fuzzy α-open operators. Further, some important properties of α-openness, α-continuity, α-irresoluteness, and α-compactness degree were extended to the setting of L-fuzzy pretopology based on graded concepts. Moreover, it presents a systematic discussion on the relationship among the new concepts.

2 Preliminaries

In the sequel, X ≠ ∅, and L refers to a completely distributive De Morgan algebra (briefly, CDDA). Let 1_L and 0_L denote the greatest and smallest elements of L, respectively. For each u, v ∈ L, the element u is wedge below v [15], written u ⊲ v, if for each 𝓓 ⊆ L, ⋁ 𝓓 ≥ v yields to w ≥ u for some w ∈ 𝓓. We say the complete lattice L is completely distributive (briefly, CD) if and only if v = ⋁ {u ∈ L | u ⊲ v} for any v ∈ L. A member u ∈ L is said to be co-prime if u ≤ v ∨ w yields to u ≤ v or u ≤ w. P(L) and J(L) refer to the family of non-unit prime members and non-zero co-prime members of L respectively. The greatest minimal family and the greatest maximal family of v ∈ L are denoted by α(v) and β(v) respectively. Moreover, α^*(v) = α(v) ∩ J(L) and β^*(v) = β(v) ∩ P(L). By L^X we refer to the set of all L-subsets on X. 2^𝓤 denotes the collection of all finite sub-collections of 𝓤 ⊆ L^X. Evidently, L^X is a CDDA when it inherits the structure of the lattice L in a natural way, by defining ⋁, ⋀, ≤ and ′ pointwisely. Further, {x_u | u ∈ J(L)} denotes the collection of non-zero co-primes of L^X.

For each CDDA L, there exists an implication operation ↦ : L × L ⟶ L as the right adjoint for the meet operation ∧ is defined by

u↦v=⋁{w∈L|u∧w≤v}.

Further, the operation ↔ is given by

u↔v=(u↦v)∧(v↦u).

The following lemma lists some important properties of implication operation.

Lemma 2.1

[16] Let (L, ⋁, ⋀) be a CD lattice and ↦ be the implication operation corresponding to ∧. Then for all u, v, w ∈ L, {u_i}_i∈Γ, and {v_i}_i∈Γ ⊆ L, we have the following statements:

(u ↦ v) ≥ w ⟺ u ∧ w ≤ v.
u ≤ v ⇔ u ↦ v = 1_L.
u ↦ (v ↦ w) = (u ∧ v) ↦ w.
(w ↦ u) ∧ (u ↦ v) ≤ w ↦ v.
w ↦ u ≤ (u ↦ v) ↦ (w ↦ v).
u↦⋀i∈Γui=⋀i∈Γ(u↦ui) , hence u ↦ v ≤ u ↦ w whenever v ≤ w.
⋁i∈Γui↦v=⋀i∈Γ(ui↦v) , hence u ↦ w ≥ v ↦ w whenever u ≤ v.

An L-fuzzy inclusion [17, 18] on X is defined by the function ⊂̃ : L^X × L^X ⟶ L, where ⊂̃ (A₁, A₂) = ⋀_x∈X( A1′ (x) ∨ A₂(x)). We shall denote an L-fuzzy inclusion by [A₁ ⊂̃ A₂]. For each function f : X ⟶ Y and 𝓒 ⊆ L^Y, the next equality is defined in [19]:

⋀y∈Y{f→(A)′(y)∨⋁B∈CB(y)}=⋀x∈X{B′(x)∨⋁B∈Cf←(B)(x)}.

An L-topological space (briefly, L-ts) is a pair (X, τ), where the subfamily τ ⊆ L^X contains 0_L^X, 1_L^X, and closed for any suprema and finite infima. Elements of τ are called open L-subsets and their complements are called closed L-subsets. For an L-subset A of an L-topological space (X, τ) we denote by Ā and A^∘ the closure and the interior of A, respectively.

Definition 2.2

[16, 20, 21, 22] An L-fuzzy pretopology is given by the function σ : L^X ⟶ L satisfies the following statements:

σ(1_L^X) = σ(0_L^X) = 1_L.
σ⋁i∈IAi≥⋀i∈Iσ(Ai), ∀{Ai}i∈I⊆LX .

For any L-subset A ∈ L^X, σ(A) refers to the degree of openness of A. σ^*(A) = σ(A′) is the closeness degree of A. The pair (X, σ) is said to be an L-fuzzy pretopological space (briefly, L-pfts). A function f : (X, σ₁) ⟶ (Y, σ₂) is said to be L-fuzzy continuous with respect to L-fpts’s (X, σ₁) and (Y, σ₂) if and only if σ₁(f^← (B)) ≥ σ₂(B) for each B ∈ L^Y, where f^← (B)(x) = B(f (x)).

Definition 2.3

[1] Let σ be an L-fpt on X and let the mapping 𝒜 : L^X ⟶ L defined as follows:

A(A)=⋁B≤A{σ(B)∧⋀xu⊲A⋁xu⊲C{σ(C)∧⋀yv⊲C⋀yv≰D≥B(σ(D′))′}}.

In this case, 𝒜 is the induced L-fuzzy α-open operator by σ. 𝒜(A) is called the degree of α-openness of A and 𝒜^*(A) = 𝒜(A′) can be regarded as the α-closeness degree of A.

Corollary 2.4

If σ is an L-fpt on X and A ∈ L^X, then:

A(A)=⋁B≤A{σ(B)∧⋀xu⊲A⋁xu⊲C{σ(C)∧⋀yv⊲CClσ(B)(yv)}},

where Cl^σ refers to the L-fuzzy closure operator induced by σ (see [23]).

Theorem 2.5

[1] Let σ be an L-fpt on X, A ∈ L^X, and u ∈ J(L), then A ∈ 𝒜_[u] if and only if A is an α-open set in 𝒜_[u], where 𝒜_[u] = {A ∈ L^X | 𝒜(A) ≥ u.

Theorem 2.6

Let σ : L^X ⟶ {0_L, 1_L} be an L-pts and let 𝒜 : L^X ⟶ {0_L, 1_L} be the corresponding L-α-open operator. Then 𝒜(A) = 1_L if and only if A is α-open L-subset.

Proof

We can prove the theorem by using the following fact:

A(A)=1L⇔⋁B≤A{σ(B)∧⋀xu⊲A⋁xu⊲C{σ(C)∧⋀yv⊲CClσ(B)(yμ)}}=1L⇔∃ B≤A such that σ(B)=1L and ⋀xu⊲A⋁xu⊲C{σ(C)∧⋀yv⊲CClσ(B)(yμ)}=1L⇔∃ B≤A such that σ(B)=1L and ∀xu⊲A,∃ C with xu⊲C such that σ(C)=1L and ⋀yv⊲CClσ(B)(yμ)⇔∃ B≤A such that σ(B)=1L and ∀xu⊲A,∃ C with xu⊲C such that σ(C)=1L and ∀yv⊲C, Clσ(B)(yv)=1L⇔∃ B≤A such that σ(B)=1L and ∀xu⊲A,∃ C with xu⊲C such that σ(C)=1L and C≤Clσ(B)⇔∃ B∈σ, B≤A≤(B¯)∘⇔A is α−open L−subset.

Where ̄ and ^∘ refer to the closure and the interior operator, respectively. □

Theorem 2.7

Let σ be an L-fpt on X and let 𝒜 be its corresponding L-fuzzy α-open operator. Then σ(A) ≤ 𝒜(A) for all A ∈ L^X.

Proof

The proof can be obtained from the following inequality:

A(A)=⋁B≤A{σ(B)∧⋀xu⊲A⋁xu⊲C{σ(C)∧⋀yv⊲CClσ(B)(yμ)}}≥σ(A)∧⋀xu⊲A⋁xu⊲C{σ(C)∧⋀yv⊲CClσ(A)(yμ)}≥σ(A)∧⋀xu⊲A{σ(A)∧⋀yv⊲AClσ(A)(yμ)}=σ(A)∧σ(A)∧1L=σ(A).

□

Corollary 2.8

Let σ be an L-fpt on X and let 𝒜 be its corresponding L-fuzzy α-open operator. Then σ^*(A) ≤ 𝒜^*(A) for all A ∈ L^X.

Theorem 2.9

Let 𝒜 : L^X ⟶ L be an L-fuzzy α-open operator induced by L-fpt σ on X. Then 𝒜 satisfies the following conditions:

𝒜(0_L^X) = 𝒜(1_L^X) = 1_L.
A(⋁i∈IAi)≥⋀i∈IA(Ai) for any {A_i}_i∈I ⊆ L^X.

Proof

The proof of (1) is clear. To prove (2), suppose that w ∈ L and w ⊲ ⋀i∈I 𝒜(A_i). Then for any i ∈ I, there is B_i ≤ A_i such that

w⊲σ(Bi)andw⊲⋀xu⊲Ai⋁xu⊲Ci{σ(Ci)∧⋀yv⊲Ci⋀yv≰D≥Bi(σ(D′))′},

i.e., w ⊲ σ(B_i) and for any i ∈ I and x_u ⊲ A_i, there is C_i ∈ L^X such that x_u ⊲ C_i, w ⊲ σ(C_i) and w⊲⋀yv⊲Ci⋀yv≰D≥⋁i∈IBi(σ(D′))′ . Hence

w≤⋀i∈Iσ(Bi)≤σ⋁i∈IBi,w≤⋀i∈Iσ(Ci)≤σ⋁i∈ICi,

and

w≤⋀i∈I⋀yv⊲Ci⋀yv≰D≥⋁i∈IBi(σ(D′))′.

Since {xu|xu⊲⋁i∈IAi}=⋃i∈I{xu|xu⊲Ai} and {yv|yv⊲⋁i∈ICi}=⋃i∈I{yv|yv⊲Ci} , we have

A⋁i∈IAi=⋁B≤⋁i∈IAiσ(B)∧⋀xu⊲⋁i∈IAi⋁xu⊲Cσ(C)∧⋀yv⊲C⋀yv≰D≥B(σ(D′))′≥σ⋁i∈IBi∧⋀i∈I⋀xu⊲⋁i∈IAiσ⋁i∈ICi∧⋀yv⊲⋁i∈ICi⋀yv≰D≥⋁i∈IBi(σ(D′))′=σ⋁i∈IBi∧⋀i∈I⋀xu⊲⋁i∈IAiσ⋁i∈ICi∧⋀i∈I⋀yv⊲Ci⋀yv≰D≥⋁i∈IBi(σ(D′))′≥σ⋁i∈IBi∧⋀i∈I⋀xu⊲Aiσ⋁i∈ICi∧⋀i∈I⋀yv⊲Ci⋀yv≰D≥⋁i∈IBi(σ(D′))′≥w.

This shows A⋁i∈IAi≥⋀i∈IA(Ai) . □

In the following definition, we use L-fuzzy α-open operators to introduce generalized definitions for L-fuzzy α-open, L-fuzzy α-continuous and L-fuzzy α-irresolute functions.

Definition 2.10

If (X, σ₁) and (Y, σ₂) are L-fpts’s and f : (X, σ₁) ⟶ (Y, σ₂) is a function, then:

f is an L-fuzzy α-open function iff σ₁(A) ≤ 𝒜₂(f^→ (A)) for any A ∈ L^X.
f is an L-fuzzy α-continuous function iff σ₂(B) ≤ 𝒜₁(f^← (B)) holds for any B ∈ L^Y.
f is an L-fuzzy α-irresolute iff 𝒜₂(B) ≤ 𝒜₁(f^← (B)) holds for any B ∈ L^Y.

Corollary 2.11

If (X, σ₁) and (Y, σ₂) are L-fpts’s and f : (X, σ₁) ⟶ (Y, σ₂) is a function, then:

f is an L-fuzzy α-continuous iff σ2∗(B)≤A1∗(f←(B)) for any B ∈ L^Y.
f is an L-fuzzy α-irresolute iff A2∗(B)≤A1∗(f←(B)) for any B ∈ L^Y.

Definition 2.12

[24] For an L-fpt σ on X and an L-subset A ∈ L^X, the degree of fuzzy compactness com(A) of A is given by:

com(A)=⋀U⊆LX⋀B∈Uσ(B)∧⋀x∈XA′∨⋁B∈UB(x)↦⋁V∈2(U)⋀x∈XA′∨⋁B∈VB(x) =⋀U⊆LX⋀B∈Uσ(B)∧A⊆~⋁U↦⋁V∈2(U)A⊆~⋁V.

In this case, an L-subset A is said to be fuzzy compact if and only if com(A) = 1_L.

Definition 2.13

[1] Let σ be an L-fpt on X. An L-subset A ∈ L^X is called α-compact if

⋀B∈UA(B)∧⋀x∈X(A′∨⋁B∈UB)(x)≤⋁V∈2(U)⋀x∈X(A′∨⋁B∈VB)(x)

for every 𝓤 ⊂ L^X.

Definition 2.14

[25, 26] For an L-pt τ on X, u ∈ L \ {1_L} and A ∈ L^X, a family 𝓤 ⊆ L^X is said to be a α_u-cover of A if for each x ∈ X, we have u ∈ α(A′(x) ∨ ⋁B∈U B(x)). The family 𝓤 is said to be a strong α_u-cover of A if u∈α(⋀x∈X(A′(x)∨⋁B∈UB(x))) .

Definition 2.15

[25, 26] For an L-pt τ on X, u ∈ L \ {1_L} and A ∈ L^X, a family 𝓤 ⊆ L^X is said to be a Q_u-cover of A if for each x ∈ X, we have A′(x) ∨ ⋁B∈U B(x) ≥ u.

Definition 2.16

[25, 26] For an L-pt τ on X, u ∈ L \ {1_L} and A ∈ L^X, a family 𝓤 ⊆ L^X is called:

a u-shading of A if for each x ∈ X, (A′(x)∨⋁B∈UB(x))≰u .
a strong u-shading of A if ⋀x∈X(A′(x)∨⋁B∈UB(x))≰u .
a u-remote family of A if for each x ∈ X, (A(x)∧⋀B∈UB(x))≱u .
a strong u-remote family of A if ⋁x∈X(A(x)∧⋀B∈UB(x))≱u .

3 Degree of α-openness, α-continuity and α-irresolutness for functions between L-fpts’s

In this section, we will introduce the notions of α-openness, α-continuity, and α-irresolutness degree for functions between L-fpts’s. Further, we will discuss their properties.

Definition 3.1

If (X, σ₁) and (Y, σ₂) are L-fpts’s and f : (X, σ₁) ⟶ (Y, σ₂) is a function, then:

the α-openness degree of f with respect to σ₁ and σ₂ is defined by

αo(f)=⋀A∈LX{A1(A)↦A2(f→(A))}.
the continuity degree of f with respect to σ₁ and σ₂ is defined by

αc(f)=⋀B∈LY{σ(B)↦A1(f←(B))}.
the irresoluteness degree of f with respect to σ₁ and σ₂ is defined by

αi(f)=⋀B∈LY{A2(B)↦A1(f←(B))}.

Definition 3.2

For any two L-fpts’s (X, σ₁) and (Y, σ₂) and any bijective function f : (X, σ₁) ⟶ (Y, σ₂), the α-homomorphism degree of f with respect to σ₁ and σ₂ is given by

α-Hom(f)=αi(f)∧αo(f).

Remark 3.3

Based on (2) of Lemma 2.1, αc(f) = 1_L implies to 𝒜₁(f^← (B)) ≥ σ₂(B) for all B ∈ L^Y. This is exactly the definition of α-continuous function. The cases α o(f) = 1_L and α i(f) = 1_L can be shown similarly. Thus (2) and (3) in Definition 3.1 are precisely the α-open and α-irresolute function’s definition as in the sense of Definition 2.10.
For the identity function i : (X, σ₁) ⟶ (X, σ₁), we have α i(i) = α o(i) = α - Hom(i) = 1_L.

By using Definition 3.1 and Corollary 2.11, we can state the following corollary.

Corollary 3.4

If (X, σ₁) and (Y, σ₂) are L-fpts’s and f : (X, σ₁) ⟶ (Y, σ₂) is a function, then:

the α-continuity degree of f is characterized by

αc(f)=⋀B∈LY{σ∗(B)↦A1∗(f←(B))}.
the α-irresoluteness degree of f is characterized by

αi(f)=⋀B∈LY{A2∗(B)↦A1∗(f←(B))}.

Definition 3.5

For any function f : (X, σ₁) ⟶ (Y, σ₂) where (X, σ₁) and (Y, σ₂) are two L-fpts’s, the α-closeness degree of f is given by

αcl(f)=⋀A∈LX{A1∗(A)↦A2∗(f→(A))}.

Theorem 3.6

If f : (X, σ₁) ⟶ (Y, σ₂) and g : (Y, σ₂) ⟶ (Z, σ₃) are two functions where (X, σ₁), (Y, σ₂) and (Z, σ₃) are three L-fpts’s, then:

α i(f) ∧ α i(g) ≤ α i(g ∘ f).
α o(f) ∧ α o(g) ≤ α i(g ∘ f).
α cl(f) ∧ α i(g) ≤ α cl(g ∘ f).

Proof

Since the proof of (2) and (3) is clear, we only prove (1). By using Definition 3.1 and Lemma 2.1 (4), we obtain

αi(f)∧αi(g)=⋀B∈LY{A2(B)↦A1(f←(B))}∧⋀C∈LZ{A3(C)↦A2(g←(C))}≤⋀C∈LZ{A2(g←(C))↦A1(f←(g←(C)))}∧⋀C∈LZ{A3(C)↦A2(g←(C))}=⋀C∈LZ{(A2(g←(C))↦A1((g∘g)←(C)))∧(A3(C)↦A2(g←(C)))}≤⋀C∈LZ{A3(g←(C))↦A1((g∘f)←(C))}=αi(g∘f).

□

By using Definition 3.2 and Theorem 3.6, we have the following corollary.

Corollary 3.7

Let (X, σ₁), (Y, σ₂) and (Z, σ₃) be L-fpts’s, f : X ⟶ Y and g : Y ⟶ Z be two bijective functions. Then α-Hom(f) ∧ α-Hom(g) ≤ α-Hom(g ∘ f).

Theorem 3.8

Let (X, σ₁), (Y, σ₂) and (Z, σ₃) be L-fpts’s and g : Y ⟶ Z be a surjective function. Then:

α o(g ∘ f) ∧ α i(f) ≤ α o(g).
α cl(g ∘ f) ∧ α i(f) ≤ α cl(g).

Proof

(1) Since f is a surjective function, we have (g ∘ f)^→ (f^← (B)) = g^→ (B) for each B ∈ L^Y. By using (4) of Lemma 2.1, we get

αo(g∘f)∧αi(f)=⋀A∈LX{A1(A)↦A3((g∘f)→(A))}∧⋀B∈LY{A2(B)↦A1(f←(B))}≤⋀B∈LY{A1(f←(B))↦A3((g∘f)→(f←(B)))}∧⋀B∈LY{A2(B)↦A1(g←(B))}=⋀B∈LY{(A1(f←(B))↦A3(g→(B)))∧(A2(B)↦A1(f←(B)))}≤⋀B∈LY{A2(B)↦A3(g→(B))}=αo(g).

Analogously, we can prove (2).□

Similarly, the following theorem is true.

Theorem 3.9

Given three L-fpts’s (X, σ₁), (Y, σ₂) and (Z, σ₃). If f : (X, σ₁) ⟶ (Y, σ₂) is an injective function and g : Y ⟶ Z is any function, then

α o(g ∘ f) ∧ α i(g) ≤ α o(f).
α cl(g ∘ f) ∧ α i(g) ≤ α cl(f).

Theorem 3.10

If f : (X, σ₁) ⟶ (Y, σ₂) is a bijective function where (X, σ₁) and (Y, σ₂) are two L-fpts’s, then

α i(f) = ⋀_A∈L^X {𝒜₂(f^→ (A)) ↦ 𝒜₁(A)}.
α o(f) = ⋀_B∈L^Y {𝒜₁(f^← (B)) ↦ 𝒜₂(B)}.
α i(f⁻¹) = α o(f) = α cl(f).

Proof

The proof of (2) is similar to (1), we only prove (1) and (3).

(1) From the bijectivity of f, we get f^← (f^→ (A)) = A for any A ∈ L^X, and f^→ (f^← (B)) = B for any B ∈ L^Y. It follows that

⋀A∈LX{A2(f→(A))↦A1(A)}=⋀A∈LX{A2(f→(A))↦A1(f←(f→(A)))}≥⋀B∈LY{A2(B)↦A1(f←(B))}=⋀B∈LY{A2(f→(f←(B)))↦A1(f←(B))}≥⋀A∈LX{A2(f→(A))↦A1(A)}.

Hence

αi(f)=⋀B∈LY{A2(B)↦A1(f←(B))}=⋀A∈LX{A2(f→(A))↦A1(A)}.

(3) Since f is a bijective function, we get (f⁻¹)^← (A) = f^→ (A) and f^→ (A′) = f^→ (A)′ for any A ∈ L^X. Therefore

αi(f−1)=⋀A∈LX{A1(A)↦A2((f−1)←(A))}=⋀A∈LX{A1(A)↦A2(f→(A))}=αo(f).

and

αo(f−1)=⋀A∈LX{A1(A)↦A2(f→(A))}=⋀A∈LX{A1(A′)↦A2(f→(A′))}=⋀A∈LX{A1(A′)↦A2(f→(A)′)}=αcl(f).

The proof is completed.□

Corollary 3.11

Given a bijective function f : (X, σ₁) ⟶ (Y, σ₂) between two L-fpts’s (X, σ₁) and (Y, σ₂), then:

α-Hom(f) = α i(f) ∧ α i(f⁻¹) = α i(f) ∧ α cl(f).
α-Hom(f) = ⋀_A∈L^X{𝒜₂(f^→ (A)) ↔ 𝒜₁(A)}.
α-Hom(f) = ⋀_B∈L^Y {𝒜₁(f^← (B)) ↔ 𝒜₂(B)}.

4 A new extension of α-compactness

Nannan and Ruiying [1] introduced the notion of α-compactness in L-fuzzy topology with the help of L-fuzzy α-open operator. In the following definition, we present the degree of α-compactness based on implication operation as a new generalization of α-compactness.

Definition 4.1

Let (X, σ) be an L-fpts. For any A ∈ L^X, let

αCom(A)=⋀U⊆LX{A(U)↦([A⊂~⋁U]↦⋁V∈2(U)[A⊂~⋁V])}=⋀U⊆LX{⋀A1∈UA(A1)↦{⋀x∈X(A′∨⋁A1∈UA1)(x)↦⋁V∈2(U)⋀x∈X(A′∨⋁A1∈VA1)(x)}}.

Then α Com₍A) is said to be the degree of α-compactness of A with respect to σ. By using Theorem 2.9, we have Com_𝒜(A) = α Com(A) for any A ∈ L^X.

Theorem 4.2

Let τ be an L-pt on X and A ∈ L^X. An L-subset A is fuzzy α-compact if and only if α Com_{χ_τ} (A) = 1_L, where the mapping χ_τ : L^X ⟶ L is given by

χτ(A)=1L,if A∈τ;0L,if A∉τ.

Proof

Let τ be an L-pt on X. It is clear that χ_τ is L-fpt. An L-subset A ∈ L^X is α-open set with respect to τ if and only if 𝒜_{χ_τ}(A) = 1_L. Based on the definition of fuzzy α-compactness, we have an L-subset A ∈ L^X is fuzzy α-compact such that for any collection 𝓤 ⊆ L^X, we have that

Aχτ(U)≤A⊆~⋁U≤⋁V∈2(U)A⊆~⋁V.

By using Lemma 2.1, A is fuzzy α-compact if and only if for any collection 𝓤 ⊆ L^X, we have

Aχτ(U)↦A⊆~⋁U↦⋁V∈2(U)A⊆~⋁V=1L.

This result together with the definition of α Com_{χ_τ}(A) yields to α Com_{χ_τ}(A) = 1_L.□

Theorem 4.3

Let σ be an L-fpt on X and A ∈ L^X. An L-subset A is L-fuzzy α-compact if and only if α Com(A) = 1_L.

Proof

Based on Definition 4.1 and Lemma 2.1, the conclusion is straightforward.□

Theorem 4.3

For any L-fpt σ on X and A ∈ L^X, we have α Com(A) ≤ Com(A).

Proof

Straightforward. □

Lemma 4.5

For any L-fpt σ on X and A ∈ L^X, we have α Com(A) ≥ u if and only if

A(U)∧A⊆~⋁U∧u≤⋁V∈2(U)A⊆~⋁V,

for any 𝓤 ⊆ L^X.

Proof

For every u ∈ L, A ∈ L^X and 𝓤 ⊆ L^X, we have

αCom(A)≥u⇔⋀U⊆LXA(U)↦A⊆~⋁U↦⋁V∈2(U)A⊆~⋁V≥u⇔A(U)↦A⊆~⋁U↦⋁V∈2(U)A⊆~⋁V≥u⇔A(U)∧A⊆~⋁U↦⋁V∈2(U)A⊆~⋁V≥u⇔A(U)∧A⊆~⋁U∧u≤⋁V∈2(U)A⊆~⋁V.

□

Theorem 4.6

For any L-fpt σ on X and A ∈ L^X, we have α Com(A) ≥ u if and only if

⋁B∈MA∗(B)′∨⋁x∈XA(x)∧⋀B∈MB(x)∨u′≥⋀N∈2(M)⋁x∈XA(x)∧⋀B∈NB(x),

for each 𝓜 ⊆ L^X.

Proof

Based on the definition of 𝒜^* and Lemma 2.1, the proof is clear.□

Theorem 4.7

For any L-fpt σ on X and A ∈ L^X, we have

αCom(A)=⋁u∈L|⋀B∈UA(B)∧A⊆~⋁U∧u≤⋁V∈2(U)A⊆~⋁V,∀U⊆LX.

Proof

By using Lemma 2.1, we have α Com(A) as the upper bound of

u∈L|⋀B∈UA(B)∧A⊆~⋁U∧u≤⋁V∈2(U)A⊆~⋁V,∀U⊆LX.

By using the Definition 4.1, we have

αCom(A)≤⋀B∈UA(B)↦A⊆~⋁U↦⋁V∈2(U)A⊆~⋁V=⋀B∈UA(B)∧A⊆~⋁U↦⋁V∈2(U)A⊆~⋁V,

for each 𝓤 ⊆ L^X. By applying the properties of the operation “↦”, we have

⋀B∈UA(B)∧A⊆~⋁U∧αCom(A)≤⋁V∈2(U)A⊆~⋁V,

and hence

αCom(A)∈⋁u∈L|⋀B∈UA(B)∧A⊆~⋁U∧u≤⋁V∈2(U)A⊆~⋁V,∀U⊆LX.

Therefore, we completed the proof.□

Theorem 4.8

For any L-fpt σ on X and A₁, A₂ ∈ L^X, we have

αCom(A1∨A2)≥αCom(A1)∧αCom(A2).

Proof

We can prove the theorem by using the next inequality:

αCom(A1∨A2)=⋁u∈L|⋀B∈UA(B)∧A1∨A2⊆~⋁U∧u≤⋁V∈2(U)A1∨A2⊆~⋁V,∀U⊆LX=⋁u∈L|⋀B∈UA(B)∧A1⊆~⋁U∧A2⊆~⋁U∧u≤⋁V∈2(U)A1⊆~⋁V∧A2⊆~⋁V,∀U⊆LX≥⋁u∈L|⋀B∈UA(B)∧A1⊆~⋁U∧u≤⋁V∈2(U)A1⊆~⋁V,∀U⊆LX∧⋁u∈L|⋀B∈UA(B)∧A2⊆~⋁U∧u≤⋁V∈2(U)A2⊆~⋁V,∀U⊆LX=αCom(A1)∧αCom(A2).

□

Theorem 4.9

For any L-fpt σ on X and A₁, A₂ ∈ L^X, we have

αCom(A1∧A2)≥αCom(A1)∧A(A2′).

Proof

We can prove the theorem by using the next inequality

αCom(A1∧A2)=⋁u∈L|⋀B∈UA(B)∧A1∧A2⊆~⋁U∧u≤⋁V∈2(U)A1∧A2⊆~⋁V,∀U⊆LX=⋁u∈L|⋀B∈UA(B)∧A1⊆~A2′∨⋁U∧u≤⋁V∈2(U)A1⊆~A2′∨⋁V,∀U⊆LX≥αCom(A1)∧A(A2′).

□

Corollary 4.10

For any L-fpt σ on X and A ∈ L^X, we have

αCom(A)≥αCom(1L)∧A(A′).

Theorem 4.11

For any L-fts’s (X, σ₁) and (X, σ₂) such that σ₁ ≤ σ₂ and for any A ∈ L^X, we have α Com_σ₂(A) ≤ α Com_σ₁(A).

Corollary 4.12

For any L-fpts (X, σ) with the base or the subbase 𝓑, we have α Com(A) ≤ α Com_𝓑(A), for any A ∈ L^X.

Theorem 4.13

For any L-fpts’s (X, σ₁) and (Y, σ₂), if f : (X, σ₁) ⟶ (Y, σ₂) is an L-fuzzy α-irresolute function, then

αComσ2(fL→(C))≥αComσ1(C),

for every C ∈ L^X.

Proof

For each C ∈ L^X, we have

αComσ2(fL→(C))=⋁u∈L|A2(U)∧fL→(C)⊆~⋁U∧u≤⋁V∈2(U)fL→(C)⊆~⋁V,∀U⊆LY≥⋁u∈L|A1(fL←(U))∧C⊆~⋁fL←(U)∧u≤⋁V∈2(U)C⊆~⋁fL←(V),∀U⊆LY≥⋁u∈L|A1(P)∧C⊆~⋁P∧u≤⋁R∈2(P)C⊆~⋁R,∀R⊆LX=αComσ1(C).

□

Theorem 4.14

For any L-fpts’s (X, σ₁) and (Y, σ₂), if f : (X, σ₁) ⟶ (Y, σ₂) is an L-fuzzy α-continuous function, then

Comσ2(fL→(C))≥αComσ1(C),

for every C ∈ L^X.

Proof

For each C ∈ L^X, we have

Comσ2(fL→(C))=⋁u∈L|σ2(U)∧fL→(C)⊆~⋁U∧u≤⋁V∈2(U)fL→(C)⊆~⋁V,∀U⊆LY≥⋁u∈L|A1(fL←(U))∧C⊆~⋁fL←(U)∧u≤⋁V∈2(U)C⊆~⋁fL←(V),∀U⊆LY≥⋁u∈L|A1(P)∧C⊆~⋁P∧u≤⋁R∈2(P)C⊆~⋁R,∀R⊆LX=αComσ1(C).

□

Theorem 4.15

For any L-fpt σ on X, A ∈ L^X and u ∈ L ∖ {0_L}, the next statements are equivalent:

α Com(A) ≥ u.
For each v ∈ P(L), v ≰ u, every strong v-shading 𝓤 of A with 𝒜(𝓤) ⧸ ≤ v has a finite sub-collection 𝓥 which is a strong v-shading of A.
For each v ∈ P(L), v ≰ u, every strong v-shading 𝓤 of A with 𝒜(𝓤) ⧸ ≤ v, there exists a finite sub-collection 𝓥 of 𝓤 and w ∈ β^*(v) such that 𝓥 is a w-shading of A.
For each v ∈ P(L), v ≰ u, every strong v-shading 𝓤 of A with 𝒜(𝓤) ⧸ ≤ v, there exists a finite sub-collection 𝓥 of 𝓤 and w ∈ β^*(v) such that 𝓥 is a strong w-shading of A.
For each v ∈ J(L), v ⧸ ≤ u′, every strong v-remote collection 𝓦 of A with 𝒜^*(𝓦) ⧸ ≤ v′ has a finite sub-collection 𝓡 which is a strong v-remote collection of A.
For each v ∈ J(L), v ⧸ ≤ u′, every strong v-remote collection 𝓦 of A with 𝒜^*(𝓦) ⧸ ≤ v′, there exist a finite sub-collection 𝓡 of 𝓦 and w ∈ α^*(v) such that 𝓡 is an w-remote collection of A.
For each v ∈ J(L), v ⧸ ≤ u′, every strong v-remote collection 𝓦 of A with 𝒜^*(𝓦) ⧸ ≤ v′, there exist a finite sub-collection 𝓡 of 𝓦 and w ∈ α^*(v) such that 𝓡 is a strong w-remote collection of A.
For each v ≤ u, u ∈ α(v), v, w ≠ 0_L, every Q_v-cover 𝓤 ⊆ (𝒜)_v of A has a finite sub-collection 𝓥 which is a Q_w-cover of A.
For each v ≤ u, w ∈ α(v), v, w ≠ 0_L, every Q_v-cover 𝓤 ⊆ (𝒜)_v of A has a finite sub-collection 𝓥 which is a strong α_w-cover of A.
For each v ≤ u, w ∈ α(v), v, w ≠ 0_L, every Q_v-cover 𝓤 ⊆ (𝒜)_v of A has a finite sub-collection 𝓥 which is a α_w-cover of A.
For each v ≤ u, w ∈ α (v), v, u ≠ 0_L, every strong α_v-cover 𝓤 ⊆ (𝒜)_v of A has a finite sub-collection 𝓥 which is a Q_w-cover of A.
For each v ≤ u, w ∈ α(v), v, w ≠ 0_L, every strong α_v-cover 𝓤 ⊆ (𝒜)_v of A has a finite sub-collection 𝓥 which is a strong α_w-cover of A.
For each v ≤ u, w ∈ α(v), v, w ≠ 0_L, every strong α_v-cover 𝓤 ⊆ (𝒜)_v of A has a finite sub-collection 𝓥 which is a α_w-cover of A.

Theorem 4.16

For any L-fpt σ on X, A ∈ L^X, and u ∈ L ∖ {0_L}, if α(w ∧ s) = α(w) ∧ α(s) for each w, s ∈ L, then the next statements will be equivalent:

α Com(A) ≥ u.
For each v ∈ α(u), v ≠ 0_L, every strong α_v-cover 𝓤 of A with v ∈ α(𝒜(𝓤)) has a finite sub-collection 𝓥 which is a Q_v-cover of A.
For each v ∈ α(u), v ≠ 0_L, every strong α_v-cover 𝓤 of A with v ∈ α(𝒜(𝓤)) has a finite sub-collection 𝓥 which is a strong α_v-cover of A.
For each v ∈ α(u), v ≠ 0_L, every strong α_v-cover 𝓤 of A with v ∈ α(𝒜(𝓤)) has a finite sub-collection 𝓥 which is a α_v-cover of A.

The following theorem and its corollary verify the relationship between α-irresoluteness degree and α-compactness degree.

Theorem 4.17

If f : (X, σ₁) ⟶ (Y, σ₂) is a function between two L-fpts’s (X, σ₁) and (Y, σ₂), then

αComA1(A)∧αi(f)≤αComA2(f→(A))

for any A ∈ L^X.

Proof

Suppose that u₁ ∈ L with u₁ ⊲ α Com_𝒜₁(A) ∧ α i(f). Then

u1⊲αi(f)=⋀B∈LY{A2(B)↦A1(f←(B))},

and

u1⊲αComA1(A)=⋀U∈LX{{⋀A1∈UA1(A1)∧⋀x∈X(A′∨⋁A1∈UA1)(x)}↦⋁V∈2(U)⋀x∈X(A′∨⋁A1∈VA1)(x)}

Then for any B ∈ L^Y and 𝓤 ⊆ L^X, we have u₁ ≤ 𝒜₂(B) ↦ 𝒜₁(f^← (B)) and

u1≤{⋀A1∈UA1(A1)∧⋀x∈X(A′∨⋁A1∈UA1)(x)}↦⋁V∈2(U)⋀x∈X(A′∨⋁A1∈VA1)(x).

By Lemma 2.1 (1), we have u₁ ∧ 𝒜₂(B) ≤ 𝒜₁(f^← (B)) for any B ∈ L^Y, and

u1∧⋀W∈UA1(W)∧⋀x∈X(A′∨⋁W∈UW)(x)≤⋁V∈2(U)⋀x∈X(A′∨⋁W∈UW)(x).

To prove

u1≤αComA2(f→(A))=⋀W∈LY{(⋀B1∈WA2(B1)∧⋀y∈Y(f→(A)′∨⋁B1∈WB1)(y))↦⋁D∈2(W)⋀y∈Y(f→(A)′∨⋁B1∈WB1)(y)},

for all 𝓦 ⊆ L^Y, let f^← (𝓦) = {f^← (B₁) | B₁ ∈ 𝓦} ⊆ L^X. Then, we have

u1∧⋀B1∈WA2(B1)∧⋀y∈Y(f→(A)′∨⋁B1∈WB1)(y)≤u1∧⋀B1∈WA1(f←(B1))∧⋀y∈Y(f→(A)′∨⋁B1∈WB1)(y)=u1∧⋀B1∈WA1(f←(B1))⋀x∈X(A′∨⋁B1∈Wf←(B1))(x)=u1∧⋀A1∈f←(W)A1(A1)∧⋀x∈X(A′∨⋁A1∈f←(W)(A1))(x)≤⋁V∈2(f←(W))⋀x∈X(A′∨⋁A1∈W(A1))(x)=⋁D∈2(W)⋀x∈X(A′∨⋁B1∈Df←(B1))(x)=⋁D∈2(W)⋀y∈Y(f→(A)′∨⋁B1∈DB1)(y).

By using Lemma 2.1 (1), we know

u1≤(⋀B1∈WA2(B1)∧⋀y∈Y(f→(A)′∨⋁B1∈WB1)(y))↦⋁D∈2(W)⋀y∈Y(f→(A)′∨⋁B1∈DB1)(y).

Thus

u1≤⋀W⊆LY{(⋀B1∈WA2(B1)∧⋀y∈Y(f→(A)′∨⋁B1∈WB1)(y))↦⋁D∈2(W)⋀y∈Y(f→(A)′∨⋁B1∈DB1)(y)}=αComA2(f→(A)).

Since u₁ is arbitrary, we have α Com_𝒜₁(A) ∧ α i(f) ≤ α Com_𝒜₂(f^→ (A)). The proof is completed.□

Corollary 4.18

For any surjective function f : (X, σ₁) ⟶ (Y, σ₂) where (X, σ₁) and (Y, σ₂) are L-fpts’s, we have

αComA1(1_LX)∧αi(f)≤αComA2(1_LY).

Permanent address: Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

Permanent address: Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt; nasserfuzt@hotmail.com

Acknowledgement

The authors would like to thank Deanship of Scientific Research at Majmaah University for supporting this work under Project Number No. 1440-91. Moreover, we are extremely grateful to the editors and the anonymous referees for their valuable comments and suggestions in improving this paper.

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Received: 2018-01-17

Accepted: 2019-04-05

Published Online: 2019-06-22

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0047

Keywords for this article

L-fuzzy pretopology; L-fuzzy α-open operator; L-fuzzy α-openness degree; L-fuzzy α-continuty degree; L-fuzzy α-irresolutness degree; L-fuzzy α-compactness degree

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