L‐fuzzy pre‐proximities, L‐fuzzy filters and L‐fuzzy grills

Introduction Proximity is an important concept in topology and it can be considered either as axiomatizations of geometric notions, close to but quite independent of topology, or as convenient tools for an investigation of topological spaces. Hence proximity has close relations with topology, uniformity and metric. With the development of topology, the theory of proximity makes a massive progress. In the framework of L-topology, many authors generalized the crisp proximity to L-fuzzy setting. For example, in [1], Ghanim et al. introduced the concept of S-quasi-proximities on [0, 1] and in [2], Shi studied S-quasi-proximities on L and pointwise S-quasi-proximities. Katsaras [3–5] introduced quasi-proximity in [0,1]-fuzzy set theory. Subsequently, Liu [6], Artico and Moresco [7] extended it into L-fuzzy set theory. In recently Yue and Shi extended the proximity theory of L-topology to L-fuzzy topology, see [8]. As an extension of Katsaras’s definition, Kim and Min[9] introduced L-fuzzy proximities on strictly two-sided, commutative quantales L in view points of Höhle fuzzy topology [10, 11]. Thron [12] carried out an extensive study of proximity structures with grills playing a central role. In this paper, we introduce more properties of L-fuzzy pre-proximities , L-fuzzy grills and L-fuzzy filters. Moreover, we investigate the relations among the L-fuzzy pre-proximities , L-fuzzy grills and L-fuzzy filters. We show that there is a Galois correspondence between the category of separated L-fuzzy grill spaces and that of separated L-fuzzy pre-proximity spaces. We introduce the local function associated with L-fuzzy grill and Abstract This article gives results on fixed complete lattice L‐fuzzy pre‐proximities, L‐fuzzy grills and L‐fuzzy filters. Moreover, we investigate the relations among the L‐fuzzy pre‐prox‐ imities , L‐fuzzy grills and L‐fuzzy filters. We show that there is a Galois correspondence between the category of separated L‐fuzzy grill spaces and that of separated L‐fuzzy pre‐proximity spaces. We introduced the local function associated with L‐fuzzy grill and L‐fuzzy topology and studied some of its properties. Finally, we build an L‐fuzzy topol‐ ogy for the corresponding L‐fuzzy grill by using local function.


Preliminaries
Throughout the text we consider (L, ≤, ∨, ∧) (or L in short) as fixed complete lattice, that is a lattice in which the suprema (joins) and infima (meets) for all subfamilies K ⊆ L exist. In particular, the top ⊤ and the bottom ⊥ elements in L exist and ⊤ � = ⊥. We use notation ∨ and ∧ to denote, respectively, infima and suprema of finite families of the elements of the lattice having notation and for the case when these families are arbitrary. We will additionally request the lattice L to be completely distributive, that is satisfying the first infinite distributive law of finite meets over arbitrary joins: If a ≤ b or b ≤ a , for each a, b ∈ L , then L is called a chain. A lattice L is called an order dense chain if for each a, b ∈ L such that a < b , there exists c ∈ L such that a < c < b.
Definition 2.1 [13][14][15][16] An implicator on a lattice L is a mapping →: L × L → L defined by x → y = {z ∈ L | x ∧ z ≤ y}, such that: (1) ⊤ → x = x , x → ⊤ = ⊤ and ⊥ → x = ⊤, (2) If y ≤ z , then x → y ≤ x → z and z → x ≤ y → x, (3) x ≤ y iff x → y = ⊤ and x ∧ y ≤ z iff x ≤ y → z for x, y, z ∈ L, (6) x ∧ (x → y) ≤ y and y ≤ x → (x ∧ y) and (x → y) → y ≥ x, (7) (x → ⊥) → (y → ⊥) = y → x, (8) x ∧ y = (x → (y → ⊥)) → ⊥, and x ∨ y = (x → ⊥) → y. From (7) and (1) we have the following important double negation property: Thus x → ⊥ is an order-reversing involution on L and in the following we write x * = x → ⊥. Referring to the properties of the implicator we see that De Morgan laws Page 3 of 20 Ramadan et al. J Egypt Math Soc (2020) 28:47 hold in the lattice with involution (L, ≤, ∨, ∧, * ) determined by an implicator. In what follows (L, ≤, ∨, ∧, →) is a complete lattice endowed with an implicator. For α ∈ L, f ∈ L X , we denote (α → f ), (α ∧ f ) and A fuzzy point x t for t ∈ L ⊥ = L − {⊥} is an element of L X such that, for y ∈ X: The set of all fuzzy points in X is denoted by Pt(X). Definition 2.2 [12]A map G : L X → L is called an L-fuzzy grill on X if G satisfies the following conditions for all f , g ∈ L X : LG1 The pair (X, G) is called an L-fuzzy grill space. An L-fuzzy grill space is called: [11,17] A mapping C : L X → L X is called an L-fuzzy closure operator on X if C satisfies the following conditions: for all f , g ∈ L X The pair (X, C) is called an L-fuzzy closure space. A L-fuzzy closure space (X, C) is called: Definition 2.4 [11] A map F : L X → L is called an L-fuzzy filter on X if F satisfies the following conditions for all f , g ∈ L X : The pair (X, F) is called an L-fuzzy filter space. An L-fuzzy filter space is called: Definition 2.5 [11,16,18] A mapping I : L X → L X is called an L-fuzzy interior operator on X if I satisfies the following conditions for all f , g ∈ L X : The pair (X, I) is called an L-fuzzy interior space. An L-fuzzy interior space (X, I) is called: Lemma 2.6 Let F : L X → L and G : L X → L be two maps. For all f ∈ L X and α ∈ L, the following statements are equivalent Definition 2.7 [9] A mapping δ : L X × L X → L is called an L-fuzzy pre-proximity on X if it satisfies the following axioms.
The pair (X, δ) is called an L-fuzzy pre-proximity space.
An L-fuzzy pre-proximity is called stratified if the following hold: Let (X, δ X ) and (Y , δ Y ) be two L-fuzzy pre-proximity spaces. A mapping φ : [19,20], A mapping T : L X → L is called an L-fuzzy topology on X if it satisfies the following conditions:

Definition 2.9
The pair (Y , T ) is called an L-fuzzy topological space.

The relationships between L-fuzzy pre-proximities and L-fuzzy grills
Now, let δ be an L-fuzzy pre-proximity, we can identify the relation δ f on L X with the mapping δ f : L X → L such that It is clear that δ f is L-fuzzy grill. Let P(X) and G(X) be the families of all L-fuzzy pre-proximities and L-fuzzy grills on X, respectively. Theorem 3.1 For the mapping H : P(X) × G(X) → G(X) defined as follows: We have the following properties: (LG3) Let f , g ∈ L X . Then we have (2) It is clear from the definition.
(4) Let α ∈ L and f ∈ L X . If δ and G are stratified, then we have Thus, H(δ, G) is stratified.
If δ and G are Alexandrov , then we have Then we have the following properties.
From the following theorem, we obtain an L-fuzzy pre-proximity induced by an L-fuzzy grill.
, for all x ∈ X. Then we have the following properties.
Thus, (P4) For every f 1 , f 2 , g 1 , g 2 ∈ L X , we have and Hence, δ G is an L-fuzzy pre-proximity on X.
. (2) If G is a stratified, we have and for each, f , g ∈ L X and α ∈ L. and Thus, δ G is Alexandroff. (1) δ G is an L-fuzzy pre-proximity.
The relationships between L-fuzzy pre-proximities and filters Now, let δ be an L-fuzzy pre-proximity, we can identify the relation F f on L X with the mapping F f : L X → L such that It is clear that F f is L-fuzzy filter. Let F(X) be the family of all L-fuzzy filters on X.

Theorem 4.1 For the mapping H : P(X) × F(X) → F(X) defined as follows:
Then we have the following properties: (1) H(δ, F) ∈ F(X), If δ and F are stratified, then H(δ, F) is stratified.
(LF2) Easily proved (LF3) Let f , g ∈ L X . Then we have (2) It is clear from the definition (1) (X, I F ) is an L-fuzzy interior space (2) If F is stratified, then I F is stratified.
(3) If F is separated (resp., Alexandrov), then so is I F . (1) (X, F I ) is an L-fuzzy filter space with If I is stratified, then F I is stratified.
(3) If I is separated (resp., Alexandrov), then so is F I , (4) F I F ≤ F and I F I ≤ I. (1) δ F is an L-fuzzy pre-proximity, Hence, δ F is an L-fuzzy pre-proximity.
. (5) It is easily proved from definitions.
Example 4.5 (1) Define C 1 : L X → L X as C 1 (f )(x) = x∈X f (x) and G 1 : L X → L as G 1 (f ) = x∈X f (x). Hence C 1 is L-fuzzy closure operator on X and G 1 is L-fuzzy grill on X. Since C 1 (⊤ * x ) = ⊤ X and G 1 (⊤ * x ) = ⊤ X , C 1 and G 1 and are not separated. Theorems 3.2 and 3.3, C G C 1 ≥ C 1 and G C G 1 ≥ G 1 . By Theorem 3.4 , we have (2) Define C 2 : L X → L X as C 2 (f )(x) = f (x) and G 2 : L X → L as G 2 (f ) = f , then C 2 is L-fuzzy closure operator on X and G 2 is L-fuzzy grill on X. Since C 2 (⊤ * x )(x) = ⊤ * x and G 2 (⊤ * x ) = ⊤ * x = ⊥, then C 2 and G 2 are separated. From Theorems 3.2 and 3.3, C G C 2 ≥ C 1 and G C G 2 ≥ G 1 . By Theorem 3.4 , we have Hence I 1 is L-fuzzy interior operator on X and F 1 is L-fuzzy filter on X. Since I 1 (⊤ x ) = ⊥ X and F 1 (⊤ x ) = ⊥ , I 1 and F 1 are not separated. By Theorems 4.2 and 4.3 we obtain I F I 1 ≤ I 1 and F I F 1 ≤ F 1 . By Theorem 4.4 , we have (4) Define I 2 : L X → L X as I 2 (f )(x) = f (x) and F 2 : L X → L as I 2 (f ) = f (x). Hence, I 2 is L-interior operator on X and F 2 is L-fuzzy filter. Since I 2 (⊤ x ) = ⊤ x and F 2 (⊤ x ) = ⊤ , I 2 and F 2 are separated. By Theorem 4.4, we obtain L-fuzzy preproximities δ I 2 as

Galois correspondences
Theorem 5.1 Let (X, G X ) and (Y , G Y ) be L-fuzzy grill spaces and φ : X → Y be a map.
Proof For each f ∈ L Y , we have g(y)).
). Theorem 5.2 Let (X, C X ) and (Y , C Y ) be L-fuzzy closure spaces and φ : X → Y be a map. If a map φ : is an LF-grill map.
Proof For each f ∈ L Y , we have Proof For each f ∈ L Y , we have is an LF-filter map.
Proof For each f ∈ L Y , we have Theorem 5.5 Let (X, G X ) and (Y , G Y ) be L-fuzzy grill spaces and φ : (X, G X ) → (Y , G Y ) be an LF-grill map. Then φ : Proof Since G X (φ ← (g)) ≤ G Y (g) , we have Theorem 5.6 Let (X, F X ) and (Y , F Y ) be L-fuzzy filter spaces and φ : Definition 5.7 [21,22] Suppose that F : D → C, G : C → D are concrete functors. The pair (F, G) is called a Galois correspondence between C and D if for each Y ∈ C, id Y : F • G(Y ) → Y is a C-morphism, and for each X ∈ D , id X : If (F, G) is a Galois correspondence, then it is easy to check that F is a left adjoint of G, or equivalently that G is a right adjoint of F. The category of separated L-fuzzy closure spaces with LF-closure mappings as morphisms is denoted by SCS.
The category of separated L-fuzzy interior spaces with LF-interior mappings as morphisms is denoted by SIS.
The category of separated L-fuzzy filter spaces (resp. separated L-fuzzy grill spaces) with L-filter mappings (resp. L-grill maps) as morphisms is denoted by SFF (resp. SFG).
From Theorems 3.2 and 5.1, we obtain a concrete functor ϒ : SFG → SCS defined as From Theorems 3.2 and 5.2, we obtain a concrete functor : SCS → SFG defined as �(X, C) = (X, G C ), �(φ) = φ.  (4), if G X is an separated L-fuzzy grill on a set X, then ϒ(�(G X )) = G C G X ≥ G X . Hence, the identity map id X : (X, G X ) → (X, G C X ) = (X, ϒ(�(F X ))) is an LF-closure map. Moreover, if C Y is a separated L-fuzzy closure on a set Y, by Theorem 3.3(4), �(ϒ(C Y )) = C G C Y ≥ C Y . Hence the identity map id Y : (Y , G C G Y ) → (Y , δ Y ) is LF-closure map. Therefore (ϒ, �) is a Galois correspondence. Proof By Theorem 4.3(4), if F X is a separated L-fuzzy filter on a set X, then �(Ŵ(F X )) = G I F X ≤ F X . Hence, the identity map id X : (X, F X ) → (X, G I F X ) = (X, �(Ŵ(F X ))) is an LF-filter map. Moreover, if δ Y is a separated L-fuzzy preproximity on a set Y, by Theorem 4.3(4), Ŵ(�(I Y )) = I F I Y ≤ I Y . Hence the identity map id Y : (Y , Ŵ(�(I Y ))) → (Y , I Y ) is an LF-interior map. Therefore (�, Ŵ) is a Galois correspondence.

L-fuzzy grill fuzzy topological space
In this section, we assume that L is an order dense chain. Let T (x t , r) = {g ∈ L X : x t ∈ g, T (g) ≥ r}. Definition 6.1 Let (X, T ) be an L-fuzzy topological space and G be an L-fuzzy grill on X. Then, the triplet (X, T , G) is called an L-fuzzy grill fuzzy topological space. Definition 6.2 Let (X, T , G) be an L-fuzzy grill fuzzy topological space. The operator G,T : L X × L ⊥ → L X which defined by: is called the local function associated with L-fuzzy grill G and L-fuzzy topology T , simply we denote it by � G (f , r) .  x t ∈ P t (X) : G(f ∧ g) ≥ r, for each g ∈ T (x t , r)