The Properties of L-lower Approximation Operators

(1)

Vol. 14, No. 1, March 2014, pp. 57-65 http://dx.doi.org/10.5391/IJFIS.2014.14.1.57

The Properties of L-lower Approximation Operators

Yong Chan Kim

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

Abstract

In this paper, we investigate the properties of L-lower approximation operators as a generalization of fuzzy rough set in complete residuated lattices. We study relations lower (upper, join meet, meet join) approximation operators and Alexandrov L-topologies. Moreover, we give their examples as approximation operators induced by various L-fuzzy relations.

Keywords: Complete residuated lattices, L-upper approximation operators, Alexandrov L-topologies

Received: Dec. 10, 2013 Revised : Mar. 18, 2014 Accepted: Mar. 19, 2014

Correspondence to: Yong Chan Kim ([email protected])

©The Korean Institute of Intelligent Systems

This is an Open Access article dis-cc tributed under the terms of the Creative Commons Attribution Non-Commercial Li- cense (http://creativecommons.org/licenses/

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

1. Introduction

Pawlak [1, 2] introduced rough set theory as a formal tool to deal with imprecision and uncertainty in data analysis. H´ajek [3] introduced a complete residuated lattice which is an algebraic structure for many valued logic. Radzikowska and Kerre [4] developed fuzzy rough sets in complete residuated lattice. Bˇelohl´avek [5] investigated information systems and decision rules in complete residuated lattices. Lai and Zhang [6, 7] introduced Alexandrov L-topologies induced by fuzzy rough sets. Kim [8, 9] investigate relations between lower approximation operators as a generalization of fuzzy rough set and Alexandrov L-topologies.

Algebraic structures of fuzzy rough sets are developed in many directions [4, 8, 10]

In this paper, we investigate the properties of L-lower approximation operators as a generalization of fuzzy rough set in complete residuated lattices. We study relations lower (upper, join meet, meet join) approximation operators and Alexandrov L-topologies. Moreover, we give their examples as approximation operators induced by various L-fuzzy relations.

Definition 1.1. [3, 5] An algebra (L, ∧, ∨, , →, ⊥, >) is called a complete residuated lattice if it satisfies the following conditions:

(C1) L = (L, ≤, ∨, ∧, ⊥, >) is a complete lattice with the greatest element > and the least element ⊥;

(C2) (L, , >) is a commutative monoid;

(C3) x y ≤ z iff x ≤ y → z for x, y, z ∈ L

Remark 1.2. [3, 5] (1) A completely distributive lattice L = (L, ≤, ∨, ∧ = , →, 1, 0) is a complete residuated lattice defined by

x → y =_

{z | x ∧ z ≤ y}.

(2)

(2) The unit interval with a left-continuous t-norm , ([0, 1], ∨, ∧, , →, 0, 1),

is a complete residuated lattice defined by x → y =_

{z | x z ≤ y}.

In this paper, we assume (L, ∧, ∨, , →,^∗⊥, >) is a complete residuated lattice with the law of double negation;i.e.

x^∗∗= x. For α ∈ L, A, >x∈ L^X,

(α → A)(x) = α → A(x), (α A)(x) = α A(x) and

>x(x) = >, >x(y) = ⊥, otherwise.

Lemma 1.3. [3, 5] For each x, y, z, xi, yi ∈ L, we have the following properties.

(1) If y ≤ z, (x y) ≤ (x z), x → y ≤ x → z and z → x ≤ y → x.

(2) x y ≤ x ∧ y ≤ x ∨ y.

(3) x → (V

i∈Γy_i) =V

i∈Γ(x → y_i) and (W

i∈Γx_i) → y = V

i∈Γ(xi→ y).

(4) x → (W

i∈Γy_i) ≥W

i∈Γ(x → y_i) (5) (V

i∈Γxi) → y ≥W

i∈Γ(xi→ y).

(6) (x y) → z = x → (y → z) = y → (x → z).

(7) x (x → y) ≤ y, x → y ≤ (y → z) → (x → z) and x → y ≤ (z → x) → (z → y).

(8) y ≤ x → (x y) and x ≤ (x → y) → y.

(9) x → y ≤ (x z) → (y z).

(10) (x → y) (y → z) ≤ x → z.

(11) x → y = > iff x ≤ y.

(12) x → y = y^∗→ x^∗.

(13) (x y)^∗= x → y^∗= y → x^∗and x → y = (x y^∗)^∗. (14) V

i∈Γx^∗_i = (W

i∈Γx_i)^∗andW

i∈Γx^∗_i = (V

i∈Γx_i)^∗. Definition 1.4. [8, 9]

(1) A map H : L^X → L^Xis called an L-upper approximation operatoriff it satisfies the following conditions (H1) A ≤ H(A),

(H2) H(α A) = α H(A) where α(x) = α for all x ∈ X,

(H3) H(W

i∈IA_i) =W

i∈IH(Ai).

(2) A map J : L^X→ L^Xis called an L-lower approximation operatoriff it satisfies the following conditions

(J1) J (A) ≤ A,

(J2) J (α → A) = α → J (A), (J3) J (V

i∈IA_i) =V

i∈IJ (A_i).

(3) A map K : L^X → L^Xis called an L-join meet approximation operatoriff it satisfies the following conditions (K1) K(A) ≤ A^∗,

(K2) K(α A) = α → K(A), (K3) K(W

i∈IAi) =V

i∈IK(Ai).

(4) A map M : L^X → L^Xis called an L-meet join approximation operatoriff it satisfies the following conditions (M1) A^∗≤ M(A),

(M2) M(α → A) = α M(A), (M3) M(V

i∈IA_i) =W

i∈IM(A_i).

Definition 1.5. [6, 9] A subset τ ⊂ L^Xis called an Alexandrov L-topology if it satisfies:

(T1) ⊥_X, >_X ∈ τ where >_X(x) = > and ⊥X(x) = ⊥ for x ∈ X.

(T2) If Ai∈ τ for i ∈ Γ,W

i∈ΓA_i,V

i∈ΓA_i∈ τ . (T3) α A ∈ τ for all α ∈ L and A ∈ τ . (T4) α → A ∈ τ for all α ∈ L and A ∈ τ . Theorem 1.6. [8, 9]

(1) τ is an Alexandrov topology on X iff τ_∗= {A^∗∈ L^X | A ∈ τ } is an Alexandrov topology on X.

(2) If H is an L-upper approximation operator, then τ_H = {A ∈ L^X | H(A) = A} is an Alexandrov topology on X.

(3) If J is an L-lower approximation operator, then τ_J = {A ∈ L^X | J (A) = A} is an Alexandrov topology on X.

(4) If K is an L-join meet approximation operator, then τK = {A ∈ L^X| K(A) = A^∗} is an Alexandrov topology on X.

(5) If M is an L-meet join operator, then τM= {A ∈ L^X | M(A) = A^∗} is an Alexandrov topology on X.

Definition 1.7. [8, 9] Let X be a set. A function R : X × X → L is called:

(R1) reflexive if R(x, x) = > for all x ∈ X, (R2) symmetric if R(x, x) = > for all x ∈ X,

(R3) transitive if R(x, y)R(y, z) ≤ R(x, z), for all x, y, z ∈ X.

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(R4) Euclidean if R(x, z)R(y, z) ≤ R(x, y), for all x, y, z ∈ X.

If R satisfies (R1) and (R3), R is called a L-fuzzy preorder.

If R satisfies (R1), (R2) and (R3), R is called a L-fuzzy equivalence relation

2. The Properties of L-lower Approximation Op- erators

Theorem 2.1. Let J : L^X → L^X be an L-lower approximation operator. Then the following properties hold.

(1) For A ∈ L^X, J (A)(y) =V

x∈X(J^∗(>^∗_x)(y) → A(x)).

(2) Define HJ(B) = V{A | B ≤ J (A)}. Then HJ : L^X → L^Xwith

HJ(B)(x) = _

y∈X

(J^∗(>^∗_x)(y) B(y))

is an L-upper approximation operator such that (HJ, J ) is a residuated connection;i.e.,

HJ(B) ≤ A iff B ≤ J (A).

Moreover, τJ = τH_J.

(3) If J (J (A)) = J (A) for A ∈ L^X, then HJ(HJ(A)) = HJ(A) for A ∈ L^X such that τJ = τHJ with

τ_J = {J (A) = ^

x∈X

(J^∗(>^∗_x)(y) → A(x)) | A ∈ L^X},

τH_J = {HJ(A)(x)

= _

y∈X

(J^∗(>^∗_x)(y) A(y)) | A ∈ L^X}.

(4) If J (J^∗(A)) = J^∗(A) for A ∈ L^X, then J (J (A)) = J (A) such that

{J^∗(A) =W

x∈X(A^∗(x) J^∗(>^∗_x)) | A ∈ L^X}

= τ_J = (τ_J)_∗.

(5) Define Hs(A) = J (A^∗)^∗. Then Hs: L^X → L^Xwith Hs(B)(x) = _

y∈X

(J^∗(>^∗_y)(x) B(y))

is an L-upper approximation operator. Moreover, τ_H_s= (τ_J)_∗= (τ_H_J)_∗.

(6) If J (J (A)) = J (A) for A ∈ L^X, then Hs(Hs(A)) = Hs(A)

for A ∈ L^Xsuch that τHs = (τ_J)_∗= (τ_H_J)_∗. with τ_H_s = {Hs(A) = _

y∈X

(J^∗(>^∗_y) A(y)) | A ∈ L^X}.

(7) If J (J^∗(A)) = J^∗(A) for A ∈ L^X, then Hs(H_s^∗(A)) = H^∗_s(A)

such that

{H_s^∗(A) =V

y∈X(A(y) → J (>^∗_y)) | A ∈ L^X}

= τ_H_s= (τ_H_s)_∗.

(8) Define KJ(A) = J (A^∗). Then KJ : L^X → L^Xwith KJ(A) = ^

y∈X

(A(y) → J (>^∗_y))

is an L-join meet approximation operator.

(9) If J (J (A)) = J (A) for A ∈ L^X, then KJ(K^∗_J(A)) = K^∗_J(A) for A ∈ L^Xsuch that τ_K_J = (τ_J)_∗with

τ_K_J = {K^∗_J(A) = _

y∈X

(J^∗(>^∗_y) A(y)) | A ∈ L^X}.

(10) If J (J^∗(A)) = J^∗(A) for A ∈ L^X, then K_J(K_J(A)) = K^∗_J(A)

such that

{KJ(A) =V

y∈X(A(y) → J (>^∗_y)) | A ∈ L^X}

= τ_K_J = (τ_K_J)_∗.

(11) Define MJ(A) = (J (A))^∗. Then MJ : L^X → L^X with

MJ(A)(y) = _

x∈X

(A^∗(x) J^∗(>^∗_x)(y))

is an L-meet join approximation operator. Moreover, τM_J = τJ.

(12) If J (J (A)) = J (A) for A ∈ L^X, then MJ(M^∗_J(A)) =

(4)

MJ(A) for A ∈ L^Xsuch that τMJ = (τJ)∗with {M^∗_J(A)(y) = ^

x∈X

(J^∗(>^∗_x)(y) → A(x)) | A ∈ L^X}

= τM_J = (τJ)∗.

(13) If J (J^∗(A)) = J^∗(A) for A ∈ L^X, then MJ(MJ(A)) = M^∗_J(A)

such that τM_J = (τM_J)∗

= (

M_J(A)(y) = _

x∈X

(A^∗(x) J^∗(>^∗_x)(y)) |

A ∈ L^Xo .

(14) Define KHJ(A) = (H_J(A))^∗. Then KHJ : L^X → L^X with

KHJ(A)(y) = ^

x∈X

(A(x) → J (>^∗_y)(x))

is an L-meet join approximation operator. Moreover, τK_HJ = τJ.

(15) If J (J (A)) = J (A) for A ∈ L^X, then KH_J(K^∗_H_J(A)) = KH_J

for A ∈ L^Xsuch that τ_K_HJ = (τ_J)_∗with τ_K_HJ = {K_H^∗_J(y)

= _

x∈X

(J^∗(>^∗_y)(x) A^∗(x)) | A ∈ L^X}.

(16) If HJ(H^∗_J(A)) = H^∗_J(A) for A ∈ L^X, then KH_J(KH_J) = K^∗_H_J(A)

such that

τ_K_HJ = (τ_K_HJ)_∗

= {KH_J(A)(y)

= ^

x∈X

(A(x) → J (>^∗_y)(x)) | A ∈ L^X}.

(17) Define MHJ(A) = H_J(A^∗). Then MHJ : L^X → L^X

with

MH_J(A)(y) = _

x∈X

(A^∗(x) J^∗(>^∗_y)(x))

is an L-join meet approximation operator. Moreover, τ_M_HJ = (τ_J)_∗.

(18) If J (J (A)) = J (A) for A ∈ L^X, then MH_J(M^∗_H_J(A)) = MH_J(A) for A ∈ L^Xsuch that τ_M_HJ = (τ_J)_∗with

τ_M_HJ = {M^∗_H_J(A)(y)

= ^

x∈X

(J^∗(>^∗_y)(x) → A(x)) | A ∈ L^X}.

(19) If HJ(H^∗_J(A)) = H^∗_J(A) for A ∈ L^X, then MH_J(MH_J(A)) = M^∗_H_J(A)

such that

τ_M_HJ = (τ_M_HJ)_∗

== _

x∈X

(A^∗(x) J^∗(>^∗_y)(x)) | A ∈ L^X}.

(20) (KHJ, KJ) is a Galois connection;i.e, A ≤ KH_J(B) iff B ≤ KJ(A).

Moreover, τKJ = (τ_K_HJ)_∗.

(21) (MJ, M_H_J) is a dual Galois connection;i.e, MH_J(A) ≤ B iff MJ(B) ≤ A.

Moreover, τMJ = (τ_M_HJ)_∗. Proof.

(1) Since A =V

x∈X(A^∗(x) → >^∗_x), by (J2) and (J3), J (A)(y) = ^

x∈X

(A^∗(x) → J (>^∗_x)(y))

= ^

x∈X

(J^∗(>^∗_x)(y) → A(x)).

(2) Since B(y) ≤ J (A)(y) =V

x∈X(J^∗(>^∗_x)(y) → A(x)) iffW

y∈X(J^∗(>^∗_x)(y) B(y)) ≤ A(x), we have HJ(B)(x) = _

y∈Y

(J^∗(>^∗_x)(y) B(y)).

(5)

(H1) Since HJ(A) ≤ HJ(A) iff A ≤ J (HJ(A)), we have A ≤ J (HJ(A)) ≤ HJ(A).

(H2) aA ≤ J (H_J(a A))

iff A ≤ a → J (HJ(a A))

= J (a → J (H_J(a A))) iff HJ(A) ≤ a → H_J(a A) iff a HJ(A) ≤ H_J(a A).

A ≤ J (H_J(A))

≤ J (a → a HJ(A)) = a → J (a HJ(A)) iff a A ≤ J (a HJ(A))

iff HJ(a A) ≤ a HJ(A).

(H3) By the definition of HJ, since HJ(A) ≤ HJ(B) for B ≤ A, we have

_

i∈Γ

HJ(Ai) ≤ HJ(_

i∈Γ

Ai).

Since J (W

i∈ΓHJ(A_i)) ≥ J (H_J(A_i)) ≥ A_i, then J (W

i∈ΓHJ(Ai)) ≥W

i∈ΓAi. Thus _

i∈Γ

HJ(A_i) ≥ H_J(_

i∈Γ

A_i).

Thus HJ : L^X → L^X is an L-upper approximation operator. By the definition of HJ, we have

HJ(B) ≤ A iff B ≤ J (A).

Since A ≤ J (A) iff HJ(A) ≤ A, we have τHJ = τ_J. (3) Let J (J (A)) = J (A) for A ∈ L^X. Since J (B) ≥

HJ(A) iff J (J (B)) = J (B) ≥ A from the definition of HJ, we have

HJ(HJ(A)) =V{B | J (B) ≥ HJ(A)}

=V{B | J (J (B)) = J (B) ≥ A}

= H_J(A).

(4) Let J^∗(A) ∈ τ_J. Since J (J^∗(A)) = J^∗(A),

J (J (A)) = J (J^∗(J^∗(A))) = (J (J^∗(A)))^∗= J (A).

Hence J (A) ∈ τ_J; i.e. J^∗(A) ∈ (τ_J)_∗. Thus, τJ ⊂ (τ_J)_∗.

Let A ∈ (τ_J)_∗. Then A^∗ = J (A^∗). Since J (A) = J (J^∗(A^∗)) = J^∗(A^∗) = A, then A ∈ τJ. Thus, (τ_J)_∗⊂ τ_J.

(5) (H1) Since J (A^∗) ≤ A^∗, Hs(A) = J (A^∗)^∗≥ A.

(H2) Hs(α A) = (J ((α A)^∗)^∗

= (J (α → A^∗))^∗

= (α → J (A^∗))^∗

= α J (A^∗)^∗

= α Hs(A).

(H3) Hs(_

i∈Γ

Ai) = (J (_

i∈Γ

Ai)^∗)^∗

= (J (^

i∈Γ

A^∗_i))^∗

= (^

i∈Γ

J (A^∗_i))^∗

=_

i∈Γ

(J (A^∗_i))^∗

=_

i∈Γ

H_s(A_i).

Hence Hsis an L-upper approximation operator such that

Hs(B)(x) = (J (B^∗)(x))^∗= _

y∈X

(J^∗(>^∗_y)(x)B(y)).

Moreover, τ_H_s= (τ_J)_∗from:

A = H_s(A) iff A = J (A^∗)^∗ iff A^∗= J (A^∗).

(6) Let J (J (A)) = J (A) for A ∈ L^X. Then

H_s(H_s(A)) = J^∗(H^∗_s(A)) = (J (J (A^∗)))^∗

= J^∗(A^∗) = Hs(A).

Hence τ_H_s= {Hs(A) =W

y∈X(J^∗(>^∗_y)A(y)) | A ∈ L^X}.

(7) Let J (J^∗(A)) = J^∗(A) for A ∈ L^X. Then Hs(H^∗_s(A)) = J^∗(Hs(A)) = (J (J^∗(A^∗)))^∗

= (J^∗(A^∗))^∗= H_s^∗(A).

Hence τ_H_s = {H^∗_s(A) = V

y∈X(A(y) → J (>^∗_y)) | A ∈ L^X}.

Hs(Hs(A)) = Hs(H^∗_s(H_s^∗(A)))

= H^∗_s(H^∗_s(A)) = H_s(A).

By a similar method in (4), τHs= (τHs)∗. (8) It is similarly proved as (5).

(9) If J (J (A)) = J (A) for A ∈ L^X, then KJ(K^∗_J(A)) =

(6)

KJ(A)

KJ(K^∗_J(A)) = KJ(J^∗(A^∗)) = J (J (A^∗))

= J (A^∗) = K_J(A).

(10) If J (J^∗(A)) = J^∗(A) for A ∈ L^X, then K_J(K_J(A)) = K^∗_J(A)

K_J(K_J(A)) = J (K_J^∗(A)) = J (J^∗(A^∗))

= J^∗(A^∗) = K^∗_J(A).

Since KJ(KJ(A)) = K_J^∗(A),

KJ(K^∗_J(A)) = KJ(KJ(KJ(A)))

= K^∗_J(KJ(A)) = KJ(A).

Hence τKJ = {KJ(A) | A ∈ L^X} = (τKJ)∗.

(11) , (12), (13) and (14) are similarly proved as (5), (9), (10) and (5), respectively.

(15) If J (J (A)) = J (A) for A ∈ L^X, then HJ(HJ(A)) = HJ(A). Thus, KHJ(K^∗_H

J(A)) = K_H_J(A) KHJ(K^∗_H_J(A)) = KHJ(HJ(A))

= (HJ(HJ(A)))^∗= (HJ(A))^∗ = KH_J(A).

Since J (A) = A iff H_J(A) = A iff KHJ(A) = A^∗, τK_HJ = (τJ)∗with

τ_K_HJ = {K^∗_H

J(A)(y)

= _

x∈X

(J^∗(>^∗_y)(x) A(x)) | A ∈ L^X}.

(16) If HJ(H^∗_J(A)) = H^∗_J(A) for A ∈ L^X, then KH_J(KH_J(A)) = K^∗_H_J(A)

K_H_J(K_H_J(A)) = K_H_J(K^∗_J(A)) = H^∗_J(H^∗_J(A))

= HJ(A) = K^∗_H

J(A).

(17) , (18) and (19) are similarly proved as (14), (15) and (16), respectively.

(20) (KHJ, KJ) is a Galois connection;i.e, A ≤ KH_J(B) iff A ≤ (HJ(B))^∗ iff HJ(B) ≤ A^∗ iff B ≤ J (A^∗) = K_J(A) Moreover, since A^∗≤ KJ(A) iff A ≤ KH_J(A^∗), τKJ = (τ_K_HJ)_∗.

(21) (MJ, MHJ) is a dual Galois connection;i.e, MH_J(A) ≤ B iff HJ(A^∗) ≤ B iff A^∗≤ J (B) iff MJ(B) = (J (B))^∗≤ A.

Since MH_J(A^∗) ≤ A iff MJ(A) ≤ A^∗, τ_M_J = (τ_M_HJ)_∗.

Let R ∈ L^X×Xbe an L-fuzzy relation. Define operators as follows

HR(A)(y) =W

x∈X(A(x) R(x, y)), J_R(A)(y) =V

x∈X(R(x, y) → A(x)), KR(A)(y) =V

x∈X(A(x) → R(x, y)) M_R(A)(y) =W

x∈X(A^∗(x) R(x, y)).

Example 2.2. Let R be a reflexive L-fuzzy relation. Define JR: L^X→ L^X as follows:

JR(A)(y) = ^

x∈X

(R(x, y) → A(x)).

(1) (J1) J_R(A)(y) ≤ R(y, y) → A(y) = A(y). J_Rsatisfies the conditions (J1) and (J2) from:

JR(a → A)(y) =V

x∈X(R(x, y) → (a → A)(x))

= a →V

x∈X(R(x, y) → A(x)), JR(V

i∈ΓAi)(y) =V

x∈X(R(x, y) →V

i∈ΓAi(x))

=V

i∈Γ

V

x∈X(R(x, y) → A_i(x)).

Hence JRis an L-lower approximation operator.

(2) Define H_J_R(B) =W{A | B ≤ JR(A)}. Since B(y) ≤ JR(A)(y) iff B(y) ≤ ^

x∈X

(R(x, y) → A(x)) iff _

y∈X

(B(y) R(x, y)) ≤ A(x),

then

H_J_R(B)(x) = _

y∈X

(R(x, y) B(y)) = H_R−1(B)(x).

By Theorem 2.1(2), HJR= H_R−1is an L-upper approximation operator such that (HJ_R, JR) is a residuated connection;i.e.,

HJ_R(A) ≤ B iff A ≤ JR(B).

Moreover, τH_JR = τ_J_R.

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(3) If R is an L-fuzzy preorder, then R⁻¹ is an L-fuzzy preorder. Since

JR(J_R(A))(z) = ^

y∈X

(R(y, z) → J_R(A)(y))

= ^

y∈X

(R(y, z) → ^

x∈X

(R(x, y) → A(x)))

= ^

x∈X

^

y∈X

(R(y, z) R(x, y) → A(x)))

= ^

x∈X

(_

y∈X

(R(y, z) R(x, y)) → A(x))

= ^

x∈X

(R(x, z) → A(x))

= JR(A)(z),

By Theorem 2.1(3), HJ_R(HJ_R(A)) = HJ_R(A). By The- orem 2.1(3), τH_JR = τ_J_Rwith

{H_R⁻¹(A) = _

x∈X

(R(−, x) A(x)) | A ∈ L^X}

= τ_H_JR = τ_H

R−1, τ_J_R= {J_R(A) = ^

x∈X

(R(x, −) → A(x)) | A ∈ L^X}.

(4) Let R be a reflexive and Euclidean L-fuzzy relation.

Since R(x, z) R(y, z) A^∗(x) ≤ R(x, y) A^∗(x) iff R(x, z) A^∗(x) ≤ R(y, z) → R(x, y) ≤ A^∗(x),

J_R(J_R^∗(A))(z)

= ^

y∈X

(R(y, z) → J_R^∗(A)(y))

= ^

y∈X

(R(y, z) → _

x∈X

R(x, y) A^∗(x)))

≥ _

x∈X

R(x, z) A^∗(x))).

Thus, JR(J_R^∗(A)) = J_R^∗(A).

By Theorem 2.1(4), JR(J_R(A)) = J_R(A) for A ∈ L^X. Thus, τ_J_R= (τ_J_R)_∗with

τ_J_R = (

J_R^∗(A) = _

x∈X

(R(x, −) A^∗(x)) = M_R(A)

| A ∈ L^X )

.

(5) Define Hs(A) = J_R(A^∗)^∗. By Theorem 2.1(5), Hs=

HRis an L-upper approximation operator such that Hs(A)(y) = (^

x∈X

R(x, y) → A^∗(x))^∗

= _

x∈X

(R(x, y) A(x)).

Moreover, τHs= τHR= (τH_JR)∗.

(6) If R is an L-fuzzy preorder, then JR(JR(A)) = JR(A) for A ∈ L^X. By Theorem 2.1(6), then Hs(Hs(A)) = H_s(A) for A ∈ L^Xsuch that τ_H_s = (τ_J_R)_∗= (τ_H_JR)_∗ with

τ_H_s = {H_s(A) = _

y∈X

(R(y, −) A(y)) | A ∈ L^X}.

(7) If R is a reflexive and Euclidean L-fuzzy relation, then JR(J_R^∗(A)) = J_R^∗(A) for A ∈ L^X. By Theorem 2.1(7), H_s(H^∗_s(A)) = H^∗_s(A) such that

τ_H_s= (τ_H_s)_∗

= {H^∗_s(A)

= ^

y∈X

(A(y) → R^∗(y, −))

= KR^∗(A) | A ∈ L^X}.

(8) Define KJ_R(A) = JR(A^∗). Then KJ_R : L^X → L^X with

KJ_R(A)(y) = ^

x∈X

(R(x, y) → A^∗(x)) = KR^∗(y)

is an L-join meet approximation operator. Moreover, τ_K_JR = (τ_J_R)_∗.

(9) R is an L-fuzzy preorder, then JR(JR(A)) = JR(A) for A ∈ L^X. By Theorem 2.1(9), KJR(K_J^∗

R(A)) = K_J_R(A) for A ∈ L^Xsuch that τ_K_JR = (τ_J_R)_∗with

τK_JR = {K^∗_J_R(A)

= _

x∈X

(R(x, −) A(x))

= HR(A) | A ∈ L^X}.

(10) If R is a reflexive and Euclidean L-fuzzy relation, then JR(J_R^∗(A)) = J_R^∗(A) for A ∈ L^X. By Theorem 2.1(10), KJ_R(KJ_R(A)) = K^∗_J

R(A) such that {K_J_R(A) =V

x∈X(A(x) → R^∗(x, −) | A ∈ L^X}

= τK_JR = (τK_JR)∗.

(11) Define MJR(A) = (J_R(A))^∗. Then MJR : L^X → L^X

(8)

with

MJ_R(A)(y) = _

x∈X

(A^∗(x) R(x, y)) = MR(A)(y) is an L-join meet approximation operator. Moreover, τ_M_JR = τ_J_R.

(12) If R is an L-fuzzy preorder, then JR(JR(A)) = JR(A) for A ∈ L^X. By Theorem 2.1(12), MJ_R(M^∗_J

R(A)) = MJR(A) for A ∈ L^Xsuch that τM_JR = τJRwith

τ_M_JR = {M^∗_J

R(A) =V

x∈X(R(x, −) → A(x))

= JR(A) | A ∈ L^X}.

(13) If R is a reflexive and Euclidean L-fuzzy relation, then JR(J_R^∗(A)) = J_R^∗(A) for A ∈ L^X. By Theorem 2.1(13), MJ_R(MJ_R(A)) = M^∗_J

R(A) such that τ_M_JR = {MJ_R(A) =W

x∈X(A(x) R(x, −))

= HJ_R(A) | A ∈ L^X} = (τM_JR)∗. (14) Define KH_JR(A) = (HJ_R(A))^∗. Then

KH_JR : L^X → L^X with

KH_JR(A)(y) = ^

x∈X

(A(x) → R^∗(y, x))

= K_R^−1∗(A)(y)

is an L-join meet approximation operator. Moreover, τ_K

R−1 = τ_J_R= τ_H

R−1.

(15) If R is an L-fuzzy preorder, then JR(JR(A)) = JR(A) for A ∈ L^X. By Theorem 2.1(15), KR⁻¹(K^∗_R−1(A)) = K_R−1(A) for A ∈ L^Xsuch that τ_K

R−1 = τ_J_R= τ_H

R−1

with τ_K

R−1∗ = {K^∗_R−1∗(A)(y) =W

x∈X(R(y, x) A(x))

= H_R⁻¹(A)(y) | A ∈ L^X}.

(16) Let R⁻¹be a reflexive and Euclidean L-fuzzy relation.

Since

R⁻¹(x, z) R⁻¹(y, z) ≤ R⁻¹(x, y)

iff R⁻¹(y, z) ≤ R⁻¹(x, z) → R⁻¹(x, y) iff R^−1∗(y, z) ≥ R⁻¹(x, z) R^−1∗(x, y), we have

(A(x) → R^−1∗(x, y)) A(x) R⁻¹(x, z)

≤ R⁻¹(x, y) R^−1∗(x, z) ≤ R^−1∗(y, z).

Thus,

A(x)R⁻¹(x, z) ≤ (A(x) → R^−1∗(x, y))

→ R⁻¹(x, y) R^−1∗(x, z) ≤ R^−1∗(y, z).

Hence

K_R^−1∗(K_R^−1∗(A))(z)

= ^

y∈X

(K_R^−1∗(A)(y) → R^−1∗(y, z))

= ^

y∈X

(^

x∈X

(A(x) → R^−1∗(x, y)) → R^−1∗(y, z))

≤ _

x∈X

(A(x) R^−1∗(x, z)) = K_R−1∗(A)(z)

By (K1), K_R^−1∗(K_R^−1∗(A)) = K_R^∗−1∗(A) such that {K_R^−1∗(A) = ^

x∈X

(A(x) → R^∗(−, x)) | A ∈ L^X}

= τ_K

R−1 = (τ_K

R−1)_∗.

(17) Define MH_JR(A) = H_J_R(A^∗). Then MH_JR : L^X→ L^X

is an L-meet join approximation operator as follows:

M_H_JR(A)(y) = _

x∈X

(R(y, x) A^∗(x))

= M_R−1(A)(y).

Moreover, τM_HJ

R

= (τ_J_R)_∗.

(18) If R is an L-fuzzy preorder, then J_R(J_R(A)) = J_R(A) for A ∈ L^X. By Theorem 2.1(18),

M_H_JR(M^∗_H

JR(A)) = M_H_JR(A) for A ∈ L^Xsuch that τM_HJ

R

= (τJ)∗with

τ_M_HJ

R

= (

M^∗_H

JR(A)(y) = ^

x∈X

(R(y, x) → A(x))

= J_R−1(A)(y) | A ∈ L^X )

.

(19) Let R⁻¹ be a reflexive and Euclidean L-fuzzy relation.

Since

(R(y, x) →A(x)) R(z, y) R(z, x)

≤ R(y, x) → A(x)) R(y, x) ≤ A(x), then (R(y, x) → A(x)) R(z, y) ≤ R(z, x) → A(x).

(9)

Thus,

M_R−1(M_R−1(A))(z)

= _

y∈X

(M_R−1(A)(y) R(z, y))

= _

y∈X

(^

x∈X

(R(y, x) → A(x)) R(z, y))

≤ ^

x∈X

(R(z, x) → A(x)) = M_R−1(A)(z).

By (M1), M_R−1(M_R−1(A)) = M^∗_R−1(A) such that {M_R⁻¹(A) =W

x∈X(A^∗(x) R(−, x)) | A ∈ L^X}

= τ_M

R−1 = (τ_M

R−1)_∗.

(20) (KH_JR = K_R−1∗, K_J_R = K_R^∗) is a Galois connection;i.e, A ≤ KH_JR(B) iff B ≤ KJ_R(A). Moreover, τ_K_JR = (τ_K_HJ

R

)_∗.

(21) (MJ_R= MR, MH_JR = M_R−1) is a dual Galois connection;i.e, MH_JR(A) ≤ B iff MJ_R(B) ≤ A. More- over, τ_M_JR = (τ_M_HJ

R

)_∗.

3. Conclusions

In this paper, L-lower approximation operators induce L-upper approximation operators by residuated connection. We study relations lower (upper, join meet, meet join) approximation operators, Galois (dual Galois, residuated, dual residuated) connections and Alexandrov L-topologies. Moreover, we give their examples as approximation operators induced by various L-fuzzy relations.

Conflict of Interest

No potential conflict of interest relevant to this article was reported.

Acknowledgements

This work was supported by the Research Institute of Natural Science of Gangneung-Wonju National University.

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Yong Chan Kim received the B.S., M.S. and Ph.D. degrees in Mathematics from Yonsei University, Seoul, Korea, in 1982, 1984 and 1991, respectively. He is currently Profes- sor of Gangneung-Wonju University, his research interests is a fuzzy topology and fuzzy logic.

E-mail: [email protected]