INITIAL SOFT L-FUZZY PREPROXIMITIES

Abstract

In this paper, we introduce the notions of soft L-fuzzy preproximities in complete residuated lattices. We prove the existence of initial soft L-fuzzy preproximities. From this fact, we define subspaces and product spaces for soft L-fuzzy preproximity spaces. Moreover, we give their examples.

1. INTRODUCTION

Hájek [5] introduced a complete residuated lattice which is an algebraic structure for many valued logic. It is an important mathematical tool for algebraic structures [6, 7-9]. Recently, Molodtsov [11] introduced the soft set as a mathematical tool for dealing information as the uncertainty of data in engineering, physics, computer sciences and many other diverse field. Presently, the soft set theory is making progress rapidly [1, 4]. Pawlak’s rough set [12, 13] can be viewed as a special case of soft rough sets [4]. The topological structures of soft sets have been developed by many researchers [2, 7-9, 15-17].

Čimoka et.al [3] introduced L-fuzzy syntopogenous structures as fundamentals and application to L-fuzzy topologies, L-fuzzy proximities and L-fuzzy uniformities in a complete residuated lattice. Kim [7] introduced a fuzzy soft F : A → LU as an extension as the soft F : A → P(U) where L is a complete residuated lattice. Kim [7-9] introduced the soft topological structures, soft L-fuzzy quasi-uniformities and soft L-fuzzy topogenous orders in complete residuated lattices.

In this paper, we prove the existence of initial soft L-fuzzy preproximities. From this fact, we define subspaces and product spaces for soft L-fuzzy preproximity spaces. Moreover, we give their examples.

 

2. PRELIMINARIES

Definition 2.1 ([5, 6]). An algebra (L,∧, ∨,⨀,→, 0, 1) is called a complete residuated lattice if it satisfies the following conditions:

(C1) L = (L,≤,∨,∧,1, 0) is a complete lattice with the greatest element 1 and the least element 0; (C2) (L,⨀, 1) is a commutative monoid; (C3) x ⨀ y ≤ z iff x ≤ y → z for x, y, z ∈ L.

In this paper, we assume that (L, ≤, ⨀, →, ⊕, *) is a complete residuated lattice with an order reversing involution * which is defined by x ⊕ y = (x* ⨀ y*)* and x* = x → 0.

Lemma 2.2 ([5, 6]). For each x, y, z, xi, yi,w ∈ L, we have the following properties.

(1) 1 → x = x, 0 ⨀ x = 0, (2) If y ≤ z, then x⨀y ≤ x⨀z, x⊕y ≤ x⊕z, x → y ≤ x → z and z → x ≤ y → x, (3) x ⨀ y ≤ x ∧ y ≤ x ∨ y ≤ x ⊕ y, (4) , , (5) , (6) , (7) , (8) , (9) , (10) , (11) (x ⨀ y) → z = x → (y → z) = y → (x → z), (12) x ⨀ (x → y) ≤ y and x → y ≤ (y → z) → (x → z), (13) (x → y) ⨀ (z → w) ≤ (x ⨀ z) → (y ⨀ w), (14) (x → y) ⨀ (z → w) ≤ (x ⊕ z) → (y ⊕ w), (15) x → y ≤ (x ⨀ z) → (y ⨀ z) and (x → y) ⨀ (y → z) ≤ x → z, (16) x ⨀ y ⨀ (z ⨀ w) ≤ (x ⨀ z) ⊕ (y ⨀ w). (17) x → y = y* → x*.

Definition 2.3 ([7-9]). Let X be an initial universe of objects and E the set of parameters (attributes) in X. A pair (F, A) is called a fuzzy soft set over X, where A ⊂ E and F : A → LX is a mapping. We denote S(X, A) as the family of all fuzzy soft sets under the parameter A.

Definition 2.4 ([7-9]). Let (F, A) and (G, A) be two fuzzy soft sets over a common universe X.

(1) (F, A) is a fuzzy soft subset of (G, A), denoted by (F, A) ≤ (G, A) if F(ϵ) ≤ G(ϵ), for each ϵ ∈ A. (2) (F, A) ∧ (G, A) = (F ∧ G, A) if (F ∧ G)(ϵ) = F(ϵ) ∧ G(ϵ) for each ϵ ∈ A. (3) (F, A) ∨ (G, A) = (F ∨ G, A) if (F ∨ G)(ϵ) = F(ϵ) ∨ G(ϵ) for each ϵ ∈ A. (4) (F, A) ⨀ (G, A) = (F ⨀ G, A) if (F ⨀ G)(ϵ) = F(ϵ) ⨀ G(ϵ) for each ϵ ∈ A. (5) (F, A)* = (F*, A) if F*(ϵ) = (F(ϵ))* for each ϵ ∈ A. (6) (F, A)⊕(G, A) = (F ⊕G, A) if (F ⊕ G)(ϵ) = (F*(ϵ)⨀G*(ϵ))* for each ϵ ∈ A.

Definition 2.5 ([8, 9]). Let S(X, A) and S(Y, B) be the families of all fuzzy soft sets over X and Y , respectively. The mapping fϕ : S(X, A) → S(Y, B) is a soft mapping where f : X → Y and ϕ : A → B are mappings.

(1) The image of (F, A) ∈ S(X, A) under the mapping fϕ is denoted by fϕ((F, A)) = (fϕ(F), B) where (2) The inverse image of (G, B) ∈ S(Y, B) under the mapping fϕ is denoted by where (3) The soft mapping fϕ : S(X, A) → S(Y, B) is called injective (resp. surjective, bijective) if f and ϕ are both injective (resp. surjective, bijective).

Lemma 2.6 ([8, 9]). Let fϕ : S(X, A) → S(Y, B) be a soft mapping. Then we have the following properties. For (F, A), (Fi, A) ∈ S(X, A) and (G, B), (Gi, B) ∈ S(Y, B),

(1) with equality if f is surjective, (2) with equality if f is injective, (3) , (4) , (5) , (6) with equality if f is injective, (7) (8) (9) fϕ((F1, A)⨀(F2, A)) ≤ fϕ((F1, A))⨀fϕ((F2, A)) with equality if f is injective. (10) fϕ((F1, A) ⊕ (F2, A)) ≤ fϕ((F1, A)) ⊕ fϕ((F2, A)).

Definition 2.7. A function δ : LX × LX → L is called a soft L-fuzzy pre-proximity on X if it satisfies the following conditions:

(SP1) δ((1X, A), (0X, A)) = 0 and δ((0X, A), (1X, A)) = 0.

(SP2) If (F, A) ≤ (F1, A) and (G, A) ≤ (G1, A), then δ((F, A), (G, A)) ≤ δ((F1, A), (G1, A)).

(SP3) If δ((F, A), (G, A)) ≠ 1, then (F, A) ≤ (G, A)*.

(SP4) δ((F1, A) ⨀ (F2, A), (H1, A) ⊕ (H2, A) ≤ δ((F1, A), (H1, A)) ⊕ δ((F2, A), (H2, A)).

The triple (X, A, δ) is said to be a soft L-fuzzy pre-proximity space.

A soft L-fuzzy pre-proximity space is called a soft L-fuzzy quasi-proximity if (SQ)

A soft L-fuzzy pre-proximity space is called perfect if (P)

Let (X, A, δ1) and (X, A, δ2) be soft L-fuzzy pre-proximity spaces. We say that δ1 is finer than δ2 (δ2 is coarser than δ1) if δ1((F, A), (G, A)) ≤ δ2((F, A), (G, A)) for all (F, A), (G, A) ∈ S(X, A).

Let (X, A, δX) and (Y, B, δY ) be soft L-fuzzy pre-proximity spaces and fϕ : X → Y be a soft map. Then f is called a fuzzy proximity soft map if ∀(F, A), (G, A) ∈ S(X, A), δX((F, A), (G, A)) ≤ δX((fϕ((F, A)), (fϕ((G, A))).

Remark 2.8. (1) If a complete residuated lattice (L, ≤, ⨀, ⊕, * ) is a completely distributive lattice (L, ≤, ∧, ∨, * ) with a strong negation * with ⨀ = ∧ and ⊕ = ∨, the above definition coincide with that in the sense [3].

(2) Let (X, A, δ) be a soft L-fuzzy pre-proximity space. By (SP4), we have

where K = {σ | σ : {1, 2, ..., p} → {1, 2, ..., p} is a bijective function}.

(3) Let L be an idempotent complete residuated lattice, that is, x ⨀ x = x, for each x ∈ L. Since (F, A) ⨀ (F, A) = (F, A) and (G, A) ⊕ (G, A) = (G, A), then δ((F, A), (G1, A) ⊕ (G2, A)) ≤ δ((F, A), (G1, A)) ⊕ δ((F, A), (G2, A)) and δ((F1, A) ⨀(F2, A), (G, A)) ≤ δ((F1, A), (G, A)) ⊕ δ((F2, A), (G, A)).

 

3. INITIAL SOFT L-FUZZY PREPROXIMITIES

Theorem 3.1. Let {(Xi, Ai, δi) | i ∈ Г } be a family of soft L-fuzzy pre-proximity spaces. Let X be a set and, for each i ∈ Г, fi : X → Xi and ϕi : A → Ai mappings. Define the function δ : S(X, A) × S(X, A) → L on X by

where the first is taken over all two finite families , and

Then:

(1) δ is the coarsest soft L-fuzzy pre-proximity on X which all (fi)ϕi ,i ∈ Г, are fuzzy proximity soft maps. (2) If {(Xi, Ai, δi) | i ∈ Г } is a family of soft L-fuzzy quasi-proximity spaces, δ is a soft L-fuzzy quasi-proximity on X. (3) A map fϕ : (Y, B, δ0) → (X, A, δ) is a fuzzy proximity soft map iff each (fi)ϕi ∘ fϕ : (Y, B, δ0) → (Xi, Ai, δi) is a fuzzy proximity soft map.

Proof. (1) First, we will show that δ is a soft L-fuzzy pre-proximity on X.

(SP1) Since δ((F, A), (0X, A)) ≤ δi((fi)ϕi((F, A)), (0Xi , Ai)) = 0 for all (F, A) ∈ S(X, A), it is clear.

(SP2) It follows from the definition of δ.

(SP3) We will show that if (F, A) ≰ (G, A)* , then δ((F, A), (G, A)) = 1.

Let (F, A) ≰ (G, A)*. Then, for every two finite families and and σ ∈ K, there exist j0, σ(j0), x0 such that (Fj0 , A)(x0) ≰ (Gσ(j0), A)(x0)*: It follows that, for all i ∈ Г,

Since δi is a soft L-fuzzy pre-proximity on Xi,for each i ∈ Г, by (SP3),

So, . By Lemma 2.2(3), it follows

for every two finite families and and σ ∈ K . Hence δ((F, A), (G, A)) = 1.

(SP4) Suppose there exist (Fi, A), (Gi, A) ∈ S(X, A) such that

By the definition of δ((F1, A), (G1, A)) and Lemma 2.2(6), there exist two finite families and with a bijective function σ, we have

Again, by the definition of δ((F2, A), (G2,A)) and Lemma 2.2(6), there exist two finite families and with a bijective function ϵ, we have

By Lemma 2.2(6), for each j, σ(j) and k, ϵ(k), there exist ij , ik ∈ Г such that

On the other hand, since

for a bijective function σ ∪ ϵ, we have

It is a contradiction. Hence the condition (SP4) holds.

Second, from the definition of δ, for two families {(F, A) | (F, A) = (F, A)} and {(G, A) | (G, A) = (G, A)}, since

for each i ∈ Г, (fi)ϕi : (X, A, δ) → (Xi, Ai, δi) is a fuzzy proximity soft map.

If all (fi)ϕi : (X, A, δ0) → (Xi, Ai, δi) are fuzzy proximity soft maps, then, for all two finite families and σ ∈ K,

Thus, δ0((F, A), (G, A)) ≤ δ((F, A), (G, A)) for each (F, A), (G, A) ∈ S(X, A).

(2) Let {(Xi, Ai, δi) | i ∈ Г } be a family of soft L-fuzzy quasi-proximity spaces. We will show that δ is an soft L-fuzzy quasi-proximity on X.

Suppose there exist (F, A), (G, A) ∈ S(X, A) such that

By the definition of δ, there are finite families and and a bijective function σ such that

It follows that for any j, σ(j), there exists an ij ∈ Г such that

Since δij is a soft L-fuzzy quasi-proximity on Xij , by (SQ), there exists (Hij , Aij ) ∈ S(Xij , Aij ) such that

(B)

On the other hand, put . Since

for the identity σ(j) = j, then

Since , for σ ∈ K, we have

It implies

It is a contradiction for (B). Thus, the result follows.

(3) Necessity of the composition condition is clear since the composition of fuzzy proximity soft maps is a fuzzy proximity soft map.

Each two finite families and and each σ ∈ K, we have

It follows

Since (fi)ϕi ∘ fϕ is a fuzzy proximity soft map and for any j, σ(j),

Since , we have, for all j, σ(j) and i ∈ Г,

Hence δ0((F, A), (G, A)) ≤ δ(fϕ((F, A)), fϕ((G, A))).      □

From Remark 2.8(3) and Theorem 3.1, we obtain the following corollary.

Corollary 3.2. Let (L, ⨀, ≤) be an idempotent complete residuated lattice. Let {(Xi, Ai, δi) | i ∈ Г } be a family of soft L-fuzzy pre-proximity spaces. Let X be a set and, for each i ∈ Г, fi : X → Xi a mapping. Define the function δ : S(X, A) × S(X, A) → L on X by

where the first is taken over all two finite families and . Then δ is the coarsest soft L-fuzzy pre-proximity on X which for each i ∈ Г, (fi)ϕi is a fuzzy proximity soft map.

Let SPROX be a category with object (X, A, δX) where δX is a soft L-fuzzy preproximity with a morphism fϕ : (X, A, δX) → (Y, B, δY ) is a fuzzy proximity soft map. Let SET be a category with object (X, f) where X is a set with a morphism f : X → Y is a function.

Theorem 3.3. The forgetful functor U : SPROX → Set defined by U(X, A, δ) = X and U(f) = f is topological.

Proof. From Theorem 3.1, every U-structured source (fi : X → U(Xi, Ai, δi))i∈Г has a unique U-initial lift (fi : (X, A, δ) → (Xi, δi))i∈Г where δ in Theorem 3.1.      □

Corollary 3.4. Let (Y, B, δY ) be a soft L-fuzzy pre-proximity space. Let X be a set , f : X → Y and ϕ : A → B mappings. Define the function δ : S(X, A) × S(X, A) → L on X by

where the first is taken over all two finite families , and

Then δ is the coarsest soft L-fuzzy pre-proximity on X which fϕ is a fuzzy proximity soft map such that

Proof. From Theorem 3.1 and the definition of δ((F, A), (G, A)), we only show:

Suppose . Then there exist two finite families , and σ ∈ K such that

On the other hand, since and from Lemma 2.6(9,10), we have

It is a contradiction. Hence the result holds.      □

Definition 3.5. Let (X, A, δX) be a soft L-fuzzy pre-proximity space , Z ⊂ X and C ⊂ A. The pair (Z, C, δ) is said to be a subspace of (X, A, δX) if it is endowed with the initial soft L-fuzzy pre-proximity with respect to (Z, i, (X, δX)) where i is the inclusion function. From Corollary 3.8, we define the function δ : LZ × LZ → L on A by

Definition 3.6. Let X = ∏i∈Г Xi be the product of the sets from family {(Xi, Ai, δi)| i∈Г } of soft L-fuzzy pre-proximity spaces. The initial soft L-fuzzy pre-proximity δ = ⊗δi on X with respect to the family {πi : X → (Xi, Ai, δi) | i∈Г } of all projection maps is called the product soft L-fuzzy pre-proximity of {δi | i∈Г }, and (X, ∏i∈Г Ai, ­⊗δi) is called the product soft L-fuzzy pre-proximity space.

Example 3.7. Let U = {hi | i = {1, ..., 6}} with hi=house and E = {e, b, w, c, i} with e=expensive,b= beautiful, w=wooden, c= creative, i=in the green surroundings. Define a binary operation ⨀ on [0, 1] by

Then ([0, 1], ∧,→, 0, 1) is a complete residuated lattice (ref. [5, 6]). Let A = {b, c, i}⊂ E and X = {h1, h4, h5, h6}. Put (H, A) be a fuzzy soft set as follow:

(1) We define soft L-fuzzy preproximities δ1, δ2 : S(X, A) × S(X, A) → L as

But δi for i = 1, 2, is not a soft L-fuzzy quasi-proximity because

(2) By Theorem 3.1, let f1 = f2 : X → S and ϕ1 = ϕ2 : A → A be identity maps. We obtain the coarsest soft L-fuzzy preproximity δ : S(X, A) × S(X, A) → L which is finer than δi, i = 1, 2, as follows