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The Properties of L-lower Approximation Operators
The Properties of L-lower Approximation Operators
International Journal of Fuzzy Logic and Intelligent Systems. 2014. Mar, 14(1): 57-65
Copyright © 2014, Korean Institute of Intelligent Systems
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.
  • Received : December 10, 2013
  • Accepted : March 19, 2014
  • Published : March 25, 2014
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Yong Chan Kim

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
1. Introduction
Pawlak [1 , 2] introduced rough set theory as a formal tool to deal with imprecision and uncertainty in data analysis. Hájek [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. Bělohlávek [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
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(2) The unit interval with a left-continuous t-norm ⊙,
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is a complete residuated lattice defined by
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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 ,
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and
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Lemma 1.3. [3 , 5] For each x, y, z, xi, yi L , we have the following properties.
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Definition 1.4. [8 , 9]
(1) A map H : LX LX is called an L-upper approximation operator iff it satisfies the following conditions
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(2) A map 𝒥 : LX LX is called an L-lower approximation operator iff it satisfies the following conditions
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(3) A map K : LX LX is called an L-join meet approximation operator iff it satisfies the following conditions
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(4) A map M : LX LX is called an L-meet join approximation operator iff it satisfies the following conditions
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Definition 1.5. [6 , 9] A subset 𝜏 ⊂ LX is called an Alexandrov L-topology if it satisfies:
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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 𝜏 H = { A LX | H ( A ) = A } is an Alexandrov topology on X .
(3) If 𝒥 is an L -lower approximation operator, then 𝜏 𝒥 = { A LX | 𝒥 ( A ) = A } is an Alexandrov topology on X .
(4) If K is an L -join meet approximation operator, then 𝜏 K = { A LX | K ( A ) = A *} is an Alexandrov topology on X .
(5) If M is an L -meet join operator, then 𝜏 M = { 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:
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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 ofL-lower Approximation Operators
Theorem 2.1. Let 𝒥 : LX LX be an L -lower approximation operator. Then the following properties hold.
(1) For A LX ,
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.
(2) Define HJ ( B ) = ∧{ A | B ≤ 𝒥 ( A )}. Then HJ : LX LX with
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is an L -upper approximation operator such that ( HJ ,𝒥 )
is a residuated connection;i.e.,
HJ (B) ≤ A iff B ≤ 𝒥 (A).
Moreover, 𝜏 𝒥 = 𝜏 HJ .
(3) If 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A LX , then HJ ( HJ ( A )) = HJ ( A ) for A LX such that 𝜏 𝒥 = 𝜏 HJ with
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(4) If 𝒥 (𝒥 *( A )) = 𝒥 *( A ) for A LX , then 𝒥 (𝒥 ( A )) = 𝒥 ( A ) such that
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(5) Define Hs ( A ) = 𝒥 ( A *)*. Then Hs : LX LX with
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is an L -upper approximation operator. Moreover, 𝜏 Hs = (𝜏 𝒥 ) * = (𝜏 HJ ) * .
(6) If 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A LX , then
Hs(Hs(A)) = Hs(A)
for A LX such that 𝜏 Hs = (𝜏 𝒥 ) * = (𝜏 HJ ) * . with
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(7) If 𝒥 (𝒥 *( A )) = 𝒥 *( A ) for A LX , then
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such that
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(8) Define KJ ( A ) = 𝒥 ( A *). Then KJ : LX LX with
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is an L -join meet approximation operator.
(9) If 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A LX , then
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for A LX such that 𝜏 KJ = (𝜏 𝒥 ) * with
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(10) If 𝒥 (𝒥 *( A )) = 𝒥 *( A ) for A LX , then
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such that
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(11) Define MJ ( A ) = (𝒥 ( A ))*. Then MJ : LX LX with
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is an L -meet join approximation operator. Moreover, 𝜏 MJ = 𝜏 𝒥 .
(12) If 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A LX , then
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for A LX such that 𝜏 MJ = (𝜏 𝒥 ) * with
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(13) If 𝒥 (𝒥 *( A )) = 𝒥 *( A ) for A LX , then
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such that
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(14) Define KHJ ( A ) = ( HJ ( A ))*. Then KHJ : LX LX with
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is an L -meet join approximation operator. Moreover, 𝜏 KHJ = 𝜏 𝒥 .
(15) If 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A LX , then
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for A LX such that 𝜏 KHJ = (𝜏 𝒥 ) * with
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(16) If
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for A LX , then
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such that
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(17) Define MHJ ( A ) = HJ ( A *). Then MHJ : LX LX
with
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is an L -join meet approximation operator. Moreover, 𝜏 MHJ = (𝜏 𝒥 ) * .
(18) If 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A LX , then
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for A LX such that 𝜏 MHJ = (𝜏 𝒥 ) * with
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(19) If
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for A LX , then
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such that
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(20) ( KHJ , KJ ) is a Galois connection;i.e,
AKHJ (B) iff BKJ (A).
Moreover, 𝜏 KJ = (𝜏 KHJ ) * .
(21) ( MJ , MHJ ) is a dual Galois connection;i.e,
MHJ (A) ≤ B iff MJ (B) ≤ A.
Moreover, 𝜏 MJ = (𝜏 MHJ ) * .
Proof.
(1) Since
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, by (J2) and (J3),
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(2) Since
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iff
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, we have
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(H1) Since HJ ( A ) ≤ HJ ( A ) iff A ≤ 𝒥 ( HJ ( A )), we have A ≤ 𝒥 ( HJ ( A )) ≤ HJ ( A ).
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(H3) By the definition of HJ , since HJ ( A ) ≤ HJ ( B ) for B A , we have
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Since 𝒥 (∨ i∈𝚪 HJ ( Ai )) ≥ 𝒥 ( HJ ( Ai )) ≥ Ai , then
𝒥 (∨ i∈𝚪 HJ ( Ai )) ≥ ∨ i∈𝚪 Ai . Thus
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Thus HJ : LX LX 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 𝜏 HJ = 𝜏 𝒥 .
(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
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(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, 𝜏 𝒥 ⊂ (𝜏 𝒥 ) * .
Let A ∈ (𝜏 𝒥 ) * . Then A * = 𝒥 ( A *). Since 𝒥 ( A ) = 𝒥 (𝒥 *( A *)) = 𝒥 *( A *) = A , then A ∈ 𝜏 𝒥 . Thus, (𝜏 𝒥 ) * ⊂ 𝜏 𝒥 .
(5) (H1) Since 𝒥 ( A *) ≤ A *, Hs ( A ) = 𝒥 ( A *)* ≥ A .
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Hence Hs is an L -upper approximation operator such that
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Moreover, 𝜏 Hs = (𝜏 𝒥 ) * from:
A = Hs(A) iff A = 𝒥 (A*)* iff A* = 𝒥 (A*).
(6) Let 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A LX . Then
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Hence
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(7) Let 𝒥 (𝒥 *( A )) = 𝒥 *( A ) for A LX . Then
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Hence
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By a similar method in (4), 𝜏 Hs = (𝜏 Hs ) * .
(8) It is similarly proved as (5).
(9) If 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A LX , then
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(10) If 𝒥 (𝒥 *( A )) = 𝒥 *( A ) for A LX , then
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Since
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,
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Hence 𝜏 KJ = { KJ ( A ) | A LX } = (𝜏 KJ ) * .
(11) , (12), (13) and (14) are similarly proved as (5), (9), (10) and (5), respectively.
(15) If 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A LX , then HJ ( HJ ( A )) = HJ ( A ). Thus,
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Since 𝒥 ( A ) = A iff HJ ( A ) = A iff KHJ ( A ) = A *, 𝜏 KHJ = (𝜏 𝒥 ) * with
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(16) If
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for A ∈ LX , then
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(17) , (18) and (19) are similarly proved as (14), (15) and (16), respectively.
(20) ( KHJ , KJ ) is a Galois connection;i.e,
AKHJ (B) iff A ≤ (HJ (B))*
iff HJ (B) ≤ A* iff B ≤ 𝒥 (A*) = KJ (A)
Moreover, since A * ≤ KJ ( A ) iff A KHJ ( A *), 𝜏 KJ = (𝜏 KHJ ) * .
(21) ( MJ , MHJ ) is a dual Galois connection;i.e,
MHJ (A) ≤ B iff HJ (A*) ≤ B
iff A* ≤ 𝒥 (B) iff MJ (B) = (𝒥 (B))* ≤ A.
Since MHJ ( A *) ≤ A iff MJ ( A ) ≤ A *, 𝜏 MJ = (𝜏 MHJ ) * .
Let R L X × X be an L -fuzzy relation. Define operators as follows
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Example 2.2. Let R be a reflexive L -fuzzy relation. Define 𝒥 R : LX LX as follows:
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(1) (J1) 𝒥 R ( A )( y ) ≤ R ( y, y ) → A ( y ) = A ( y ): 𝒥 R satisfies the conditions (J1) and (J2) from:
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Hence 𝒥 R is an L -lower approximation operator.
(2) Define HJR ( B ) = ∨ { A | B ≤ 𝒥 R ( A )}. Since
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then
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By Theorem 2.1(2), HJR = H R-1 is an L -upper approximation operator such that ( HJR ,𝒥 R ) is a residuated connection;i.e.,
HJR(A) ≤ B iff A ≤ 𝒥R(B).
Moreover, 𝜏 HJR = 𝜏 𝒥R .
(3) If R is an L -fuzzy preorder, then R -1 is an L -fuzzy preorder. Since
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By Theorem 2.1(3), HJR ( HJR ( A )) = HJR ( A ): By Theorem 2.1(3), 𝜏 HJR = 𝜏 𝒥R with
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(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 ),
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Thus,
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.
By Theorem 2.1(4), 𝒥 R (𝒥 R ( A )) = 𝒥 R ( A ) for A LX .
Thus, 𝜏 𝒥R = (𝜏 𝒥R ) * with
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(5) Define Hs ( A ) = 𝒥 R ( A *)*. By Theorem 2.1(5), Hs = HR is an L -upper approximation operator such that
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Moreover, 𝜏 Hs = 𝜏 HR = (𝜏 HJR ) * .
(6) If R is an L -fuzzy preorder, then 𝒥 R (𝒥 R ( A )) = 𝒥 R ( A ) for A LX . By Theorem 2.1(6), then Hs ( Hs ( A )) = Hs ( A ) for A LX such that 𝜏 Hs = (𝜏 𝒥R ) * = (𝜏 HJR ) * with
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(7) If R is a reflexive and Euclidean L -fuzzy relation, then
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for A LX . By Theorem 2.1(7),
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such that
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(8) Define KJR ( A ) = 𝒥 R ( A *). Then KJR : LX LX with
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is an L -join meet approximation operator. Moreover, 𝜏 KJR = (𝜏 𝒥R ) * .
(9) R is an L -fuzzy preorder, then 𝒥 R (𝒥 R ( A )) = 𝒥 R ( A ) for A LX . By Theorem 2.1(9),
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for A LX such that 𝜏 KJR = (𝜏 𝒥R ) * with
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(10) If R is a reflexive and Euclidean L -fuzzy relation, then
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for A LX . By Theorem 2.1(10),
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such that
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(11) Define MJR ( A ) = (𝒥 R ( A ))*. Then MJR : LX LX with
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is an L -join meet approximation operator. Moreover, 𝜏 MJR = 𝜏 𝒥R .
(12) If R is an L -fuzzy preorder, then 𝒥 R (𝒥 R ( A )) = 𝒥 R ( A ) for A LX . By Theorem 2.1(12),
PPT Slide
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for A LX such that 𝜏 MJR = 𝜏 𝒥R with
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(13) If R is a reflexive and Euclidean L -fuzzy relation, then
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for A LX . By Theorem 2.1(13),
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such that
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(14) Define KHJR ( A ) = ( HJR ( A ))*. Then
KHJR : LXLX
with
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is an L -join meet approximation operator. Moreover, 𝜏 KR-1 = 𝜏 𝒥R = 𝜏 HR-1 .
(15) If R is an L -fuzzy preorder, then 𝒥 R (𝒥 R ( A )) = 𝒥 R ( A ) for A LX . By Theorem 2.1(15),
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for A LX such that 𝜏 KR-1 = 𝜏 𝒥R = 𝜏 HR-1 with
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(16) Let R -1 be a reflexive and Euclidean L -fuzzy relation. Since
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we have
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Thus,
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Hence
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By (K1),
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such that
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(17) Define MHJR ( A ) = HJR ( A *). Then
MHJR : LXLX
is an L -meet join approximation operator as follows:
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Moreover, 𝜏 MHJR = (𝜏 𝒥R ) * .
(18) If R is an L -fuzzy preorder, then 𝒥 R (𝒥 R ( A )) = 𝒥 R ( A ) for A LX . By Theorem 2.1(18),
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for A LX such that 𝜏 MHJR = (𝜏 𝒥 ) * with
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(19) Let R -1 be a reflexive and Euclidean L -fuzzy relation.
Since
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then ( R ( y, x ) → A ( x )) ⊙ R ( z, y ) ≤ R ( z, x ) → A ( x ).
Thus,
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By (M1),
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such that
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(20) ( KHJR = K R-1* , KJR = K R* ) is a Galois connection; i.e, A KHJR ( B ) iff B KJR ( A ): Moreover, 𝜏 KJR = (𝜏 KHJR ) * .
(21) ( MJR = MR , MHJR = M R-1 ) is a dual Galois connection; i.e, MHJR ( A ) ≤ B iff MJR ( B ) ≤ A . Moreover, 𝜏 MJR = (𝜏 MHJR ) * .
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.
BIO
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: yck@gwnu.ac.kr
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