STRUCTURES INDUCED BY ALEXANDROV FUZZY TOPOLOGIES

Abstract

In this paper, we investigate the properties of Alexandrov fuzzy topologies and meet-join approximation operators. We study fuzzy preorder, Alexandrov topologies and meet-join approximation operators induced by Alexandrov fuzzy topologies. We give their examples.

1. Introduction

Hájek [2] introduced a complete residuated lattice which is an algebraic structure for many valued logic. Höhle [3] introduced L-fuzzy topologies and L-fuzzy interior operators on complete residuated lattices. Pawlak [8, 9] introduced rough set theory as a formal tool to deal with imprecision and uncertainty in data analysis. Radzikowska [10] developed fuzzy rough sets in complete residuated lattice. Bělohlávek [1] investigated information systems and decision rules in complete residuated lattices. Zhang [6, 7] introduced Alexandrov L-topologies induced by fuzzy rough sets. Kim [5] investigated the properties of Alexandrov topologies in complete residuated lattices.

In this paper, we investigate the properties of Alexandrov fuzzy topologies and meet-join approximation operators in a sense as Höhle [3]. We study fuzzy preorder, Alexandrov topologies and meet-join approximation operators induced by Alexandrov fuzzy topologies. We give their examples.

 

2. Preliminaries

Definition 2.1 ([1-3]). A structure (L,∨,∧,⊙, →, ⊥,⊤) is called a complete residuated lattice iff it satisfies the following properties:

(L1) (L,∨,∧,⊥,⊤) is a complete lattice where ⊥ is the bottom element and ⊤ is the top element; (L2) (L, ⊙, ⊤) is a monoid; (L3) It has an adjointness,i.e.

x ≤ y → z iff x ⊙ y ≤ z.

An operator * : L→ L defined by a* = a → ⊥ is called strong negations if a** = a.

In this paper, we assume that (L, ∨, ∧, ⊙, →, *, ⊥, ⊤) be a complete residuated lattice with a strong negation *.

Definition 2.2 ([6, 7]). Let X be a set. A function eX : X × X → L is called a fuzzy preorder if it satisfies the following conditions

(E1) reflexive if eX (x, x) = 1 for all x ∈ X, (E2) transitive if eX (x, y) ⊙ eX (y, z) ≤ eX (x, z), for all x, y, z ∈ X’

Example 2.3. (1) We define a function eL : L × L → L as eL (x, y) = x → y. Then eL is a fuzzy preorder on L. (2) We define a function eLX : LX × LX → L as Then eLX is a fuzzy preorder from Lemma 2.4 (9).

Lemma 2.4 ([1, 2]). Let (L,∨,∧,⊙, →,*, ⊥,⊤) be a complete residuated lattice with a strong negation *. For each x, y, z, xi, yi ∈ L, the following properties hold.

(1) If y ≤ z, then x ⊙ y ≤ x ⊙ z. (2) If y≤ z, then x → y ≤ x → z and z → x ≤ y → x. (3) x → y = ⊤ iff x ≤ y. (4) x → ⊤ = ⊤ and ⊤ → x = x. (5) x ⊙ y ≤ x ∧ y. (6) and . (7) and . (8) and . (9) (x → y) ⊙ x ≤ y and (y → z) ⊙ (x → y) ≤ (x → z). (10) x → y ≤ (y → z) → (x→ z) and x → y ≤ (z → x) → (z →y). (11) and (12) (x ⊙ y) → z = x → (y → z) = y → (x → z) and (x ⊙ y)* = x→ y*. (13) x* → y* = y → x and (x → y)* = x ⊙ y*. (14) y → z ≤ x ⊙ y → x ⊙ z.

Definition 2.5 ([5]). A map : LX → LY is called an meet-join approximation operator if it satisfies the following conditions, for all A, Ai ∈ LX, and α ∈ L,

(M1)where (α → A)(x) = α →A(x) for each x ∈ X, (M2) (M3) A* ≤ (A), (M4) (*(A))≤ (A).

Definition 2.6 ([4]). An operator T : LX → L is called an Alexandrov fuzzy topology on X iff it satisfies the following conditions, for all A, Ai ∈ LX, and α∈ L,

(T1) T(αX) = ⊤, where α X (x) = α for each x ∈ X,(T2) T(Ai) ≥ T(Ai) and T( Ai) ≥ T(Ai),(T3) T(α ⊙ A) ≥ T(A), where (α ⊙ A)(x) = α ⊙ A(x) for each x ∈ X,(T4) T(α → A) ≥ T(A).

Definition 2.7 ([5]). A subset τ ⊂ LX is called an Alexandrov topology if it satisfies satisfies the following conditions.

(O1) αX ∈ τ.(O2) If Ai ∈ τ for i ∈ Γ,Ai , Ai ∈ τ .(O3) α ⊙ A ∈ τ for all α ∈ L and A ∈ τ .(O4) α → A ∈ τ for all α ∈ L and A ∈ τ .

Remark 2.8. (1) If T : LX → L is an Alexandrov fuzzy topology. Define T*(A) =T(A*). Then T* is an Alexandrov fuzzy topology. (2) If T be an Alexandrov fuzzy topology on X, = {A ∈ LX | T(A) ≥ r} is an Alexandrov topology on X and for s ≤ r ∈ L.

 

3. Structures Induced by Alexandrov Fuzzy Topologies

Theorem 3.1. If is a meet-join approximation operator, then = {A ∈ LX | (A) = A*} is an Alexandrov topology on X.

Proof. (O1) Since ⊤X ≤ (⊥X) and (⊤X) =(⊥X → A) = ⊥X ⊙ (A) = ⊥, ⊥X =(⊤X) and ⊤X =(⊤X). Then ⊥X;⊤X ∈ . (O2) For Ai ∈ for each i ∈ Γ , by (M2), So, Ai ∈ . Since Thus, Ai ∈ (O3) For A ∈ , since α ⊙(α⊙A) =(α→ (α⊙A)) ≥(A),(α⊙A) ≥ α →(A) = (α ⊙ A)*. Then α ⊙ A ∈ .(O4) For A ∈ , by (M4), (α → A) = α ⊙(A) = α ⊙ A*. Hence α → A∈ .

Theorem 3.2. Let T be an Alexandorv fuzzy topology on X. Define

We have the following properties. (1) is a fuzzy preorder with ≤ for each s ≤ r. (2) is a fuzzy preorder with ≤ for each s ≤ r and = (x, y) = *(x, y) (3) Define as follows

Then  is a meet-join approximation operator on X with  for each s ≤ r.

(4) (5) is a meet-join approximation operator on X such that (6) (7) for all A ∈ LX and r ∈ L. Moreover, , for each x, y ∈ X. (8) for all A ∈ LX and r ∈ L. Moreover,   for each x, y ∈ X. (9) If = B for all i ∈ Γ≠ , then with s = ri. (10) If for all i ∈ Γ≠ , then with s = ri.

Proof. (1) Since T(B) ≥ r* iff then

Since and Hence is a fuzzy preorder.

For s ≤ r, since T(B) ≥ s* ≥ r*, we have ≤ (2) By a similar method as (1), is a fuzzy preorder. Moreover,(3) (M1) (M2)(M3)(M4)

For s ≤ r, since ≤ r, since , then (4) Since ; i.e. T(A) ≥ r*, = ⊙ A*(x) ≤ (A*(x) → A*(y)) ⊙ A*(x) ≤ A*(y), by M(3), So, Thus Let ; i.e. Let Then

Since and we have . Hence ⊂*.

(5) It is similarly proved as (4).(6) Let since A ∈ *,

Hence (A) = A*; i.e. . Thus

Let ; i.e. Then

Since and we have Hence

(7) For each A ∈ LX with A* ≤ Ai, T(Ai) ≥ r*, since then So, Since A ≥

Since and So, ,≤ . Hence = for all A∈ LX and r ∈ L.

(8) It is proved in a similar way as (7).(9) Let = B for all i ∈ Γ ≠ . Since

then where Since then So, Thus Since s ≤ ri, Thus                            ⧠

Theorem 3.3. Let  T be an Alexandorv fuzzy topology on X. We have the following properties.

(1) Define : LX → L as Then = T* is an Alexandrov fuzzy topology on X.

(2) Define : LX → L as Then = T* is an Alexandrov fuzzy topology on X.

(3) for all A, B ∈ LX.(4) There exists an Alexandrov fuzzy topology  Tr such that

If r ≤ s, then Tr ≤ Ts for all A ∈ LX.

(5) There exists an Alexandrov fuzzy topology  T*r such thatT*r(A) =eLX( (A) , A*). Moreover, T*r(A) = Tr(A*) for all A ∈ LX. If r ≤ s, then  T*r ≤ T*s for all A ∈ LX.

(6) Define TM : LX → L as Then TM = T* = TMT  is an Alexandrov fuzzy topology on X.

(7) Define TM* : LX → L as Then TM* = T = TMT* is an Alexandrov fuzzy topology on X.

Proof. (1) We only show that TMT = T*. Let = A*. Then form Theorem 3.3 (6). So T* (A) = T(A*) = Thus, Since T*(A) ≥ (T(A))* then with s = T(A). Thus, Hence TMT = T*.

(4) (T1) Since (T2) Since we have (T3) Since then Thus(T4)

Hence Tr is an Alexandrov fuzzy topology. Since for r ≤ s , Ts(A) = = Tr(A).

(5) From a similar method as (4), T*r is an Alexandrov fuzzy topology. By (3), Tr(A*) = = T*r(A) for all A ∈ LX.(6) Since Tr(A) = iff by (9), (2) and (7) are similarly proved as (1) and (6), respectively.                           ⧠

Example 3.4. Let (L = [0, 1], ⊙, →, * ) be a complete residuated lattice with a strong negation.

(1) Let X = {x, y, z} be a set. Define a map T : [0, 1]X → [0, 1] asT(A) = A(x) → A(z). Trivially, T(αX) = 1

Since α ⊙ A(x) → α ⊙ A(z) ≥ A(x) → A(z) from Lemma 2.4 (14), T(α ⊙ A) ≥ T(A). Since (α → A(x)) → (α →A(z)) ≥ A(x) → A(z) from Lemma 2.4 (10), T(α → A) ≥ T(A). By Lemma 2.4 (8), T( Ai) ≥ T(Ai) and T( Ai) ≥ T(Ai). Hence T is an Alexandrov fuzzy topology.

If T(A) = A(x) → A(z) ≥ r*, then A(z) ≥ A(x) ⊙ r*. Put A(x) = 1, A(y) = 0. So, and similarly, we can obtain

By Theorem 3.2(3), we obtain such that

If A*(x) ⊙ r* ≤ A*(z), then Thus . Moreover, since T*(A) = A*(x)→ A*(z) ≥ r* iff A*(z) ≥ A*(x) ⊙ r*, iff . So, From Theorem 3.3(1), we have Moreover, we obtain Hence TM = TMT = T*.

Since B(x) = 1 and T(B) = 1 → B(z) = B(z) ≥ r*, then

Since B(z) = 1 and T(B) = B(x) → 1= 1, then Then

(2) By (1), we obtain a map T* : [0, 1]Y → [0, 1] as T*(A) = A*(x) →A*(z) = A(z) →A(x).

Since T*(A) = A(z) → A(x) ≥ r*, then A(x) ≥ A(z) ⊙ r*. Put A(z) = 1, A(y) = 0. So, and

Moreover, for all x, y ∈ X.

If A*(z) ⊙ r* ≤ A*(x), then If , then A*(z) ⊙ r* ≤ A*(x). Moreover, since T(A) = A(x) → A(z) ≥ r* iff A*(z) ⊙ r* ≤ A*(z), iff Thus

Moreover, we obtain

Hence TM* = TMT* = T.

Since B(x) = 1 and T*(B) = B(z) →1 = 1, then

Since B(z) = 1 and T*(B) = 1 → B(x) = B(x) ≥ r*, then B(x) ≥ r*. We have

Then

(3) Let (L = [0, 1], ⊙, →, * ) be a complete residuated lattice with a strong negation defined by, for each n ∈ N,

By (1) and (2), we obtain

Since we have