FUZZY RELATIONS AND ALEXANDROV L-TOPOLOGIES

Abstract

In this paper, we investigate the relationships between fuzzy relations and Alexandrov L-topologies in complete residuated lattices. Moreover, we give their examples.

1. INTRODUCTION

Pawlak [9, 10] introduced rough set theory as a formal tool to deal with impre- cision and uncertainty in data analysis. Hájek [3] introduced a complete residuated lattice which is an algebraic structure for many valued logic. Radzikowska [11] de- veloped fuzzy rough sets in complete residuated lattice. Bělohlávek [1] investigated information systems and decision rules in complete residuated lattices. Lai [7, 8] in- troduced Alexandrov L-topologies induced by fuzzy rough sets. Algebraic structures of fuzzy rough sets are developed in many directions [1-13].

In this paper, we investigate the relationships between fuzzy relations and Alexan- drov L-topologies in complete residuated lattices. Moreover, we give their examples.

 

2. PRELIMINARIES

Definition 2.1 ([1, 3]). An algebra (L,∧,∨,⊙, →⊥,⊤) is called a complete residuated lattice if it satisfies the following conditions:

(L1) L = (L, ≤ , ∨,∧,⊥,⊤) is a complete lattice with the greatest element ⊤ and the least element ⊥; (L2) (L,⊙,⊤) is a commutative monoid; (L3) x ⊙ y ≤ z iff x ≤ y → z for x, y, z ∈ L.

In this paper, we assume (L,∧,∨,⊙, →, *⊥,⊤) is a complete residuated lattice with the law of negation;i.e. x** = x. For α ∈ L, A,⊤x ∈ LX, (α → A)(x) = α → A(x), (α ⊙ A)(x) = α ⊙ A(x) and ⊤x(x) = ⊤,⊤x(x) = ⊥, otherwise.

Definition 2.2 ([1, 7]). Let X be a set. A function R : X × X → L is called a fuzzy relation. A fuzzy relation R is called a fuzzy preorder if satisfies (R1) and (R2).

(R1) reflexive if R(x, x) = ⊤ for all x ∈ X, (R2) transitive if R(x, y) ⊙ R(y, z) ≤ R(x, z), for all x, y, z ∈ X. We denote

Lemma 2.3 ([1, 3]). Let (L,∨,∧,⊙, →, *⊥,⊤) be a complete residuated lattice with a 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) and (4) and (5) (x → y) ⊙ x ≤ y and (y→ z) ⊙ (x →y) ≤ (x → z). (6) (x ⊙ y) → z = x → (y → z) = y → (x → z) and (x ⊙ y)* = x → y*. (7) x* → y* = y → x and (x → y)* = x ⊙ y*.

Definition 2.4 ([5-7]). A subset τ ⊂ LX is called an Alexandrov topology if it satisfies satisfies the following conditions.

(T1) ⊥X, ⊤X ∈ τ where ⊤X(x) = ⊤ and ⊥X(x) = ⊥ for x ∈ X (T2) If (T3) α ⊙ A ∈ τ for all α ∈ L and A ∈ τ . (T4) α → A∈ τ for all α ∈ L and A ∈ τ .

Definition 2.5 ([7]). Let R ∈ LX×X be a fuzzy relation. A set A ∈ LX is called extensional if A(x) ⊙ R(x, y) ≤ A(y) for all x, y ∈ X.

 

3. FUZZY RELATIONS AND ALEXANDROV L-TOPOLOGIES

Theorem 3.1. Let R ∈ LX×X and R−1 ∈ LX×X with R−1(x, y) = R(x, y).

(1) τ is an Alexandrov topology on X iff τ* = {A* ∈ LX | A ∈ τ} is an Alexandrov topology on X. (2) τR = {A ∈ LX | A(x) ⊙ R(x, y) ≤ A(y), x, y ∈ X} is an Alexandrov topology on X. Moreover, (3) If is the smallest fuzzy preorder such that R ≤ then

where Rr(x, y) = Δ∨ R(x, y) and Δ (x, y) = ⊤ if x = y and Δ(x, y) = ⊥ if x ≠ y.

Moreover,

(4) where for each y ∈ X. (5) where for each x ∈ X. (6) where for each y ∈ X. (7) where for each x ∈ X. (8) Moreover, CτR(A) ∈ τR. (9) Moreover, IτR(A) ∈τR. (10) A ∈ τR iff A = CτR(A) = IτR(A). (11) CτR(A) = (IτR−1 (A*))* for all A ∈ LX.

Proof. (1) Let A* ∈ τ* for A ∈ τ . Since α⊙A* = (α →A)* and α → A* = (α⊙A)*, τ* is an Alexandrov topology on X.

(2) (T1) Since ⊤X(x) ⊙ R(x, y) ≤ ⊤X(y) = ⊤ and ⊥X(x) ⊙ R(x, y) = ⊥ = ⊥X(y), Then⊥X,⊤X ∈ τR.

(T2) For Ai ∈ τR for each i ∈ Γ, since Similarly,

(T3) For A ∈ τR, α ⊙ A ∈ τR.

(T4) For A ∈ τR, by Lemma 2.3(5), since α ⊙ (α → A(x)) ⊙ R(x, y) ≤ A(x) ⊙ R(x, y) ≤ A(y), (α → A(x)) ⊙ R(x, y) ≤ α → A(y). Then α → A ∈ τR. Moreover A ∈ τR iff A* ∈ τR−1 from:

A(x) ⊙ R(x, y) ≤ A(y) iff R(x, y) → A* ≥ A*(y) iff A*(y) ⊙ R(x, y) ≤ A*(x) iff A*(y) ⊙ R−1(y, x) ≤ A*(x).

(3) Define Then RτR is a fuzzy preorder. Since B ∈ τR and B(x) ⊙ R(x, y) ≤ B(y), then R(x, y) ≤ B(x) → B(y). Hence R(x, y) ≤ RτR. If P is a fuzzy preorder with R ≤ P, for Pw(x) = P(w, x), then Pw(x) ⊙ R(x, y) ≤ Pw(x) ⊙ P(x, y) ≤ Pw(y). Hence Pw ∈ τR. Thus RτR(x, y) = Thus,

Since Rr(x, y) = Δ∨ R(x, y), we have (Rr)n(x, x) = ⊤ for each n ∈ N. So Since

then Hence is a fuzzy preorder. If R ≤ P and P is fuzzy preorder, then Rr ≤ P and (Rr)n ≤ Pn = P, thus, Hence

(4) Put and Since A ∈ τR, Hence . Since Thus, A ∈ τR.

Let A ∈ τ. Since Thus, A ∈ τR.

Let A ∈ τ. Then Put A(x) = ax. Then

Let Then

Thus, D ∈ τ. Hence τR = τ = τ1.

(5) Put and Since A ∈ τR, RτR(x, y) → A(y) = → A(y) ≥ (A(x) → A(y)) → A(y) ≥ A(x). Hence Since , Thus, A ∈ η.

Let A ∈ η. Since Thus, R(x, y) → A(y) ≥ A(x) iff A(x) ⊙ R(x, y) ≤ A(y). So, A ∈ τR.

Let A ∈ η Then Put A(y) = by. Then

Let Then

Thus, A ∈ η. Hence τR = η = η1.

(6) It follows from iff

(7) It follows from iff

(8) Put Then B ∈ τR from:

If A ≤ E and E ∈ τR, then B ≤ E from:

Hence CτR = B.

(9) Let from

If E ≤ A and E ∈ τR, then E ≤ B from:

Hence IτR = B.

(11)

Theorem 3.2. Let RX and RY be fuzzy relations and f : X → Y a map with RX(x, y) ≤ RY (f(x), f(y)) for all x, y ∈ X. Then the following equivalent conditions hold.

(1) f−1(B) ∈ τRX for all B ∈ τRY. (2) for all (3) RτRX(x, y) ≤ RτRY (f(x), f(y)) for all x, y ∈ X. (4) for all x, y ∈ X. (5) f(CτRX(A)) ≤ CτRY(f(A)) for all A ∈ LX. (6) for all A ∈ LX. (7) CτRX(f−1(B)) ≤ f−1(CτRX(B)) for all B ∈ LY. (8) for all B ∈ LY. (9) f−1(IτRX(B)) ≤ IτRY(f−1(B)) for all B ∈ LY. (10) for all B ∈ LY.

Proof. (1) For all B ∈ τRY, f−1(B) ∈ τRX from:

f−1(B)(x) ⊙ RX(x, y) ≤ B(f(x)) ⊙ RY (f(x), f(y)) ≤ B(f(y)) = f−1(B)(y).

(1) ⇔ (2) It follows from (1) and Theorem 3.1(2).

(1) ⇒ (3)

(1) ⇒ (5)

(3) ⇒ (5)

(5) ⇒ (7) By (5), put A = f−1(B). Since f(CτRX(f−1(B))) ≤ CτRY(f(f−1(B))) ≤ CτRY(B), we have CτRX(f−1(B)) ≤ f−1(CτRX (B)).

(7) ⇒ (1) For all B ∈ τRY, CτY (B) = B. Since CτRX (f−1(B)) ≤ f−1(CτRX (B)) = f−1(B), f−1(B) ∈ τRX.

(1) ⇒ (9)

(9) ⇒ (1) For all B ∈ τRY, IτY (B) = B. Since IτRX (f−1(B)) ≥ f−1(IτRX (B)) = f−1(B), f−1(B) ∈ τRX.

Other cases are similarly proved.

Example 3.3. Let (L = [0, 1],⊙,→,* ) be a complete residuated lattice with the law of double negation defined by

x ⊙ y = (x + y − 1) ∨ 0, x → y = (1 − x + y) ∧ 1, x* = 1 − x.

Let X = {a, b, c}, Y = {x, y, z} be sets and f : X → Y as follows:

f(a) = x, f(b) = y, f(c) = z.

(1) Define RX ∈ LX×X, RY ∈ LY ×Y as follows

Then RX(a, b) ≤ RY (f(a), f(b)) for all a, b ∈ X.

For n ≥ 2, and as follows:

Then

Moreover,

Then RτRX (a, b) ≤ RτRX (f(a), f(b)) for all a, b ∈ X.

(2)

where ai ∈ L and

For

For

where bi ∈ L and

For

For

(3)

where ai ∈ L and

where bi ∈ L and

(4) For A = (0.2. 0.8, 0.6) ∈ LX,