The Properties of L-lower Approximation Operators
Original Article
ISSN(Print) 1598-2645 ISSN(Online) 2093-744X
International Journal of Fuzzy Logic and Intelligent Systems 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
1.
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
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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 noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
57 |
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}.
http://dx.doi.org/10.5391/IJFIS.2014.14.1.57
(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 ∈ LX , (α → A)(x) = α → A(x), (α A)(x) = α A(x) and >x (x) = >, >x (y) = ⊥, otherwise.
(J1) J (A) ≤ A, (J2) J (α → A) = α → J (A), V V (J3) J ( i∈I Ai ) = i∈I J (Ai ). (3) A map K : LX → LX is called an L-join meet approximation operator iff it satisfies the following conditions (K1) K(A) ≤ A∗ , (K2) K(α A) = α → K(A), W V (K3) K( i∈I Ai ) = i∈I K(Ai ). (4) A map M : LX → LX is called an L-meet join approximation operator iff it satisfies the following conditions (M1) A∗ ≤ M(A), (M2) M(α → A) = α M(A), V W (M3) M( i∈I Ai ) = i∈I M(Ai ).
Lemma 1.3. [3, 5] For each x, y, z, xi , yi ∈ L, we have the following properties.
Definition 1.5. [6, 9] A subset τ ⊂ LX is called an Alexandrov L-topology if it satisfies:
(1) If y ≤ z, (x y) ≤ (x z), x → y ≤ x → z and z → x ≤ y → x.
(T1) ⊥X , >X ∈ τ where >X (x) = > and ⊥X (x) = ⊥ for x ∈ X. W V (T2) If Ai ∈ τ for i ∈ Γ, i∈Γ Ai , i∈Γ Ai ∈ τ .
(2) x y ≤ x ∧ y ≤ x ∨ y. V V W (3) x → ( i∈Γ yi ) = i∈Γ (x → yi ) and ( i∈Γ xi ) → y = V i∈Γ (xi → y). W W (4) x → ( i∈Γ yi ) ≥ i∈Γ (x → yi ) V W (5) ( i∈Γ xi ) → y ≥ 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 ∗ )∗ . V W W V (14) i∈Γ x∗i = ( i∈Γ xi )∗ and i∈Γ x∗i = ( i∈Γ xi )∗ . Definition 1.4. [8, 9] X
X
(1) A map H : L → L is called an L-upper approximation operator iff it satisfies the following conditions (H1) A ≤ H(A), (H2) H(α A) = α H(A) where α(x) = α for all x ∈ X, W W (H3) H( i∈I Ai ) = i∈I H(Ai ). X
X
(2) A map J : L → L is called an L-lower approximation operator iff it satisfies the following conditions www.ijfis.org
(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∗ ∈ LX | A ∈ τ } is an Alexandrov topology on X. (2) If H is an L-upper approximation operator, then τH = {A ∈ LX | H(A) = A} is an Alexandrov topology on X. (3) If J is an L-lower approximation operator, then τJ = {A ∈ LX | J (A) = A} is an Alexandrov topology on X. (4) If K is an L-join meet approximation operator, then τK = {A ∈ LX | K(A) = A∗ } is an Alexandrov topology on X. (5) If M is an L-meet join operator, then τM = {A ∈ LX | 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. The Properties of L-lower Approximation Operators
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International Journal of Fuzzy Logic and Intelligent Systems, vol. 14, no. 1, March 2014
(R4) Euclidean if R(x, z) R(y, z) ≤ R(x, y), for all x, y, z ∈ X.
(6) If J (J (A)) = J (A) for A ∈ LX , then Hs (Hs (A)) = Hs (A)
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
for A ∈ LX such that τHs = (τJ )∗ = (τHJ )∗ . with τHs = {Hs (A) =
_
(J ∗ (>∗y ) A(y)) | A ∈ LX }.
y∈X
2.
The Properties of L-lower Approximation Operators
(7) If J (J ∗ (A)) = J ∗ (A) for A ∈ LX , then Hs (Hs∗ (A)) = Hs∗ (A)
Theorem 2.1. Let J : LX → LX be an L-lower approximation operator. Then the following properties hold. such that X
V
∗
(>∗x )(y)
(1) For A ∈ L , J (A)(y) = x∈X (J → A(x)). V (2) Define HJ (B) = {A | B ≤ J (A)}. Then HJ : LX → LX with _
HJ (B)(x) =
(J ∗ (>∗x )(y) B(y))
V {Hs∗ (A) = y∈X (A(y) → J (>∗y )) | A ∈ LX } = τHs = (τHs )∗ . (8) Define KJ (A) = J (A∗ ). Then KJ : LX → LX with
y∈X
KJ (A) = is an L-upper approximation operator such that (HJ , J ) is a residuated connection;i.e., HJ (B) ≤ A iff B ≤ J (A).
is an L-join meet approximation operator. (9) If J (J (A)) = J (A) for A ∈ LX , then KJ (KJ∗ (A)) = KJ∗ (A)
(3) If J (J (A)) = J (A) for A ∈ LX , then HJ (HJ (A)) = HJ (A) for A ∈ LX such that τJ = τHJ with τJ = {J (A) =
(A(y) → J (>∗y ))
y∈X
Moreover, τJ = τHJ .
^
^
for A ∈ LX such that τKJ = (τJ )∗ with τKJ = {KJ∗ (A) =
(J ∗ (>∗x )(y) → A(x)) | A ∈ LX },
_
(J ∗ (>∗y ) A(y)) | A ∈ LX }.
y∈X
x∈X
(10) If J (J ∗ (A)) = J ∗ (A) for A ∈ LX , then τHJ = {HJ (A)(x) _ = (J ∗ (>∗x )(y) A(y)) | A ∈ LX }.
KJ (KJ (A)) = KJ∗ (A)
y∈X
such that (4) If J (J ∗ (A)) = J ∗ (A) for A ∈ LX , then J (J (A)) = J (A) such that W {J ∗ (A) = x∈X (A∗ (x) J ∗ (>∗x )) | A ∈ LX } = τJ = (τJ )∗ .
V {KJ (A) = y∈X (A(y) → J (>∗y )) | A ∈ LX } = τKJ = (τKJ )∗ . (11) Define MJ (A) = (J (A))∗ . Then MJ : LX → LX with
(5) Define Hs (A) = J (A∗ )∗ . Then Hs : LX → LX with MJ (A)(y) = Hs (B)(x) =
_
(J ∗ (>∗y )(x) B(y))
y∈X
is an L-upper approximation operator. Moreover, τHs = (τJ )∗ = (τHJ )∗ . 59 | Yong Chan Kim
_
(A∗ (x) J ∗ (>∗x )(y))
x∈X
is an L-meet join approximation operator. Moreover, τMJ = τJ . (12) If J (J (A)) = J (A) for A ∈ LX , then MJ (M∗J (A)) =
http://dx.doi.org/10.5391/IJFIS.2014.14.1.57
MJ (A) for A ∈ LX such that τMJ = (τJ )∗ with ^
{M∗J (A)(y) =
with
(J ∗ (>∗x )(y) → A(x)) | A ∈ LX }
MHJ (A)(y) =
x∈X
(A∗ (x) J ∗ (>∗y )(x))
x∈X
= τMJ = (τJ )∗ . (13) If J (J ∗ (A)) = J ∗ (A) for A ∈ LX , then
is an L-join meet approximation operator. Moreover, τMHJ = (τJ )∗ . (18) If J (J (A)) = J (A) for A ∈ LX , then
MJ (MJ (A)) = M∗J (A)
MHJ (M∗HJ (A)) = MHJ (A)
such that
for A ∈ LX such that τMHJ = (τJ )∗ with
τMJ = (τMJ )∗ ( =
_
_
MJ (A)(y) =
τMHJ = {M∗HJ (A)(y) ^ = (J ∗ (>∗y )(x) → A(x)) | A ∈ LX }.
(A∗ (x) J ∗ (>∗x )(y)) |
x∈X
x∈X X
A∈L
o
.
(19) If HJ (HJ∗ (A)) = HJ∗ (A) for A ∈ LX , then MHJ (MHJ (A)) = M∗HJ (A)
(14) Define KHJ (A) = (HJ (A))∗ . Then KHJ : LX → LX with such that KHJ (A)(y) =
^
(A(x) → J (>∗y )(x))
τMHJ = (τMHJ )∗ _ == (A∗ (x) J ∗ (>∗y )(x)) | A ∈ LX }.
x∈X
is an L-meet join approximation operator. Moreover, τKHJ = τJ . (15) If J (J (A)) = J (A) for A ∈ LX , then
x∈X
(20) (KHJ , KJ ) is a Galois connection;i.e, A ≤ KHJ (B) iff B ≤ KJ (A).
∗ KHJ (KH (A)) = KHJ J
for A ∈ LX such that τKHJ = (τJ )∗ with
Moreover, τKJ = (τKHJ )∗ . (21) (MJ , MHJ ) is a dual Galois connection;i.e,
∗ τKHJ = {KH (y) J _ = (J ∗ (>∗y )(x) A∗ (x)) | A ∈ LX }. x∈X
(16) If HJ (HJ∗ (A)) = HJ∗ (A) for A ∈ LX , then
MHJ (A) ≤ B iff MJ (B) ≤ A. Moreover, τMJ = (τMHJ )∗ . Proof. (1) Since A =
V
x∈X (A
∗
∗ KHJ (KHJ ) = KH (A) J
^
J (A)(y) = such that
(A∗ (x) → J (>∗x )(y))
x∈X
τKHJ = (τKHJ )∗ = {KHJ (A)(y) ^ = (A(x) → J (>∗y )(x)) | A ∈ LX }. x∈X
(17) Define MHJ (A) = HJ (A∗ ). Then MHJ : LX → LX www.ijfis.org
(x) → >∗x ), by (J2) and (J3),
^
=
(J ∗ (>∗x )(y) → A(x)).
x∈X
V (2) Since B(y) ≤ J (A)(y) = x∈X (J ∗ (>∗x )(y) → A(x)) W iff y∈X (J ∗ (>∗x )(y) B(y)) ≤ A(x), we have HJ (B)(x) =
_
(J ∗ (>∗x )(y) B(y)).
y∈Y
The Properties of L-lower Approximation Operators
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International Journal of Fuzzy Logic and Intelligent Systems, vol. 14, no. 1, March 2014
(H1) Since HJ (A) ≤ HJ (A) iff A ≤ J (HJ (A)), we have A ≤ J (HJ (A)) ≤ HJ (A). (H2) a A ≤ J (HJ (a A))
(5) (H1) Since J (A∗ ) ≤ A∗ , Hs (A) = J (A∗ )∗ ≥ A. (H2) Hs (α A) = (J ((α A)∗ )∗ = (J (α → A∗ ))∗
iff A ≤ a → J (HJ (a A))
= (α → J (A∗ ))∗
= J (a → J (HJ (a A)))
= α J (A∗ )∗
iff HJ (A) ≤ a → HJ (a A) iff a HJ (A) ≤ HJ (a A).
(H3)
Hs (
_
= α Hs (A). _ Ai ) = (J ( Ai )∗ )∗
i∈Γ
A ≤ J (HJ (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).
i∈Γ
=(
HJ (Ai ) ≤ HJ (
i∈Γ
=
W Since J ( i∈Γ HJ (Ai )) ≥ J (HJ (Ai )) ≥ Ai , then W W J ( i∈Γ HJ (Ai )) ≥ i∈Γ Ai . Thus _
HJ (Ai ) ≥ HJ (
i∈Γ
_
J (A∗i ))∗
_
(J (A∗i ))∗
i∈Γ
_
Hs (Ai ).
i∈Γ
Ai ).
i∈Γ
^
i∈Γ
= _
A∗i ))∗
i∈Γ
(H3) By the definition of HJ , since HJ (A) ≤ HJ (B) for B ≤ A, we have _
^
= (J (
Hence Hs is an L-upper approximation operator such that Hs (B)(x) = (J (B ∗ )(x))∗ =
_
(J ∗ (>∗y )(x) B(y)).
y∈X
Ai ).
i∈Γ
Thus HJ : LX → LX is an L-upper approximation operator. By the definition of HJ , we have
Moreover, τHs = (τJ )∗ from: A = Hs (A) iff A = J (A∗ )∗ iff A∗ = J (A∗ ). (6) Let J (J (A)) = J (A) for A ∈ LX . Then
HJ (B) ≤ A iff B ≤ J (A). Hs (Hs (A)) Since A ≤ J (A) iff HJ (A) ≤ A, we have τHJ = τJ . (3) Let J (J (A)) = J (A) for A ∈ LX . 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} = HJ (A).
(4) Let J ∗ (A) ∈ τJ . Since J (J ∗ (A)) = J ∗ (A), J (J (A)) = J (J ∗ (J ∗ (A))) = (J (J ∗ (A)))∗ = J (A).
= J ∗ (Hs∗ (A)) = (J (J (A∗ )))∗ = J ∗ (A∗ ) = Hs (A).
Hence τHs = {Hs (A) = LX }.
Hence J (A) ∈ τJ ; i.e. J (A) ∈ (τJ )∗ . Thus, τJ ⊂ (τJ )∗ . Let A ∈ (τJ )∗ . Then A∗ = J (A∗ ). Since J (A) = J (J ∗ (A∗ )) = J ∗ (A∗ ) = A, then A ∈ τJ . Thus, (τJ )∗ ⊂ τJ . 61 | Yong Chan Kim
y∈X (J
∗
(>∗y ) A(y)) | A ∈
(7) Let J (J ∗ (A)) = J ∗ (A) for A ∈ LX . Then Hs (Hs∗ (A))
= J ∗ (Hs (A)) = (J (J ∗ (A∗ )))∗ = (J ∗ (A∗ ))∗ = Hs∗ (A).
Hence τHs = {Hs∗ (A) = A ∈ LX }. Hs (Hs (A))
∗
W
V
y∈X (A(y)
→ J (>∗y )) |
= Hs (Hs∗ (Hs∗ (A))) = Hs∗ (Hs∗ (A)) = Hs (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 ∈ LX , then KJ (KJ∗ (A)) =
http://dx.doi.org/10.5391/IJFIS.2014.14.1.57
KJ (A)
(21) (MJ , MHJ ) is a dual Galois connection;i.e,
KJ (KJ∗ (A))
= KJ (J ∗ (A∗ )) = J (J (A∗ )) = J (A∗ ) = KJ (A).
MHJ (A) ≤ B iff HJ (A∗ ) ≤ B iff A∗ ≤ J (B) iff MJ (B) = (J (B))∗ ≤ A.
(10) If J (J ∗ (A)) = J ∗ (A) for A ∈ LX , then KJ (KJ (A)) = KJ∗ (A) KJ (KJ (A))
= J (KJ∗ (A)) = J (J ∗ (A∗ )) = J ∗ (A∗ ) = KJ∗ (A).
Since KJ (KJ (A)) = KJ∗ (A), KJ (KJ∗ (A))
= KJ (KJ (KJ (A))) = KJ∗ (KJ (A)) = KJ (A).
Hence τKJ = {KJ (A) | A ∈ LX } = (τKJ )∗ . (11) , (12), (13) and (14) are similarly proved as (5), (9), (10) and (5), respectively.
Since MHJ (A∗ ) ≤ A iff MJ (A) ≤ A∗ , τMJ = (τMHJ )∗ . Let R ∈ LX×X be an L-fuzzy relation. Define operators as follows W HR (A)(y) = x∈X (A(x) R(x, y)), V JR (A)(y) = x∈X (R(x, y) → A(x)), V KR (A)(y) = x∈X (A(x) → R(x, y)) W MR (A)(y) = x∈X (A∗ (x) R(x, y)). Example 2.2. Let R be a reflexive L-fuzzy relation. Define JR : LX → LX as follows:
(15) If J (J (A)) = J (A) for A ∈ L , then HJ (HJ (A)) = ∗ HJ (A). Thus, KHJ (KH (A)) = KHJ (A) J ∗ KHJ (KH (A)) = KHJ (HJ (A)) J = (HJ (HJ (A)))∗ = (HJ (A))∗ = KHJ (A).
Since J (A) = A iff HJ (A) = A iff KHJ (A) = A∗ , τKHJ = (τJ )∗ with ∗ τKHJ = {KH (A)(y) J _ = (J ∗ (>∗y )(x) A(x)) | A ∈ LX }. x∈X
(16) If HJ (HJ∗ (A)) = HJ∗ (A) for A ∈ LX , then
^
JR (A)(y) =
X
(1) (J1) JR (A)(y) ≤ R(y, y) → A(y) = A(y). JR satisfies the conditions (J1) and (J2) from: V = x∈X (R(x, y) → (a → A)(x)) V = a → x∈X (R(x, y) → A(x)), V V V JR ( i∈Γ Ai )(y) = x∈X (R(x, y) → i∈Γ Ai (x)) V V = i∈Γ x∈X (R(x, y) → Ai (x)). JR (a → A)(y)
Hence JR is an L-lower approximation operator. W (2) Define HJR (B) = {A | B ≤ JR (A)}. Since B(y) ≤ JR (A)(y) iff B(y) ≤
KHJ (KHJ (A))
= =
(17) , (18) and (19) are similarly proved as (14), (15) and (16), respectively. (20) (KHJ , KJ ) is a Galois connection;i.e, A ≤ KHJ (B) iff A ≤ (HJ (B))∗ iff HJ (B) ≤ A∗ iff B ≤ J (A∗ ) = KJ (A) Moreover, since A∗ ≤ KJ (A) iff A ≤ KHJ (A∗ ), τKJ = (τKHJ )∗ . www.ijfis.org
^
(R(x, y) → A(x))
x∈X
∗ (A) KHJ (KHJ (A)) = KH J
KHJ (KJ∗ (A)) = HJ∗ (HJ∗ (A)) ∗ HJ (A) = KH (A). J
(R(x, y) → A(x)).
x∈X
iff
_
(B(y) R(x, y)) ≤ A(x),
y∈X
then HJR (B)(x) =
_
(R(x, y) B(y)) = HR−1 (B)(x).
y∈X
By Theorem 2.1(2), HJR = HR−1 is an L-upper approximation operator such that (HJR , JR ) is a residuated connection;i.e., HJR (A) ≤ B iff A ≤ JR (B). Moreover, τHJR = τJR . The Properties of L-lower Approximation Operators
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International Journal of Fuzzy Logic and Intelligent Systems, vol. 14, no. 1, March 2014
(3) If R is an L-fuzzy preorder, then R−1 is an L-fuzzy preorder. Since
HR is an L-upper approximation operator such that ^
Hs (A)(y) = ( ^
JR (JR (A))(z) =
(R(y, z) → JR (A)(y))
x∈X
=
y∈X
^
=
(R(y, z) →
y∈X
(R(y, z) R(x, y) → A(x)))
x∈X y∈X
^ _ = ( (R(y, z) R(x, y)) → A(x)) x∈X y∈X
^
=
(R(x, z) → A(x))
x∈X
_
(R(−, x) A(x)) | A ∈ LX }
x∈X
= τHJR = τHR−1 , ^
τJR = {JR (A) =
(R(x, −) → A(x)) | A ∈ LX }.
x∈X
^
(R(y, z) → (R(y, z) →
_
y∈X
_
≥
R(x, y) A∗ (x)))
x∈X
R(x, z) A∗ (x))).
x∈X
Thus,
JR (JR∗ (A))
=
By Theorem 2.1(4), JR (JR (A)) = JR (A) for A ∈ LX . Thus, τJR = (τJR )∗ with ( τJR =
JR∗ (A) =
= {Hs∗ (A) ^ = (A(y) → R∗ (y, −)) y∈X
(8) Define KJR (A) = JR (A∗ ). Then KJR : LX → LX with ^
(R(x, y) → A∗ (x)) = KR∗ (y)
is an L-join meet approximation operator. Moreover, τKJR = (τJR )∗ . (9) R is an L-fuzzy preorder, then JR (JR (A)) = JR (A) for A ∈ LX . By Theorem 2.1(9), KJR (KJ∗R (A)) = KJR (A) for A ∈ LX such that τKJR = (τJR )∗ with τKJR = {KJ∗R (A) _ = (R(x, −) A(x))
JR∗ (A).
_
τHs = (τHs )∗
x∈X
JR∗ (A)(y))
y∈X
=
(R(y, −) A(y)) | A ∈ LX }.
(7) If R is a reflexive and Euclidean L-fuzzy relation, then JR (JR∗ (A)) = JR∗ (A) for A ∈ LX . By Theorem 2.1(7), Hs (Hs∗ (A)) = Hs∗ (A) such that
KJR (A)(y) =
JR (JR∗ (A))(z) =
_
= KR∗ (A) | A ∈ LX }.
(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),
^
(6) If R is an L-fuzzy preorder, then JR (JR (A)) = JR (A) for A ∈ LX . By Theorem 2.1(6), then Hs (Hs (A)) = Hs (A) for A ∈ LX such that τHs = (τJR )∗ = (τHJR )∗ with
y∈X
By Theorem 2.1(3), HJR (HJR (A)) = HJR (A). By Theorem 2.1(3), τHJR = τJR with {HR−1 (A) =
(R(x, y) A(x)).
Moreover, τHs = τHR = (τHJR )∗ .
τHs = {Hs (A) =
= JR (A)(z),
_ x∈X
(R(x, y) → A(x)))
x∈X
^ ^
=
^
R(x, y) → A∗ (x))∗
(R(x, −) A∗ (x)) = MR (A)
x∈X
x∈X
= HR (A) | A ∈ LX }. (10) If R is a reflexive and Euclidean L-fuzzy relation, then JR (JR∗ (A)) = JR∗ (A) for A ∈ LX . By Theorem 2.1(10), KJR (KJR (A)) = KJ∗R (A) such that
) | A∈L
X
.
(5) Define Hs (A) = JR (A∗ )∗ . By Theorem 2.1(5), Hs = 63 | Yong Chan Kim
V {KJR (A) = x∈X (A(x) → R∗ (x, −) | A ∈ LX } = τKJR = (τKJR )∗ . (11) Define MJR (A) = (JR (A))∗ . Then MJR : LX → LX
http://dx.doi.org/10.5391/IJFIS.2014.14.1.57
Thus,
with MJR (A)(y) =
_
(A∗ (x) R(x, y)) = MR (A)(y)
A(x) R−1 (x, z) ≤ (A(x) → R−1∗ (x, y)) → R−1 (x, y) R−1∗ (x, z) ≤ R−1∗ (y, z).
x∈X
is an L-join meet approximation operator. Moreover, τMJR = τJR . (12) If R is an L-fuzzy preorder, then JR (JR (A)) = JR (A) for A ∈ LX . By Theorem 2.1(12), MJR (M∗JR (A)) = MJR (A) for A ∈ LX such that τMJR = τJR with τMJR = {M∗JR (A) = = JR (A) | A ∈ LX }.
V
x∈X (R(x, −) → A(x))
Hence KR−1∗ (KR−1∗ (A))(z) ^ = (KR−1∗ (A)(y) → R−1∗ (y, z)) y∈X
=
y∈X x∈X
≤ (13) If R is a reflexive and Euclidean L-fuzzy relation, then JR (JR∗ (A)) = JR∗ (A) for A ∈ LX . By Theorem 2.1(13), MJR (MJR (A)) = M∗JR (A) such that W
τMJR = {MJR (A) = x∈X (A(x) R(x, −)) = HJR (A) | A ∈ LX } = (τMJR )∗ . (14) Define KHJR (A) = (HJR (A))∗ . Then
^ ^ ( (A(x) → R−1∗ (x, y)) → R−1∗ (y, z)) _
(A(x) R−1∗ (x, z)) = KR−1∗ (A)(z)
x∈X ∗ By (K1), KR−1∗ (KR−1∗ (A)) = KR −1∗ (A) such that
{KR−1∗ (A) =
= τKR−1 = (τKR−1 )∗ . (17) Define MHJR (A) = HJR (A∗ ). Then MHJR : LX → LX
with ^
is an L-meet join approximation operator as follows: _ MHJR (A)(y) = (R(y, x) A∗ (x))
(A(x) → R∗ (y, x))
x∈X
x∈X
= KR−1∗ (A)(y)
= MR−1 (A)(y).
is an L-join meet approximation operator. Moreover, τKR−1 = τJR = τHR−1 . (15) If R is an L-fuzzy preorder, then JR (JR (A)) = JR (A) ∗ for A ∈ LX . By Theorem 2.1(15), KR−1 (KR −1 (A)) = X KR−1 (A) for A ∈ L such that τKR−1 = τJR = τHR−1 with ∗ {KR −1∗ (A)(y) = X
τKR−1∗ = = HR−1 (A)(y) | A ∈ L }. (16) Let R Since
−1
(A(x) → R∗ (−, x)) | A ∈ LX }
x∈X
KHJR : LX → LX KHJR (A)(y) =
^
W
x∈X (R(y, x)
A(x))
Moreover, τMHJ = (τJR )∗ . R (18) If R is an L-fuzzy preorder, then JR (JR (A)) = JR (A) for A ∈ LX . By Theorem 2.1(18), MHJR (M∗HJ (A)) = MHJR (A) R
for A ∈ LX such that τMHJ = (τJ )∗ with R
( τMHJ = M∗HJ (A)(y) = R
be a reflexive and Euclidean L-fuzzy relation.
^
R
(R(y, x) → A(x))
x∈X
) X
= JR−1 (A)(y) | A ∈ L
.
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
(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),
(A(x) → R−1∗ (x, y)) A(x) R−1 (x, z) ≤ R−1 (x, y) R−1∗ (x, z) ≤ R−1∗ (y, z). www.ijfis.org
then (R(y, x) → A(x)) R(z, y) ≤ R(z, x) → A(x). The Properties of L-lower Approximation Operators
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International Journal of Fuzzy Logic and Intelligent Systems, vol. 14, no. 1, March 2014
Thus, MR−1 (MR−1 (A))(z) _ = (MR−1 (A)(y) R(z, y)) y∈X
_ ^ = ( (R(y, x) → A(x)) R(z, y)) y∈X x∈X
≤
^
(R(z, x) → A(x)) = MR−1 (A)(z).
x∈X
By (M1), MR−1 (MR−1 (A)) = M∗R−1 (A) such that W {MR−1 (A) = x∈X (A∗ (x) R(−, x)) | A ∈ LX } = τMR−1 = (τMR−1 )∗ . (20) (KHJR = KR−1∗ , KJR = KR∗ ) is a Galois connection;i.e, A ≤ KHJR (B) iff B ≤ KJR (A). Moreover, τKJR = (τKHJ )∗ . R (21) (MJR = MR , MHJR = MR−1 ) is a dual Galois connection;i.e, MHJR (A) ≤ B iff MJR (B) ≤ A. Moreover, τMJR = (τMHJ )∗ . 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.
References [1] Z. Pawlak, “Rough sets,” International Journal of Computer & Information Sciences, vol. 11, no. 5, pp. 341-356, Oct. 1982. http://dx.doi.org/10.1007/BF01001956
65 | Yong Chan Kim
[2] Z. Pawlak, “Rough probability,” Bulletin of Polish Academy of Sciences: Mathematics, vol. 32, no. 9-10, pp. 607-615, 1984. [3] P. H´ajek,Metamathematics of Fuzzy Logic, Dordrecht, The Netherlands: Kluwer, 1998. [4] A. M. Radzikowska and E. E. Kerre, “A comparative study of fuzzy rough sets,” Fuzzy Sets and Systems, vol. 126, no. 2, pp. 137-155, Mar. 2002. http://dx.doi.org/10.1016/ S0165-0114(01)00032-X [5] R. Bˇelohl´avek, Fuzzy Relational Systems: Foundations and Principles, New York, NY: Kluwer Academic/Plenum Publishers, 2002. [6] H. Lai and D. Zhang, “Fuzzy preorder and fuzzy topology,” Fuzzy Sets and Systems, vol. 157, no. 14, pp. 1865-1885, Jul. 2006. http://dx.doi.org/10.1016/j.fss.2006.02.013 [7] H. Lai and D. Zhang, “Concept lattices of fuzzy contexts: formal concept analysis vs. rough set theory,” International Journal of Approximate Reasoning, vol. 50, no. 5, pp. 695-707, May. 2009. http://dx.doi.org/10.1016/j.ijar. 2008.12.002 [8] Y. C. Kim, “Alexandrov L-topologiesand L-join meet approximatin operators,” International Journal of Pure and Applied Mathematics, vol. 91, no. 1, pp. 113-129, 2014. http://dx.doi.org/10.12732/ijpam.v91i1.12 [9] Y. C. Kim, “Alexandrov L-topologies,” International Journal of Pure and Applied Mathematics, 2014 [in press]. [10] Y. H. She and G. J. Wang, “An axiomatic approach of fuzzy rough sets based on residuated lattices,” Computers & Mathematics with Applications, vol. 58, no. 1, pp. 189201, Jul. 2009. http://dx.doi.org/10.1016/j.camwa.2009. 03.100 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 Professor of Gangneung-Wonju University, his research interests is a fuzzy topology and fuzzy logic. E-mail: [email protected]
ISSN(Print) 1598-2645 ISSN(Online) 2093-744X
International Journal of Fuzzy Logic and Intelligent Systems 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
1.
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
cc
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 noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
57 |
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}.
http://dx.doi.org/10.5391/IJFIS.2014.14.1.57
(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 ∈ LX , (α → A)(x) = α → A(x), (α A)(x) = α A(x) and >x (x) = >, >x (y) = ⊥, otherwise.
(J1) J (A) ≤ A, (J2) J (α → A) = α → J (A), V V (J3) J ( i∈I Ai ) = i∈I J (Ai ). (3) A map K : LX → LX is called an L-join meet approximation operator iff it satisfies the following conditions (K1) K(A) ≤ A∗ , (K2) K(α A) = α → K(A), W V (K3) K( i∈I Ai ) = i∈I K(Ai ). (4) A map M : LX → LX is called an L-meet join approximation operator iff it satisfies the following conditions (M1) A∗ ≤ M(A), (M2) M(α → A) = α M(A), V W (M3) M( i∈I Ai ) = i∈I M(Ai ).
Lemma 1.3. [3, 5] For each x, y, z, xi , yi ∈ L, we have the following properties.
Definition 1.5. [6, 9] A subset τ ⊂ LX is called an Alexandrov L-topology if it satisfies:
(1) If y ≤ z, (x y) ≤ (x z), x → y ≤ x → z and z → x ≤ y → x.
(T1) ⊥X , >X ∈ τ where >X (x) = > and ⊥X (x) = ⊥ for x ∈ X. W V (T2) If Ai ∈ τ for i ∈ Γ, i∈Γ Ai , i∈Γ Ai ∈ τ .
(2) x y ≤ x ∧ y ≤ x ∨ y. V V W (3) x → ( i∈Γ yi ) = i∈Γ (x → yi ) and ( i∈Γ xi ) → y = V i∈Γ (xi → y). W W (4) x → ( i∈Γ yi ) ≥ i∈Γ (x → yi ) V W (5) ( i∈Γ xi ) → y ≥ 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 ∗ )∗ . V W W V (14) i∈Γ x∗i = ( i∈Γ xi )∗ and i∈Γ x∗i = ( i∈Γ xi )∗ . Definition 1.4. [8, 9] X
X
(1) A map H : L → L is called an L-upper approximation operator iff it satisfies the following conditions (H1) A ≤ H(A), (H2) H(α A) = α H(A) where α(x) = α for all x ∈ X, W W (H3) H( i∈I Ai ) = i∈I H(Ai ). X
X
(2) A map J : L → L is called an L-lower approximation operator iff it satisfies the following conditions www.ijfis.org
(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∗ ∈ LX | A ∈ τ } is an Alexandrov topology on X. (2) If H is an L-upper approximation operator, then τH = {A ∈ LX | H(A) = A} is an Alexandrov topology on X. (3) If J is an L-lower approximation operator, then τJ = {A ∈ LX | J (A) = A} is an Alexandrov topology on X. (4) If K is an L-join meet approximation operator, then τK = {A ∈ LX | K(A) = A∗ } is an Alexandrov topology on X. (5) If M is an L-meet join operator, then τM = {A ∈ LX | 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. The Properties of L-lower Approximation Operators
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International Journal of Fuzzy Logic and Intelligent Systems, vol. 14, no. 1, March 2014
(R4) Euclidean if R(x, z) R(y, z) ≤ R(x, y), for all x, y, z ∈ X.
(6) If J (J (A)) = J (A) for A ∈ LX , then Hs (Hs (A)) = Hs (A)
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
for A ∈ LX such that τHs = (τJ )∗ = (τHJ )∗ . with τHs = {Hs (A) =
_
(J ∗ (>∗y ) A(y)) | A ∈ LX }.
y∈X
2.
The Properties of L-lower Approximation Operators
(7) If J (J ∗ (A)) = J ∗ (A) for A ∈ LX , then Hs (Hs∗ (A)) = Hs∗ (A)
Theorem 2.1. Let J : LX → LX be an L-lower approximation operator. Then the following properties hold. such that X
V
∗
(>∗x )(y)
(1) For A ∈ L , J (A)(y) = x∈X (J → A(x)). V (2) Define HJ (B) = {A | B ≤ J (A)}. Then HJ : LX → LX with _
HJ (B)(x) =
(J ∗ (>∗x )(y) B(y))
V {Hs∗ (A) = y∈X (A(y) → J (>∗y )) | A ∈ LX } = τHs = (τHs )∗ . (8) Define KJ (A) = J (A∗ ). Then KJ : LX → LX with
y∈X
KJ (A) = is an L-upper approximation operator such that (HJ , J ) is a residuated connection;i.e., HJ (B) ≤ A iff B ≤ J (A).
is an L-join meet approximation operator. (9) If J (J (A)) = J (A) for A ∈ LX , then KJ (KJ∗ (A)) = KJ∗ (A)
(3) If J (J (A)) = J (A) for A ∈ LX , then HJ (HJ (A)) = HJ (A) for A ∈ LX such that τJ = τHJ with τJ = {J (A) =
(A(y) → J (>∗y ))
y∈X
Moreover, τJ = τHJ .
^
^
for A ∈ LX such that τKJ = (τJ )∗ with τKJ = {KJ∗ (A) =
(J ∗ (>∗x )(y) → A(x)) | A ∈ LX },
_
(J ∗ (>∗y ) A(y)) | A ∈ LX }.
y∈X
x∈X
(10) If J (J ∗ (A)) = J ∗ (A) for A ∈ LX , then τHJ = {HJ (A)(x) _ = (J ∗ (>∗x )(y) A(y)) | A ∈ LX }.
KJ (KJ (A)) = KJ∗ (A)
y∈X
such that (4) If J (J ∗ (A)) = J ∗ (A) for A ∈ LX , then J (J (A)) = J (A) such that W {J ∗ (A) = x∈X (A∗ (x) J ∗ (>∗x )) | A ∈ LX } = τJ = (τJ )∗ .
V {KJ (A) = y∈X (A(y) → J (>∗y )) | A ∈ LX } = τKJ = (τKJ )∗ . (11) Define MJ (A) = (J (A))∗ . Then MJ : LX → LX with
(5) Define Hs (A) = J (A∗ )∗ . Then Hs : LX → LX with MJ (A)(y) = Hs (B)(x) =
_
(J ∗ (>∗y )(x) B(y))
y∈X
is an L-upper approximation operator. Moreover, τHs = (τJ )∗ = (τHJ )∗ . 59 | Yong Chan Kim
_
(A∗ (x) J ∗ (>∗x )(y))
x∈X
is an L-meet join approximation operator. Moreover, τMJ = τJ . (12) If J (J (A)) = J (A) for A ∈ LX , then MJ (M∗J (A)) =
http://dx.doi.org/10.5391/IJFIS.2014.14.1.57
MJ (A) for A ∈ LX such that τMJ = (τJ )∗ with ^
{M∗J (A)(y) =
with
(J ∗ (>∗x )(y) → A(x)) | A ∈ LX }
MHJ (A)(y) =
x∈X
(A∗ (x) J ∗ (>∗y )(x))
x∈X
= τMJ = (τJ )∗ . (13) If J (J ∗ (A)) = J ∗ (A) for A ∈ LX , then
is an L-join meet approximation operator. Moreover, τMHJ = (τJ )∗ . (18) If J (J (A)) = J (A) for A ∈ LX , then
MJ (MJ (A)) = M∗J (A)
MHJ (M∗HJ (A)) = MHJ (A)
such that
for A ∈ LX such that τMHJ = (τJ )∗ with
τMJ = (τMJ )∗ ( =
_
_
MJ (A)(y) =
τMHJ = {M∗HJ (A)(y) ^ = (J ∗ (>∗y )(x) → A(x)) | A ∈ LX }.
(A∗ (x) J ∗ (>∗x )(y)) |
x∈X
x∈X X
A∈L
o
.
(19) If HJ (HJ∗ (A)) = HJ∗ (A) for A ∈ LX , then MHJ (MHJ (A)) = M∗HJ (A)
(14) Define KHJ (A) = (HJ (A))∗ . Then KHJ : LX → LX with such that KHJ (A)(y) =
^
(A(x) → J (>∗y )(x))
τMHJ = (τMHJ )∗ _ == (A∗ (x) J ∗ (>∗y )(x)) | A ∈ LX }.
x∈X
is an L-meet join approximation operator. Moreover, τKHJ = τJ . (15) If J (J (A)) = J (A) for A ∈ LX , then
x∈X
(20) (KHJ , KJ ) is a Galois connection;i.e, A ≤ KHJ (B) iff B ≤ KJ (A).
∗ KHJ (KH (A)) = KHJ J
for A ∈ LX such that τKHJ = (τJ )∗ with
Moreover, τKJ = (τKHJ )∗ . (21) (MJ , MHJ ) is a dual Galois connection;i.e,
∗ τKHJ = {KH (y) J _ = (J ∗ (>∗y )(x) A∗ (x)) | A ∈ LX }. x∈X
(16) If HJ (HJ∗ (A)) = HJ∗ (A) for A ∈ LX , then
MHJ (A) ≤ B iff MJ (B) ≤ A. Moreover, τMJ = (τMHJ )∗ . Proof. (1) Since A =
V
x∈X (A
∗
∗ KHJ (KHJ ) = KH (A) J
^
J (A)(y) = such that
(A∗ (x) → J (>∗x )(y))
x∈X
τKHJ = (τKHJ )∗ = {KHJ (A)(y) ^ = (A(x) → J (>∗y )(x)) | A ∈ LX }. x∈X
(17) Define MHJ (A) = HJ (A∗ ). Then MHJ : LX → LX www.ijfis.org
(x) → >∗x ), by (J2) and (J3),
^
=
(J ∗ (>∗x )(y) → A(x)).
x∈X
V (2) Since B(y) ≤ J (A)(y) = x∈X (J ∗ (>∗x )(y) → A(x)) W iff y∈X (J ∗ (>∗x )(y) B(y)) ≤ A(x), we have HJ (B)(x) =
_
(J ∗ (>∗x )(y) B(y)).
y∈Y
The Properties of L-lower Approximation Operators
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International Journal of Fuzzy Logic and Intelligent Systems, vol. 14, no. 1, March 2014
(H1) Since HJ (A) ≤ HJ (A) iff A ≤ J (HJ (A)), we have A ≤ J (HJ (A)) ≤ HJ (A). (H2) a A ≤ J (HJ (a A))
(5) (H1) Since J (A∗ ) ≤ A∗ , Hs (A) = J (A∗ )∗ ≥ A. (H2) Hs (α A) = (J ((α A)∗ )∗ = (J (α → A∗ ))∗
iff A ≤ a → J (HJ (a A))
= (α → J (A∗ ))∗
= J (a → J (HJ (a A)))
= α J (A∗ )∗
iff HJ (A) ≤ a → HJ (a A) iff a HJ (A) ≤ HJ (a A).
(H3)
Hs (
_
= α Hs (A). _ Ai ) = (J ( Ai )∗ )∗
i∈Γ
A ≤ J (HJ (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).
i∈Γ
=(
HJ (Ai ) ≤ HJ (
i∈Γ
=
W Since J ( i∈Γ HJ (Ai )) ≥ J (HJ (Ai )) ≥ Ai , then W W J ( i∈Γ HJ (Ai )) ≥ i∈Γ Ai . Thus _
HJ (Ai ) ≥ HJ (
i∈Γ
_
J (A∗i ))∗
_
(J (A∗i ))∗
i∈Γ
_
Hs (Ai ).
i∈Γ
Ai ).
i∈Γ
^
i∈Γ
= _
A∗i ))∗
i∈Γ
(H3) By the definition of HJ , since HJ (A) ≤ HJ (B) for B ≤ A, we have _
^
= (J (
Hence Hs is an L-upper approximation operator such that Hs (B)(x) = (J (B ∗ )(x))∗ =
_
(J ∗ (>∗y )(x) B(y)).
y∈X
Ai ).
i∈Γ
Thus HJ : LX → LX is an L-upper approximation operator. By the definition of HJ , we have
Moreover, τHs = (τJ )∗ from: A = Hs (A) iff A = J (A∗ )∗ iff A∗ = J (A∗ ). (6) Let J (J (A)) = J (A) for A ∈ LX . Then
HJ (B) ≤ A iff B ≤ J (A). Hs (Hs (A)) Since A ≤ J (A) iff HJ (A) ≤ A, we have τHJ = τJ . (3) Let J (J (A)) = J (A) for A ∈ LX . 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} = HJ (A).
(4) Let J ∗ (A) ∈ τJ . Since J (J ∗ (A)) = J ∗ (A), J (J (A)) = J (J ∗ (J ∗ (A))) = (J (J ∗ (A)))∗ = J (A).
= J ∗ (Hs∗ (A)) = (J (J (A∗ )))∗ = J ∗ (A∗ ) = Hs (A).
Hence τHs = {Hs (A) = LX }.
Hence J (A) ∈ τJ ; i.e. J (A) ∈ (τJ )∗ . Thus, τJ ⊂ (τJ )∗ . Let A ∈ (τJ )∗ . Then A∗ = J (A∗ ). Since J (A) = J (J ∗ (A∗ )) = J ∗ (A∗ ) = A, then A ∈ τJ . Thus, (τJ )∗ ⊂ τJ . 61 | Yong Chan Kim
y∈X (J
∗
(>∗y ) A(y)) | A ∈
(7) Let J (J ∗ (A)) = J ∗ (A) for A ∈ LX . Then Hs (Hs∗ (A))
= J ∗ (Hs (A)) = (J (J ∗ (A∗ )))∗ = (J ∗ (A∗ ))∗ = Hs∗ (A).
Hence τHs = {Hs∗ (A) = A ∈ LX }. Hs (Hs (A))
∗
W
V
y∈X (A(y)
→ J (>∗y )) |
= Hs (Hs∗ (Hs∗ (A))) = Hs∗ (Hs∗ (A)) = Hs (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 ∈ LX , then KJ (KJ∗ (A)) =
http://dx.doi.org/10.5391/IJFIS.2014.14.1.57
KJ (A)
(21) (MJ , MHJ ) is a dual Galois connection;i.e,
KJ (KJ∗ (A))
= KJ (J ∗ (A∗ )) = J (J (A∗ )) = J (A∗ ) = KJ (A).
MHJ (A) ≤ B iff HJ (A∗ ) ≤ B iff A∗ ≤ J (B) iff MJ (B) = (J (B))∗ ≤ A.
(10) If J (J ∗ (A)) = J ∗ (A) for A ∈ LX , then KJ (KJ (A)) = KJ∗ (A) KJ (KJ (A))
= J (KJ∗ (A)) = J (J ∗ (A∗ )) = J ∗ (A∗ ) = KJ∗ (A).
Since KJ (KJ (A)) = KJ∗ (A), KJ (KJ∗ (A))
= KJ (KJ (KJ (A))) = KJ∗ (KJ (A)) = KJ (A).
Hence τKJ = {KJ (A) | A ∈ LX } = (τKJ )∗ . (11) , (12), (13) and (14) are similarly proved as (5), (9), (10) and (5), respectively.
Since MHJ (A∗ ) ≤ A iff MJ (A) ≤ A∗ , τMJ = (τMHJ )∗ . Let R ∈ LX×X be an L-fuzzy relation. Define operators as follows W HR (A)(y) = x∈X (A(x) R(x, y)), V JR (A)(y) = x∈X (R(x, y) → A(x)), V KR (A)(y) = x∈X (A(x) → R(x, y)) W MR (A)(y) = x∈X (A∗ (x) R(x, y)). Example 2.2. Let R be a reflexive L-fuzzy relation. Define JR : LX → LX as follows:
(15) If J (J (A)) = J (A) for A ∈ L , then HJ (HJ (A)) = ∗ HJ (A). Thus, KHJ (KH (A)) = KHJ (A) J ∗ KHJ (KH (A)) = KHJ (HJ (A)) J = (HJ (HJ (A)))∗ = (HJ (A))∗ = KHJ (A).
Since J (A) = A iff HJ (A) = A iff KHJ (A) = A∗ , τKHJ = (τJ )∗ with ∗ τKHJ = {KH (A)(y) J _ = (J ∗ (>∗y )(x) A(x)) | A ∈ LX }. x∈X
(16) If HJ (HJ∗ (A)) = HJ∗ (A) for A ∈ LX , then
^
JR (A)(y) =
X
(1) (J1) JR (A)(y) ≤ R(y, y) → A(y) = A(y). JR satisfies the conditions (J1) and (J2) from: V = x∈X (R(x, y) → (a → A)(x)) V = a → x∈X (R(x, y) → A(x)), V V V JR ( i∈Γ Ai )(y) = x∈X (R(x, y) → i∈Γ Ai (x)) V V = i∈Γ x∈X (R(x, y) → Ai (x)). JR (a → A)(y)
Hence JR is an L-lower approximation operator. W (2) Define HJR (B) = {A | B ≤ JR (A)}. Since B(y) ≤ JR (A)(y) iff B(y) ≤
KHJ (KHJ (A))
= =
(17) , (18) and (19) are similarly proved as (14), (15) and (16), respectively. (20) (KHJ , KJ ) is a Galois connection;i.e, A ≤ KHJ (B) iff A ≤ (HJ (B))∗ iff HJ (B) ≤ A∗ iff B ≤ J (A∗ ) = KJ (A) Moreover, since A∗ ≤ KJ (A) iff A ≤ KHJ (A∗ ), τKJ = (τKHJ )∗ . www.ijfis.org
^
(R(x, y) → A(x))
x∈X
∗ (A) KHJ (KHJ (A)) = KH J
KHJ (KJ∗ (A)) = HJ∗ (HJ∗ (A)) ∗ HJ (A) = KH (A). J
(R(x, y) → A(x)).
x∈X
iff
_
(B(y) R(x, y)) ≤ A(x),
y∈X
then HJR (B)(x) =
_
(R(x, y) B(y)) = HR−1 (B)(x).
y∈X
By Theorem 2.1(2), HJR = HR−1 is an L-upper approximation operator such that (HJR , JR ) is a residuated connection;i.e., HJR (A) ≤ B iff A ≤ JR (B). Moreover, τHJR = τJR . The Properties of L-lower Approximation Operators
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International Journal of Fuzzy Logic and Intelligent Systems, vol. 14, no. 1, March 2014
(3) If R is an L-fuzzy preorder, then R−1 is an L-fuzzy preorder. Since
HR is an L-upper approximation operator such that ^
Hs (A)(y) = ( ^
JR (JR (A))(z) =
(R(y, z) → JR (A)(y))
x∈X
=
y∈X
^
=
(R(y, z) →
y∈X
(R(y, z) R(x, y) → A(x)))
x∈X y∈X
^ _ = ( (R(y, z) R(x, y)) → A(x)) x∈X y∈X
^
=
(R(x, z) → A(x))
x∈X
_
(R(−, x) A(x)) | A ∈ LX }
x∈X
= τHJR = τHR−1 , ^
τJR = {JR (A) =
(R(x, −) → A(x)) | A ∈ LX }.
x∈X
^
(R(y, z) → (R(y, z) →
_
y∈X
_
≥
R(x, y) A∗ (x)))
x∈X
R(x, z) A∗ (x))).
x∈X
Thus,
JR (JR∗ (A))
=
By Theorem 2.1(4), JR (JR (A)) = JR (A) for A ∈ LX . Thus, τJR = (τJR )∗ with ( τJR =
JR∗ (A) =
= {Hs∗ (A) ^ = (A(y) → R∗ (y, −)) y∈X
(8) Define KJR (A) = JR (A∗ ). Then KJR : LX → LX with ^
(R(x, y) → A∗ (x)) = KR∗ (y)
is an L-join meet approximation operator. Moreover, τKJR = (τJR )∗ . (9) R is an L-fuzzy preorder, then JR (JR (A)) = JR (A) for A ∈ LX . By Theorem 2.1(9), KJR (KJ∗R (A)) = KJR (A) for A ∈ LX such that τKJR = (τJR )∗ with τKJR = {KJ∗R (A) _ = (R(x, −) A(x))
JR∗ (A).
_
τHs = (τHs )∗
x∈X
JR∗ (A)(y))
y∈X
=
(R(y, −) A(y)) | A ∈ LX }.
(7) If R is a reflexive and Euclidean L-fuzzy relation, then JR (JR∗ (A)) = JR∗ (A) for A ∈ LX . By Theorem 2.1(7), Hs (Hs∗ (A)) = Hs∗ (A) such that
KJR (A)(y) =
JR (JR∗ (A))(z) =
_
= KR∗ (A) | A ∈ LX }.
(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),
^
(6) If R is an L-fuzzy preorder, then JR (JR (A)) = JR (A) for A ∈ LX . By Theorem 2.1(6), then Hs (Hs (A)) = Hs (A) for A ∈ LX such that τHs = (τJR )∗ = (τHJR )∗ with
y∈X
By Theorem 2.1(3), HJR (HJR (A)) = HJR (A). By Theorem 2.1(3), τHJR = τJR with {HR−1 (A) =
(R(x, y) A(x)).
Moreover, τHs = τHR = (τHJR )∗ .
τHs = {Hs (A) =
= JR (A)(z),
_ x∈X
(R(x, y) → A(x)))
x∈X
^ ^
=
^
R(x, y) → A∗ (x))∗
(R(x, −) A∗ (x)) = MR (A)
x∈X
x∈X
= HR (A) | A ∈ LX }. (10) If R is a reflexive and Euclidean L-fuzzy relation, then JR (JR∗ (A)) = JR∗ (A) for A ∈ LX . By Theorem 2.1(10), KJR (KJR (A)) = KJ∗R (A) such that
) | A∈L
X
.
(5) Define Hs (A) = JR (A∗ )∗ . By Theorem 2.1(5), Hs = 63 | Yong Chan Kim
V {KJR (A) = x∈X (A(x) → R∗ (x, −) | A ∈ LX } = τKJR = (τKJR )∗ . (11) Define MJR (A) = (JR (A))∗ . Then MJR : LX → LX
http://dx.doi.org/10.5391/IJFIS.2014.14.1.57
Thus,
with MJR (A)(y) =
_
(A∗ (x) R(x, y)) = MR (A)(y)
A(x) R−1 (x, z) ≤ (A(x) → R−1∗ (x, y)) → R−1 (x, y) R−1∗ (x, z) ≤ R−1∗ (y, z).
x∈X
is an L-join meet approximation operator. Moreover, τMJR = τJR . (12) If R is an L-fuzzy preorder, then JR (JR (A)) = JR (A) for A ∈ LX . By Theorem 2.1(12), MJR (M∗JR (A)) = MJR (A) for A ∈ LX such that τMJR = τJR with τMJR = {M∗JR (A) = = JR (A) | A ∈ LX }.
V
x∈X (R(x, −) → A(x))
Hence KR−1∗ (KR−1∗ (A))(z) ^ = (KR−1∗ (A)(y) → R−1∗ (y, z)) y∈X
=
y∈X x∈X
≤ (13) If R is a reflexive and Euclidean L-fuzzy relation, then JR (JR∗ (A)) = JR∗ (A) for A ∈ LX . By Theorem 2.1(13), MJR (MJR (A)) = M∗JR (A) such that W
τMJR = {MJR (A) = x∈X (A(x) R(x, −)) = HJR (A) | A ∈ LX } = (τMJR )∗ . (14) Define KHJR (A) = (HJR (A))∗ . Then
^ ^ ( (A(x) → R−1∗ (x, y)) → R−1∗ (y, z)) _
(A(x) R−1∗ (x, z)) = KR−1∗ (A)(z)
x∈X ∗ By (K1), KR−1∗ (KR−1∗ (A)) = KR −1∗ (A) such that
{KR−1∗ (A) =
= τKR−1 = (τKR−1 )∗ . (17) Define MHJR (A) = HJR (A∗ ). Then MHJR : LX → LX
with ^
is an L-meet join approximation operator as follows: _ MHJR (A)(y) = (R(y, x) A∗ (x))
(A(x) → R∗ (y, x))
x∈X
x∈X
= KR−1∗ (A)(y)
= MR−1 (A)(y).
is an L-join meet approximation operator. Moreover, τKR−1 = τJR = τHR−1 . (15) If R is an L-fuzzy preorder, then JR (JR (A)) = JR (A) ∗ for A ∈ LX . By Theorem 2.1(15), KR−1 (KR −1 (A)) = X KR−1 (A) for A ∈ L such that τKR−1 = τJR = τHR−1 with ∗ {KR −1∗ (A)(y) = X
τKR−1∗ = = HR−1 (A)(y) | A ∈ L }. (16) Let R Since
−1
(A(x) → R∗ (−, x)) | A ∈ LX }
x∈X
KHJR : LX → LX KHJR (A)(y) =
^
W
x∈X (R(y, x)
A(x))
Moreover, τMHJ = (τJR )∗ . R (18) If R is an L-fuzzy preorder, then JR (JR (A)) = JR (A) for A ∈ LX . By Theorem 2.1(18), MHJR (M∗HJ (A)) = MHJR (A) R
for A ∈ LX such that τMHJ = (τJ )∗ with R
( τMHJ = M∗HJ (A)(y) = R
be a reflexive and Euclidean L-fuzzy relation.
^
R
(R(y, x) → A(x))
x∈X
) X
= JR−1 (A)(y) | A ∈ L
.
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
(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),
(A(x) → R−1∗ (x, y)) A(x) R−1 (x, z) ≤ R−1 (x, y) R−1∗ (x, z) ≤ R−1∗ (y, z). www.ijfis.org
then (R(y, x) → A(x)) R(z, y) ≤ R(z, x) → A(x). The Properties of L-lower Approximation Operators
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International Journal of Fuzzy Logic and Intelligent Systems, vol. 14, no. 1, March 2014
Thus, MR−1 (MR−1 (A))(z) _ = (MR−1 (A)(y) R(z, y)) y∈X
_ ^ = ( (R(y, x) → A(x)) R(z, y)) y∈X x∈X
≤
^
(R(z, x) → A(x)) = MR−1 (A)(z).
x∈X
By (M1), MR−1 (MR−1 (A)) = M∗R−1 (A) such that W {MR−1 (A) = x∈X (A∗ (x) R(−, x)) | A ∈ LX } = τMR−1 = (τMR−1 )∗ . (20) (KHJR = KR−1∗ , KJR = KR∗ ) is a Galois connection;i.e, A ≤ KHJR (B) iff B ≤ KJR (A). Moreover, τKJR = (τKHJ )∗ . R (21) (MJR = MR , MHJR = MR−1 ) is a dual Galois connection;i.e, MHJR (A) ≤ B iff MJR (B) ≤ A. Moreover, τMJR = (τMHJ )∗ . 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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65 | Yong Chan Kim
[2] Z. Pawlak, “Rough probability,” Bulletin of Polish Academy of Sciences: Mathematics, vol. 32, no. 9-10, pp. 607-615, 1984. [3] P. H´ajek,Metamathematics of Fuzzy Logic, Dordrecht, The Netherlands: Kluwer, 1998. [4] A. M. Radzikowska and E. E. Kerre, “A comparative study of fuzzy rough sets,” Fuzzy Sets and Systems, vol. 126, no. 2, pp. 137-155, Mar. 2002. http://dx.doi.org/10.1016/ S0165-0114(01)00032-X [5] R. Bˇelohl´avek, Fuzzy Relational Systems: Foundations and Principles, New York, NY: Kluwer Academic/Plenum Publishers, 2002. [6] H. Lai and D. Zhang, “Fuzzy preorder and fuzzy topology,” Fuzzy Sets and Systems, vol. 157, no. 14, pp. 1865-1885, Jul. 2006. http://dx.doi.org/10.1016/j.fss.2006.02.013 [7] H. Lai and D. Zhang, “Concept lattices of fuzzy contexts: formal concept analysis vs. rough set theory,” International Journal of Approximate Reasoning, vol. 50, no. 5, pp. 695-707, May. 2009. http://dx.doi.org/10.1016/j.ijar. 2008.12.002 [8] Y. C. Kim, “Alexandrov L-topologiesand L-join meet approximatin operators,” International Journal of Pure and Applied Mathematics, vol. 91, no. 1, pp. 113-129, 2014. http://dx.doi.org/10.12732/ijpam.v91i1.12 [9] Y. C. Kim, “Alexandrov L-topologies,” International Journal of Pure and Applied Mathematics, 2014 [in press]. [10] Y. H. She and G. J. Wang, “An axiomatic approach of fuzzy rough sets based on residuated lattices,” Computers & Mathematics with Applications, vol. 58, no. 1, pp. 189201, Jul. 2009. http://dx.doi.org/10.1016/j.camwa.2009. 03.100 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 Professor of Gangneung-Wonju University, his research interests is a fuzzy topology and fuzzy logic. E-mail: [email protected]
