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

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.


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 rst 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] clari ed 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 de ne 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 rst discusses some important properties of L-fuzzy α-open operators. It then introduces αopenness, 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.

Preliminaries
In the sequel, X ≠ ∅, and L refers to a completely distributive De Morgan algebra (brie y, CDDA). Let L and 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 (brie y, 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 nonunit 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. U denotes the collection of all nite subcollections of U ⊆ L X . Evidently, L X is a CDDA when it inherits the structure of the lattice L in a natural way, by de ning , , ≤ and pointwisely. Further, {xu|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 de ned by Further, the operation ↔ is given by 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: An L-fuzzy inclusion [17,18] on X is de ned by the function⊂ : L X ×L X −→ L, where⊂(A , A ) = x∈X (A (x)∨ A (x)). We shall denote an L-fuzzy inclusion by [A ⊂A ]. For each function f : X −→ Y and C ⊆ L Y , the next equality is de ned in [19]: An L-topological space (brie y, L-ts) is a pair (X, τ), where the subfamily τ ⊆ L X contains L X , L X , and closed for any suprema and nite in ma. 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 byĀ and A • the closure and the interior of A, respectively.

De nition 2.3. [1]
Let σ be an L-fpt on X and let the mapping A : L X −→ L de ned as follows: . Corollary 2.4. If σ is an L-fpt on X and A ∈ L X , then:

In this case,
where Cl σ refers to the L-fuzzy closure operator induced by σ (see [23]). Proof. We can prove the theorem by using the following fact:

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
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(
Proof. The proof can be obtained from the following inequality: 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 i.e., w σ(B i ) and for any i ∈ I and xu De nition 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: In this case, an L-subset A is said to be fuzzy compact if and only if com(A) = L .

De nition 2.13. [1] Let σ be an
De nition 2.14. [25,26] For an L-pt τ on X, u ∈ L\{ L } and A ∈ L X , a family U ⊆ L X is said to be a αu-cover De nition 2.15. [25,26] For an L-pt τ on X, u ∈ L\{ L } and A ∈ L X , a family U ⊆ L X is said to be a Qu-cover De nition 2.16. [25,26] For an L-pt τ on X, u ∈ L\{ L } and A ∈ L X , a family U ⊆ L X is called:

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. (1) the α-openness degree of f with respect to σ and σ is de ned by (2) the continuity degree of f with respect to σ and σ is de ned by (3) the irresoluteness degree of f with respect to σ and σ is de ned by

De nition 3.2.
For any two L-fpts's (X, σ ) and (Y , σ ) and any bijective function f : (X, σ ) −→ (Y , σ ), the α-homomorphism degree of f with respect to σ and σ is given by  (1) the α-continuity degree of f is characterized by (2) the α-irresoluteness degree of f is characterized by

De nition 3.5.
For any function f : (X, σ ) −→ (Y , σ ) where (X, σ ) and (Y , σ ) are two L-fpts's, the αcloseness degree of f is given by Proof. Since the proof of (2) and (3) is clear, we only prove (1). By using De nition 3.1 and Lemma 2.1 (4), we obtain By using De nition 3.2 and Theorem 3.6, we have the following corollary.
Similarly, the following theorem is true.
(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 Hence (3) Since f is a bijective function, we get (f − ) ← (A) = f → (A) and f → (A ) = f → (A) for any A ∈ L X . Therefore and The proof is completed.

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 de nition, we present the degree of α-compactness based on implication operation as a new generalization of α-compactness.
De nition 4.1. Let (X, σ) be an L-fpts. For any A ∈ L X , let Then αCom ( 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) = αCom(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 αComχ τ (A) =
L , where the mapping χτ : L X −→ L is given by Proof. Let τ be an L-pt on X. It is clear that χτ is L-fpt. An L-subset A ∈ L X is α-open set with respect to τ if and only if Aχ τ (A) = L . Based on the de nition 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 By using Lemma 2.1, A is fuzzy α-compact if and only if for any collection U ⊆ L X , we have This result together with the de nition of αComχ τ (A) yields to αComχ τ (A) = L .  Proof. Straightforward.

Lemma 4.5. For any L-fpt σ on X and A ∈ L X , we have αCom(A) ≥ u if and only if
for any U ⊆ L X .
Proof. For every u ∈ L, A ∈ L X and U ⊆ L X , we have

Theorem 4.6. For any L-fpt σ on X and A ∈ L X , we have αCom(A) ≥ u if and only if B∈M
Proof. Based on the de nition 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 Proof. By using Lemma 2.1, we have αCom(A) as the upper bound of By using the De nition 4.1, we have for each U ⊆ L X . By applying the properties of the operation " →", we have and hence Therefore, we completed the proof. Proof. We can prove the theorem by using the next inequality: Proof. We can prove the theorem by using the next inequality   Proof. For each C ∈ L X , we have Proof. For each C ∈ L X , we have is a strong v-shading of A. sub-collection R of W and w ∈ α * (v) such that R is an w-remote collection of A. (7) For each v ∈ J(L), v ̸ ≤ u , every strong v-remote collection W of A with A * (W) ̸ ≤ v , there exist a nite sub-collection R of W and w ∈ α * (v) such that R is a strong w-remote collection of A.
which is a Qw-cover of A.
which is a strong αw-cover of A.
which is a αw-cover of A.
Theorem 4.16. For any L-fpt σ on X, A ∈ L X , and u ∈ L \ { L }, if α(w ∧ s) = α(w) ∧ α(s) for each w, s ∈ L, then the next statements will be equivalent: (2) For each v ∈ α(u), v ≠ L , every strong αv-cover U of A with v ∈ α(A (U)) has a nite sub-collection V which is a Qv-cover of A. (3) For each v ∈ α(u), v ≠ L , every strong αv-cover U of A with v ∈ α(A (U)) has a nite sub-collection V which is a strong αv-cover of A. (4) For each v ∈ α(u), v ≠ L , every strong αv-cover U of A with v ∈ α(A (U)) has a nite sub-collection V which is a αv-cover of A.
The following theorem and its corollary verify the relationship between α-irresoluteness degree and αcompactness degree. Since u is arbitrary, we have αCom A (A) ∧ αi(f ) ≤ αCom A (f → (A)). The proof is completed.