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FUZZY RELATIONS AND ALEXANDROV L-TOPOLOGIES
FUZZY RELATIONS AND ALEXANDROV L-TOPOLOGIES
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2014. Nov, 21(4): 247-256
Copyright © 2014, Korean Society of Mathematical Education
  • Received : June 16, 2014
  • Accepted : September 18, 2014
  • Published : November 30, 2014
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About the Authors
JUNG MI KO
DEPARTMENT OF MATHEMATICS, GANGNEUNG-WONJU NATIONAL, GANGNEUNG 210-702, KOREAEmail address:jmko@gwnu.ac.kr
YONG CHAN KIM
DEPARTMENT OF MATHEMATICS, GANGNEUNG-WONJU NATIONAL, GANGNEUNG 210-702, KOREAEmail address:yck@gwnu.ac.kr

Abstract
In this paper, we investigate the relationships between fuzzy relations and Alexandrov L -topologies in complete residuated lattices. Moreover, we give their examples.
Keywords
1. INTRODUCTION
Pawlak [9 , 10] introduced rough set theory as a formal tool to deal with impre- cision and uncertainty in data analysis. Hájek [3] introduced a complete residuated lattice which is an algebraic structure for many valued logic. Radzikowska [11] de- veloped fuzzy rough sets in complete residuated lattice. Bělohlávek [1] investigated information systems and decision rules in complete residuated lattices. Lai [7 , 8] in- troduced Alexandrov L -topologies induced by fuzzy rough sets. Algebraic structures of fuzzy rough sets are developed in many directions [1 - 13] .
In this paper, we investigate the relationships between fuzzy relations and Alexan- drov L -topologies in complete residuated lattices. Moreover, we give their examples.
2. PRELIMINARIES
- Definition 2.1([1,3]). An algebra (L,∧,∨,⊙, →⊥,⊤) is called acomplete residuated latticeif it satisfies the following conditions:
  • (L1)L= (L, ≤ , ∨,∧,⊥,⊤) is a complete lattice with the greatest element ⊤ and the least element ⊥;
  • (L2) (L,⊙,⊤) is a commutative monoid;
  • (L3)x⊙y≤ziffx≤y→zforx, y, z∈L.
In this paper, we assume ( L ,∧,∨,⊙, →, *⊥,⊤) is a complete residuated lattice with the law of negation;i.e. x** = x . For α L , A ,⊤ x LX , ( α A )( x ) = α A ( x ), ( α A )( x ) = α A ( x ) and ⊤ x ( x ) = ⊤,⊤ x ( x ) = ⊥, otherwise.
- Definition 2.2([1,7]). LetXbe a set. A functionR : X×X → Lis called afuzzy relation. A fuzzy relationRis called afuzzy preorderif satisfies (R1) and (R2).
  • (R1)reflexiveifR(x, x) = ⊤ for allx∈X,
  • (R2)transitiveifR(x, y) ⊙R(y, z) ≤R(x, z), for allx, y, z∈X.
  • We denote
- Lemma 2.3([1,3]).Let(L,∨,∧,⊙, →, *⊥,⊤)be a complete residuated lattice with a negation *. For each x,y,z,xi,yi∈L,the following properties hold.
  • (1)If y ≤ z, then x⊙y ≤ x⊙z.
  • (2)If y ≤ z, then x → y ≤ x → z and z → x ≤ y → x.
  • (3)and
  • (4)and
  • (5) (x → y) ⊙x ≤ y and(y→ z) ⊙ (x →y) ≤ (x → z).
  • (6) (x⊙y) →z = x→ (y → z) =y→ (x → z)and(x⊙y)* =x→y*.
  • (7)x* →y* =y→x and(x→y)* =x⊙y*.
- Definition 2.4([5-7]). A subsetτ⊂LXis called anAlexandrov topologyif it satisfies satisfies the following conditions.
  • (T1) ⊥X, ⊤X∈τwhere ⊤X(x) = ⊤ and ⊥X(x) = ⊥ forx∈X
  • (T2) If
  • (T3)α⊙A∈τfor allα∈LandA∈τ.
  • (T4)α → A∈τfor allα∈LandA∈τ.
- Definition 2.5([7]). LetR∈LX×Xbe a fuzzy relation. A setA∈LXis calledextensionalifA(x) ⊙R(x, y) ≤A(y) for allx,y∈X.
3. FUZZY RELATIONS AND ALEXANDROVL-TOPOLOGIES
- Theorem 3.1.Let R∈LX×Xand R−1∈LX×Xwith R−1(x, y) =R(x, y).
  • (1)τ is an Alexandrov topology on X iff τ*= {A*∈LX|A∈τ}is an Alexandrov topology on X.
  • (2)τR= {A∈LX|A(x) ⊙R(x, y) ≤A(y),x,y∈X}is an Alexandrov topology on X. Moreover,
  • (3)Ifis the smallest fuzzy preorder such that R≤then
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where Rr ( x, y ) = Δ∨ R ( x, y ) and Δ ( x, y ) = ⊤ if x = y and Δ( x, y ) = ⊥ if x y .
Moreover,
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  • (4)wherefor each y∈X.
  • (5)wherefor each x∈X.
  • (6)wherefor each y∈X.
  • (7)wherefor each x∈X.
  • (8)Moreover,CτR(A) ∈τR.
  • (9)Moreover,IτR(A) ∈τR.
  • (10)A∈τRiff A = CτR(A) =IτR(A).
  • (11)CτR(A) = (IτR−1(A*))*for all A∈LX.
Proof. (1) Let A * ∈ τ * for A ∈ τ . Since α A * = ( α →A )* and α → A * = ( α A )*, τ * is an Alexandrov topology on X .
(2) (T1) Since ⊤ X ( x ) ⊙ R ( x, y ) ≤ ⊤ X ( y ) = ⊤ and ⊥ X ( x ) ⊙ R ( x, y ) = ⊥ = ⊥ X ( y ), Then⊥ X ,⊤ X τR .
(T2) For Ai τR for each i ∈ Γ, since
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Similarly,
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(T3) For A τR , α A τR .
(T4) For A τR , by Lemma 2.3(5), since α ⊙ ( α → A ( x )) ⊙ R ( x, y ) ≤ A ( x ) ⊙ R ( x, y ) ≤ A ( y ), ( α → A ( x )) ⊙ R ( x, y ) ≤ α → A ( y ). Then α → A τR . Moreover A τR iff A * ∈ τ R−1 from:
  • A(x) ⊙R(x, y) ≤A(y) iffR(x, y) →A* ≥A*(y)
  • iffA*(y) ⊙R(x, y) ≤A*(x) iffA*(y) ⊙R−1(y, x) ≤A*(x).
(3) Define
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Then R is a fuzzy preorder. Since B τR and B ( x ) ⊙ R ( x, y ) ≤ B ( y ), then R ( x, y ) ≤ B ( x ) → B ( y ). Hence R (x, y ) ≤ RτR . If P is a fuzzy preorder with R ≤ P , for Pw ( x ) = P ( w, x ), then Pw ( x ) ⊙ R ( x, y ) ≤ Pw ( x ) ⊙ P ( x, y ) ≤ Pw ( y ). Hence Pw τR . Thus RτR ( x, y ) =
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Thus,
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Since Rr ( x, y ) = Δ∨ R ( x , y ), we have ( Rr ) n ( x , x ) = ⊤ for each n N . So
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Since
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then
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Hence
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is a fuzzy preorder. If R ≤ P and P is fuzzy preorder, then Rr ≤ P and ( Rr ) n Pn = P , thus,
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Hence
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(4) Put
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and
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Since A τR ,
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Hence
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. Since
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Thus, A τR .
Let A τ . Since
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Thus, A τR .
Let A τ . Then
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Put A ( x ) = ax . Then
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Let
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Then
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Thus, D τ . Hence τR = τ = τ 1 .
(5) Put
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and
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Since A τR , RτR ( x, y ) → A ( y ) =
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A ( y ) ≥ ( A ( x ) → A ( y )) → A ( y ) ≥ A ( x ). Hence
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Since
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,
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Thus, A η .
Let A η . Since
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Thus, R ( x, y ) → A ( y ) ≥ A ( x ) iff A ( x ) ⊙ R ( x, y ) ≤ A ( y ). So, A τR .
Let A η Then
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Put A ( y ) = by . Then
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Let
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Then
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Thus, A η . Hence τR = η = η 1 .
(6) It follows from
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iff
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(7) It follows from
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iff
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(8) Put
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Then B τR from:
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If A ≤ E and E τR , then B ≤ E from:
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Hence CτR = B .
(9) Let
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from
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If E ≤ A and E τR , then E ≤ B from:
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Hence IτR = B .
(11)
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Theorem 3.2. Let RX and RY be fuzzy relations and f : X → Y a map with RX ( x , y ) ≤ RY ( f ( x ), f ( y )) for all x , y X. Then the following equivalent conditions hold.
  • (1)f−1(B) ∈τRXfor all B∈τRY.
  • (2)for all
  • (3)RτRX(x,y) ≤RτRY(f(x),f(y))for all x,y∈X.
  • (4)for all x,y∈X.
  • (5)f(CτRX(A)) ≤CτRY(f(A))for all A∈LX.
  • (6)for all A∈LX.
  • (7)CτRX(f−1(B)) ≤f−1(CτRX(B))for all B∈LY.
  • (8)for all B∈LY.
  • (9)f−1(IτRX(B)) ≤IτRY(f−1(B))for all B∈LY.
  • (10)for all B∈LY.
Proof. (1) For all B τRY , f −1 ( B ) ∈ τRX from:
  • f−1(B)(x) ⊙RX(x,y) ≤B(f(x)) ⊙RY(f(x),f(y)) ≤B(f(y)) =f−1(B)(y).
(1) ⇔ (2) It follows from (1) and Theorem 3.1(2).
(1) ⇒ (3)
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(1) ⇒ (5)
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(3) ⇒ (5)
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(5) ⇒ (7) By (5), put A = f −1 ( B ). Since f ( CτRX ( f −1 ( B ))) ≤ CτRY ( f ( f −1 ( B ))) ≤ CτRY ( B ), we have CτRX ( f −1 ( B )) ≤ f −1 ( CτRX ( B )).
(7) ⇒ (1) For all B τRY , CτY ( B ) = B . Since CτRX ( f −1 (B)) ≤ f −1 ( CτRX ( B )) = f −1 ( B ), f −1 ( B ) ∈ τRX .
(1) ⇒ (9)
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(9) ⇒ (1) For all B τRY , IτY ( B ) = B . Since IτRX ( f −1 ( B )) ≥ f −1 ( IτRX ( B )) = f −1 ( B ), f −1 ( B ) ∈ τRX .
Other cases are similarly proved.
Example 3.3. Let ( L = [0, 1],⊙,→,* ) be a complete residuated lattice with the law of double negation defined by
  • x⊙y= (x + y− 1) ∨ 0,x → y= (1 −x + y) ∧ 1,x* = 1 −x.
Let X = { a , b , c }, Y = { x , y , z } be sets and f : X → Y as follows:
  • f(a) =x,f(b) =y,f(c) =z.
(1) Define RX LX×X , RY LY ×Y as follows
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Then RX ( a , b ) ≤ RY ( f ( a ), f ( b )) for all a , b X .
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For n ≥ 2,
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and
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as follows:
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Then
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Moreover,
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Then RτRX ( a, b ) ≤ RτRX ( f ( a ), f ( b )) for all a , b X .
(2)
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where ai L and
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For
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For
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where bi L and
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For
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For
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(3)
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where ai L and
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where bi L and
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(4) For A = (0.2. 0.8, 0.6) ∈ LX ,
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