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

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-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 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 topology 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)