APPROXIMATION OPERATORS AND FUZZY ROUGH SETS IN CO-RESIDUATED LATTICES

http://dx.doi.org/10.11568/kjm.2021.29.1.81

APPROXIMATION OPERATORS AND FUZZY ROUGH SETS IN CO-RESIDUATED LATTICES

Ju-Mok Oh and Yong Chan Kim∗,†

Abstract. In this paper, we introduce the notions of a distance function, Alexan-drov topology and -upper (⊕-lower) approximation operator based on complete co-residuated lattices. Under various relations, we define (⊕, )-fuzzy rough set on complete co-residuated lattices. Moreover, we study their properties and give their examples.

1. Introduction

Pawlak [15,16] introduced the rough set theory as a formal tool to deal with impre-cision and uncertainty in the data analysis. For an extension of Pawlak’s rough sets, many researchers [1-11,19,20,24] developed lower and upper approximation operators. Radzikowska et al.[17,18] investigated (I, T )-generalized fuzzy rough set where T is a t-norm and I is an implication. J.S.Mi et al.[14] investigated (S, T )-generalized fuzzy rough set where T is a t-norm and S(a, b) = 1 − T (1 − a, 1 − b) is an implication.

Ward et al.[23] introduced a complete residuated lattice which is an algebraic struc-ture for many valued logic [3-5]. It is an important mathematical tool as algebraic structures for many valued logics [1-11,19,20]. Using this concepts, fuzzy rough sets, information systems and decision rules were investigated in complete residuated lat-tices [1,2,7,20,25]. Moreover, Zheng et al.[25] introduced a complete co-residuated lattice as the generalization of t-conorm. Junsheng et al.[7] investigated (, &)-generalized fuzzy rough set on (L, ∨, ∧, , &, 0, 1) where (L, ∨, ∧, &, 0, 1) is a complete residuated lattice and (L, ∨, ∧, , 0, 1) is complete co-residuated lattice in a sense [13]. As the study of rough set theory and topological structures, many researchers [1,6-9,12,14,15,17,21] investigated the Alexandrov topology and lattice structures of fuzzy rough sets determined by lower and upper sets. In particular, Kim [8-11] introduce the notion of Alexandrov topologies as a topological viewpoint of fuzzy rough sets and studied the relations among fuzzy preorders, lower and upper approximation operators and Alexandrov topologies in complete residuated lattices.

Received December 1, 2020. Accepted March 4, 2021. Published online March 30, 2021. 2010 Mathematics Subject Classification: 03E72, 03G10, 06A15, 54F05.

Key words and phrases: Distance functions, co-residuated lattices, Alexandrov topologies , ap-proximation operators, (⊕, )-fuzzy rough set.

∗_{Corresponding author.}

† _{This work was supported by the Research Institute of Natural Science of Gangneung-Wonju}

National University.

© The Kangwon-Kyungki Mathematical Society, 2021.

This is an Open Access article distributed under the terms of the Creative commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by

-nc/3.0/) which permits unrestricted non-commercial use, distribution and reproduction in any medium, provided the original work is properly cited.

In this paper, we introduce the notions of distance functions, Alexandrov topologies and -upper (⊕-lower) approximation operators based on complete co-residuated lattices (L, ∨, ∧, ⊕, 0, 1). Under various relations, we define (⊕, )-fuzzy rough set on complete co-residuated lattices (L, ∨, ∧, ⊕, 0, 1) where is induced by ⊕. Moreover, we study their properties and give their examples.

2. Preliminaries

Definition 2.1. [7,25] An algebra (L, ∧, ∨, ⊕, 0, 1) is called a complete co-residuated lattice if it satisfies the following conditions:

(Q1) L = (L, ≤, ∨, ∧, 0, 1) is a complete lattice where 0 is the bottom element and 1 is the top element.

(Q2) a = a ⊕ 0, a ⊕ b = b ⊕ a and a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c for all a, b, c ∈ L. (Q3) (V

i∈Γai) ⊕ b =

V

i∈Γ(ai⊕ b).

Let (L, ≤, ⊕) be a complete co-residuated lattice. For each x, y ∈ L, we define x y =^{z ∈ L | y ⊕ z ≥ x}.

Then (x ⊕ y) ≥ z iff x ≥ (z y).

In this paper, we assume (L, ∧, ∨, ⊕, , 0, 1) is a complete co-residuated lattice. For α ∈ L, A ∈ LX, we denote (α A), (α ⊕ A), αX ∈ LX as (α A)(x) = α

A(x), (α ⊕ A)(x) = α ⊕ A(x), αX(x) = α.

Put N (x) = 1 x. The condition N (N (x)) = x for each x ∈ L is called a double negative law.

Remark 2.2. (1) An infinitely distributive lattice (L, ≤, ∨, ∧, ⊕ = ∨, 0, 1) is a complete co-residuated lattice. In particular, the unit interval ([0, 1], ≤, ∨, ∧, ⊕ = ∨, 0, 1) is a complete co-residuated lattice [7,25].

x y =V{z ∈ L | y ∨ z ≥ x} = 0, if y ≥ x,

x, if y 6≥ x.

Put N (x) = 1 x = 1 for x 6= 1 and N (1) = 0. Then N (N (x)) = 0 for x 6= 1 and N (N (1)) = 1. Hence N does not satisfy a double negative law.

(2) The unit interval with a right-continuous t-conorm ⊕, ([0, 1], ≤, ⊕), is a com-plete co-residuated lattice [7.25].

(3) ([1, ∞], ≤, ∨, ⊕ = ·, ∧, 1, ∞) is a complete co-residuated lattice where x y =V{z ∈ [1, ∞] | yz ≥ x}

= 1, if y ≥ x,x y, if y 6≥ x.

∞ · a = a · ∞ = ∞, ∀a ∈ [1, ∞], ∞ ∞ = 1.

Put N (x) = ∞ x = ∞ for x 6= ∞ and N (∞) = 1. Then N (N (x)) = 1 for x 6= ∞ and N (N (∞)) = ∞. Hence N does not satisfy a double negative law.

(4) ([0, ∞], ≤, ∨, ⊕ = +, ∧, 0, ∞) is a complete co-residuated lattice where y x =V{z ∈ [0, ∞] | x + z ≥ y}

=V{z ∈ [0, ∞] | z ≥ −x + y} = (y − x) ∨ 0, ∞ + a = a + ∞ = ∞, ∀a ∈ [0, ∞], ∞ ∞ = 0.

Put N (x) = ∞ x = ∞ for x 6= ∞ and N (∞) = 0. Then N (N (x)) = 0 for x 6= ∞ and N (N (∞)) = ∞. Hence N does not satisfy a double negative law.

(5) ([0, 1], ≤, ∨, ⊕, ∧, 0, 1) is a complete co-residuated lattice where x ⊕ y = (xp_{+ y}p₎1p 1 ≤ p < ∞,

x y =V{z ∈ [0, 1] | (zp_{+ y}p₎p1 ≥ x}

=V{z ∈ [0, 1] | z ≥ (xp _{− y}p₎p1} = (xp− yp₎ 1 p ∨ 0,

Put N (x) = 1 x = (1 − xp₎1p _{for 1 ≤ p < ∞. Then N (N (x)) = x for x ∈ [0, 1].}

Hence N satisfies a double negative law.

(6) Let P (X) be the collection of all subsets of X. Then (P (X), ⊂, ∪, ∩, ⊕ = ∪, ∅, X) is a complete co-residuated lattice where

A B =V{C ∈ P (X) | B ∪ C ⊃ A} = A ∩ Bc= A − B.

Put N (A) = X A = Ac _{for each A ⊂ X. Then N (N (A)) = A. Hence N satisfies a}

double negative law.

Lemma 2.3. [11] Let (L, ∧, ∨, ⊕, , 0, 1) be a complete co-residuated lattice. For each x, y, z, xi, yi ∈ L, we have the following properties.

(1) If y ≤ z, (x ⊕ y) ≤ (x ⊕ z), y x ≤ z x and x z ≤ x y. (2) (W i∈Γxi) y = W i∈Γ(xi y) and x ( V i∈Γyi) = W i∈Γ(x yi). (3) (V i∈Γxi) y ≤ V i∈Γ(xi y) (4) x (W i∈Γyi) ≤ V i∈Γ(x yi).

(5) x x = 0, x 0 = x and 0 x = 0. Moreover, x y = 0 iff x ≤ y. (6) y ⊕ (x y) ≥ x, y ≥ x (x y) and (x y) ⊕ (y z) ≥ x z. (7) x (y ⊕ z) = (x y) z = (x z) y. (8) x y ≥ (x ⊕ z) (y ⊕ z), y x ≥ (z x) (z y) and (x ⊕ y) (z ⊕ w) ≤ (x z) ⊕ (y w). (9) x ⊕ y = 0 iff x = 0 and y = 0. (10) (x ⊕ y) z ≤ x ⊕ (y z) and (x y) ⊕ z ≥ x (y z).

(11) If L satisfies a double negative law and N (x) = 1 x, then N (x ⊕ y) = N (x) y = N (y) x and x y = N (y) N (x). Moreover, N (W

i∈Γxi) = V i∈ΓN (xi) and N (V i∈Γxi) = W i∈ΓN (xi).

Definition 2.4. [11] Let (L, ∧, ∨, ⊕, , 0, 1) be a complete co-residuated lattice. Let X be a set. A function dX : X × X → L is called a distance function if it satisfies

the following conditions:

(M1) dX(x, x) = 0 for all x ∈ X,

(M2) dX(x, y) ⊕ dX(y, z) ≥ dX(x, z), for all x, y, z ∈ X,

(M3) If dX(x, y) = dX(y, x) = 0, then x = y.

The pair (X, dX) is called a distance space.

Remark 2.5. (1) We define a distance function dX : X × X → [0, ∞]. Then

(X, dX) is called a pseudo-quasi-metric space.

(2) Let (L, ∧, ∨, ⊕, , 0, 1) be a complete co-residuated lattice. Define a function dL : L × L → L as dL(x, y) = x y. By Lemma 2.3 (5) and (8), (L, dL) is a

dis-tance space. Moreover, we define a function d_LX : LX × LX → L as d_LX(A, B) =

W

x∈X(A(x) B(x)). Then (L X_{, d}

(3) We define a function d[0,∞]X : [0, ∞]X × [0, ∞]X → [0, ∞] as d_[0,∞]X(A, B) = W x∈X(A(x) B(x)) = W x∈X((B(x) − A(x)) ∨ 0). Then ([0, ∞] X_{, d} [0,∞]X) is a pseudo-quasi-space.

(4) If (X, dX) is a distance space and we define a function d−1X (x, y) = dX(y, x),

then (X, d−1_X ) is a distance space.

(5) Let (L, ∧, ∨, ⊕, , 0, 1) be a complete co-residuated lattice. Let (X, dX) be

a distance space and define (dX ] dX)(x, z) =

V

y∈X(dX(x, y) ⊕ dX(y, z)) for each

x, z ∈ X. By (M2), (dX ] dX)(x, z) ≥ dX(x, z) and (dX ] dX)(x, z) ≤ dX(x, x) ⊕

dX(x, z) = d(x, z). Hence (dX ] dX) = dX.

3. Approximation operators and fuzzy rough sets

Definition 3.1. A subset τ ⊂ LX is called an Alexandrov topology on X iff it satisfies the following conditions:

(O1) αX ∈ τ .

(O2) If Ai ∈ τ for all i ∈ I, then

W

i∈IAi,

V

i∈IAi ∈ τ .

(O3) If A ∈ τ and α ∈ L, then A α, α ⊕ A ∈ τ .

Definition 3.2. A map J : LX → LX is called an -upper approximation operator if it satisfies the following conditions, for all A, Ai ∈ LX, and α ∈ L,

(J1) J (A α) = J (A) α, (J2) J (W

i∈IAi) =

W

i∈IJ (Ai),

(J3) J (A) ≥ A and J (J (A)) = J (A).

Definition 3.3. A map H : LX → LX is called an ⊕-lower approximation operator if it satisfies the following conditions, for all A, Ai ∈ LX, and α ∈ L,

(H1) H(α ⊕ A) = α ⊕ H(A), (H2) H(V

i∈IAi) =

V

i∈IH(Ai),

(H3) H(A) ≤ A and H(H(A)) = H(A).

Let H (resp. J ) be ⊕-lower (resp. -upper) approximation operator on X. As a generalization of fuzzy rough set, the pair (H(A), J (A)) is called an (⊕, )-fuzzy rough set for A ∈ LX_.

The map α : LX _{→ L is an fuzzy accuracy measure defined, for A ∈ L}X

α(A) = _

x∈X

(J (A)(x) H(A)(x)).

Theorem 3.4. Let dX ∈ LX×X be a distance function. Define JdX, HdX : L X _→

LX as follows

JdX(B)(x) =

W

y∈X(B(y) dX(x, y)),

HdX(A)(y) =

V

x∈X(A(x) ⊕ dX(x, y)).

Then the followings hold.

(1) JdX is an -upper approximation operator.

(2) HdX is an ⊕-lower approximation operator. Moreover, (HdX(A), JdX(A)) is an

(⊕, )-fuzzy rough set for A ∈ LX_.

(3) JdX(αX) = αX, JdX(dX(x, −)) = dX(x, −) and α ⊕ JdX(A) ≥ JdX(α ⊕ A) for

each α ∈ L, A ∈ LX _{and J}

(4) HdX(αX) = αX, HdX(dX(x, −)) = dX(x, −) and HdX(A) α ≤ HdX(A α) for

each α ∈ L, A ∈ LX and HdX(A) ≤ HdX(B) for A ≤ B.

(5) HdX(A) = W{B | JdX(B) ≤ A} and JdX(α A) = α Hd−1_X (A), for all A ∈ LX.

(6) JdX(B) =V{A | HdX(A) ≥ B}.

(7) For each A, B ∈ LX_{, H}

dX(JdX(B)) = JdX(B) and JdX(HdX(A)) = HdX(A)

(8) τdX = {A ∈ L

X _{| A(x) ⊕ d}

X(x, y) ≥ A(y)} is an Alexandrov topology on X with

dX(x, −), (α dX(−, x)) ∈ τdX. Moreover,

τdX = {HdX(A) | A ∈ L

X_{} = {}V

y∈X(A(y) ⊕ dX(y, −)) | A ∈ L X_}

= {JdX(A) | A ∈ L

X_{} = {}W

y∈X(A(y) dX(−, y)) | A ∈ LX}.

Proof. (1) (J1) For each A ∈ LX _{and α ∈ L, by Lemma 2.3 (7),}

JdX(A α)(x) =

W

y∈X((A(y) α) dX(x, y))

=W

y∈X((A(y) dX(x, y)) α = JdX(A)(x) α.

(J2) For each Ai ∈ LX, by Lemma 2.3(2), JdX(

W

i∈ΓAi) =

W

i∈ΓJdX(Ai).

(J3) For each A ∈ LX, JdX(A)(x) =

W

y∈X(A(y) dX(x, y)) ≥ A(x) dX(x, x) =

A(x).

For each A ∈ LX_,

JdX(JdX(A))(x) =

W

y∈X(JdX(A)(y) dX(x, y))

=W

y∈X(

W

z∈X(A(z) dX(y, z)) dX(x, y))

=W

y∈X(

W

z∈X((A(z) dX(y, z)) dX(x, y))) (by Lemma 2.3 (2))

=W

y,z∈X(A(z) (dX(y, z) ⊕ dX(x, y))) (by Lemma 2.3 (7))

=W

z∈X(A(z)

V

y∈X(dX(y, z) ⊕ dX(x, y))) (by Lemma 2.3 (2))

=W

z∈X(A(z) dX(x, z)) = JdX(A)(x).

Hence JdX is an -upper approximation operator.

(2) (H1) HdX(α ⊕ A)(y) = V x∈X((α ⊕ A)(x) ⊕ dX(x, y)) = α ⊕ V x∈X(A(x) ⊕ dX(x, y)) = α ⊕ HdX(A)(y). (H2) HdX( V i∈ΓAi) = V x∈X( V i∈ΓAi(x)⊕dX(x, y)) = V i∈Γ( V x∈X(Ai(x)⊕dX(x, y)) = V i∈ΓHdX(Ai)(y). (H3) HdX(A)(y) = W

x∈X(A(x) ⊕ dX(x, y)) ≤ A(y) ⊕ dX(y, y) = A(y).

For all B ∈ LX_{, z ∈ X,} HdX(HdX(A))(z) = V x∈X(HdX(A)(x) ⊕ dX(x, z)) =V x∈X( V

y∈X(A(y) ⊕ dX(y, x)) ⊕ dX(x, z))

=V

y∈X(A(y) ⊕

V

x∈X(dX(y, x) ⊕ dX(x, z)))

=V

y∈X(A(y) ⊕ dX(y, z)) = HdX(A)(z).

Hence HdX is an ⊕-lower approximation operator.

(3) Since JdX(αX)(x) =

W

y∈X(αX(y) dX(x, y)) ≤ α, by (J3), JdX(αX) = αX.

For each x, z ∈ X,

JdX(dX(x, −))(z) =

_

y∈X

For each A ∈ LX _{and α ∈ L,}

α ⊕ JdX(A)(x) = α ⊕

W

y∈X(A(y) dX(x, y))

≥W

y∈X(α ⊕ (A(y) dX(x, y)))

≥W

y∈X((α ⊕ A)(y) dX(x, y)) (by Lemma 2.3 (10))

= JdX(α ⊕ A)(x). For A ≤ B, JdX(A) ≤ JdX(B). (4) Since HdX(αX)(y) = V x∈X(αX(x) ⊕ dX(x, y)) ≥ α, HdX(αX) = αX. For x, z ∈ X, HdX(dX(x, −))(z) = ^ y∈X (dX(x, y) ⊕ dX(y, z)) = dX(x, z).

For each A ∈ LX and α ∈ L,

HdX(A α)(z) = V x∈X((A α)(x) ⊕ dX(x, z)) ≥V x∈X((A(x) ⊕ dX(x, z)) α) ≥V x∈X(A(x) ⊕ dX(x, z)) α = HdX(A)(z) α. (5) By (J2), for each A ∈ LX_,

W{B(y) | JdX(B)(x) ≤ A(x)} =W{B(y) | Wy∈Y(B(y) dX(x, y)) ≤ A(x)}

=V

x∈X(dX(x, y) ⊕ A(x)) = HdX(A)(y).

For all B ∈ LX_{, x ∈ X,}

JdX(α B)(x) =

W

y∈X((α B(y)) dX(x, y))

=W

y∈X((α (B(y) ⊕ dX(x, y)) (by Lemma 2.3 (7))

= α W

y∈X((B(y) ⊕ dX(x, y))

= α H_d−1

X (B)(x)

(6) By (H2), for each B ∈ LX_,

V{A(x) | HdX(A)(y) ≥ B(y)}

=V{A(x) | V_x∈X(A(x) ⊕ dX(x, y)) ≥ B(y)}

=W

y∈Y(B(y) dX(x, y)) = JdX(B)(x).

(7) For each B ∈ LX, HdX(JdX(B))(z) = V x∈X(JdX(B)(x) ⊕ dX(x, z)) =V x∈X( W

y∈X(B(y) dX(x, y)) ⊕ dX(x, z))

≥V

x∈X

W

y∈X((B(y) dX(x, y)) ⊕ dX(x, z))

≥V

x∈X

W

y∈X(B(y) (dX(x, y) dX(x, z))) (by Lemma 2.3 (10))

≥W

y∈X(B(y)

W

x∈X(dX(x, y) dX(x, z)))

≥W

y∈X(B(y) dX(z, y)) = JdX(B)(z),

JdX(HdX(B))(x) =

W

y∈X(HdX(B)(y) dX(x, y))

=W y∈X( V z∈X(B(z) ⊕ dX(z, y)) dX(x, y)) ≤W y∈X V z∈X((B(z) ⊕ dX(z, y)) dX(x, y)) ≤W y∈X V

z∈X(B(z) ⊕ (dX(z, y) dX(x, y)) (by Lemma 2.3 (10))

≤V z∈X(B(z) ⊕ W y∈X(dX(z, y) dX(x, y)) =V z∈X(B(z) ⊕ dX(x, z)) = HdX(B)(x).

(O2) If Ai ∈ τdX for all i ∈ I, W i∈IAi(x) ⊕ dX(x, y) ≥ W i∈I(Ai(x) ⊕ dX(x, y)) ≥ W

i∈IAi(y) and

V

i∈IAi(x) ⊕ dX(x, y) =

V

i∈I(Ai(x) ⊕ dX(x, y)) ≥

V

i∈IAi(y). Hence

W

i∈IAi,

V

i∈IAi ∈ τdX.

(O3) If A ∈ τdX and α ∈ L, then dX(x, y)⊕(A(x) α)⊕α ≥ dX(x, y)⊕A(x) ≥ A(y).

Thus dX(x, y) ⊕ (A(x) α) ≥ A(y) α. So A α ∈ τdX. Easily, α ⊕ A ∈ τdX.

Since dX(x, −)(y) ⊕ dX(y, z) ≥ dX(x, −)(z), dX(x, −) ∈ τdX. Since

(α dX(−, x))(y) ⊕ dX(y, z) ⊕ dX(z, x)

≥ (α dX(−, x))(y) ⊕ dX(y, x) ≥ α,

(α dX(−, x))(y) ⊕ dX(y, z) ≥ α dX(z, x), that is, (α dX(−, x)) ∈ τdX.

For A ∈ τdX, A =

V

x∈X(A(x) ⊕ dX(x, −)) = HdX(A) ∈ τdX and A =

W

x∈X(A(x)

dX(−, x)) = JdX(A) ∈ τdX.

Theorem 3.5. (1) Let H : LX → LX be an ⊕-lower approximation operator iff there exist a distance function dH on X such that

H(A)(y) = ^

x∈X

(A(x) ⊕ dH(x, y)).

(2) If L satisfies a double negative law, then J : LX _{→ L}X _{be an -upper}

approx-imation operator iff there exist a distance function dJ on X such that

J (B)(x) = _

y∈X

(B(y) dJ(x, y)).

Proof. (1) (⇒) Put dH : X × X → L as dH(x, y) = H(0x)(y) where 0x(x) = 0 and

0x(y) = 1 for x 6= y ∈ X.

(M1) dH(x, x) = H(0x)(x) ≤ 0x(x) = 0.

(M2) Since A =V

y∈X(A(y) ⊕ 0y) and H(0x) =

V y∈X(H(0x)(y) ⊕ 0y), V y∈X(dH(x, y) ⊕ dH(y, z)) =V y∈X(H(0x)(y) ⊕ H(0y)(z)) ( by (H2)) = H(V y∈X(H(0x)(y) ⊕ 0y)(z)) = H(H(0x))(z) = H(0x)(z) = dH(x, z).

Hence dH is a distance function. Moreover,

H(A)(y) = H(V x∈X(A(x) ⊕ 0x))(y) =V x∈X(A(x) ⊕ H(0x)(y)) =V x∈X(A(x) ⊕ dH(x, y))).

(⇐) It follow from Theorem 3.4(2).

(2) (⇒) Put dJ : X × X → L as dJ(x, y) = N (J (N (0y))(x) where 0x(x) = 0 and

0x(y) = 1 for x 6= y ∈ X.

(M2) Since A =V

y∈X(A(y)⊕0y), N (A) =

V

y∈X(N (A)(y)⊕0y), by Lemma 2.3(11),

A =W y∈X(N (0y) N (A)(y)), N (V y∈X(dJ(x, y) ⊕ dJ(y, z))) =W y∈X(N (N J (N (0y))(x) ⊕ N J (N (0z))(y))) ( by (H2)) =W y∈X(J (N (0y))(x) N (J (N (0z))(y))) ( by (H2)) = J (W y∈X((N (0y))(x) N (J (N (0z))(y)))) ( by (H2)) = J (J (N (0z)))(x) = J (N (0z))(x) = N (dJ(x, z)).

Hence dJ is a distance function. Moreover,

J (B)(x) = J (W y∈Y(N (0y)(x) N (B)(y)) =W y∈Y(J (N (0y))(x) N (B)(y)) =W y∈Y(B(y) N (J (N (0y)))(x)) =W

y∈Y(B(y) dJ(x, y))).

(⇐) It follow from Theorem 3.4(1).

Example 3.6. Let X = {x, y, z} be a set and (L = {0,1₄,1₂,3₄, 1}, ⊕, , 0, 1) be a complete co-residuated lattice with

x ⊕ y = 1 ∧ (x + y), x y = (x − y) ∨ 0. Define d1 X, dX : X × X → L as d1_X =   0 1 1₄ 1 2 0 1 4 3 4 1 2 0  , dX =   0 3₄ 1₄ 1 2 0 1 4 3 4 1 2 0   Since d1 X(x, z) ⊕ d1X(z, y) = 1 4+ 1 2 6≥ d 1

X(x, y) = 1, d1X is not a distance function. Since

dX ] dX = dX from Remark 2.5(5), dX is a distance function.

By Theorem 3.4(8), we obtain an Alexandrov topology τdX = {HdX(C) | C ∈

LX} = {JdX(D) | D ∈ L X_{} where} Hd_τX(C) =V_x∈X(C(x) ⊕ dτX(x, −)) =   C(x) ∧ (C(y) +1₂) ∧ (C(z) + 3₄) (C(x) +3₄) ∧ C(y) ∧ (C(z) + 1₂) (C(x) +1₄) ∧ (C(y) + 1₄) ∧ C(z)   Jd_τX(D) = W x∈X(D(x) dτX(−, x)) =   D(x) ∨ (D(y) − 3₄) ∨ (D(z) − 1₄) (D(x) − 1₂) ∨ D(y) ∨ (D(z) − 1₄) (D(x) − 3₄) ∨ (D(y) − 1₂) ∨ D(z)  

The pair (HdX(A), JdX(A)) is an (⊕, )-fuzzy rough set for A ∈ L X_.

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Ju-Mok Oh

Department of Mathematics, Gangneung-Wonju National, Gangneung 25457, Korea

E-mail : jumokoh@gwnu.ac.kr Yong Chan Kim

Department of Mathematics, Gangneung-Wonju National, Gangneung 25457, Korea