FUZZY RELATIONS AND ALEXANDROV L-TOPOLOGIES
FUZZY RELATIONS AND ALEXANDROV L-TOPOLOGIES
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
PDF
e-PUB
PPT
Export by style
Article
Author
Metrics
Cited by
TagCloud
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
PPT Slide
Lager Image
where Rr ( x, y ) = Δ∨ R ( x, y ) and Δ ( x, y ) = ⊤ if x = y and Δ( x, y ) = ⊥ if x y .
Moreover,
PPT Slide
Lager Image
• (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
PPT Slide
Lager Image
Similarly,
PPT Slide
Lager Image
(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
PPT Slide
Lager Image
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 ) =
PPT Slide
Lager Image
Thus,
PPT Slide
Lager Image
Since Rr ( x, y ) = Δ∨ R ( x , y ), we have ( Rr ) n ( x , x ) = ⊤ for each n N . So
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
then
PPT Slide
Lager Image
Hence
PPT Slide
Lager Image
is a fuzzy preorder. If R ≤ P and P is fuzzy preorder, then Rr ≤ P and ( Rr ) n Pn = P , thus,
PPT Slide
Lager Image
Hence
PPT Slide
Lager Image
PPT Slide
Lager Image
(4) Put
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Since A τR ,
PPT Slide
Lager Image
Hence
PPT Slide
Lager Image
. Since
PPT Slide
Lager Image
PPT Slide
Lager Image
Thus, A τR .
Let A τ . Since
PPT Slide
Lager Image
Thus, A τR .
Let A τ . Then
PPT Slide
Lager Image
Put A ( x ) = ax . Then
PPT Slide
Lager Image
Let
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
Thus, D τ . Hence τR = τ = τ 1 .
(5) Put
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Since A τR , RτR ( x, y ) → A ( y ) =
PPT Slide
Lager Image
A ( y ) ≥ ( A ( x ) → A ( y )) → A ( y ) ≥ A ( x ). Hence
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
,
PPT Slide
Lager Image
Thus, A η .
Let A η . Since
PPT Slide
Lager Image
Thus, R ( x, y ) → A ( y ) ≥ A ( x ) iff A ( x ) ⊙ R ( x, y ) ≤ A ( y ). So, A τR .
Let A η Then
PPT Slide
Lager Image
Put A ( y ) = by . Then
PPT Slide
Lager Image
Let
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
Thus, A η . Hence τR = η = η 1 .
(6) It follows from
PPT Slide
Lager Image
iff
PPT Slide
Lager Image
(7) It follows from
PPT Slide
Lager Image
iff
PPT Slide
Lager Image
(8) Put
PPT Slide
Lager Image
Then B τR from:
PPT Slide
Lager Image
If A ≤ E and E τR , then B ≤ E from:
PPT Slide
Lager Image
Hence CτR = B .
(9) Let
PPT Slide
Lager Image
from
PPT Slide
Lager Image
If E ≤ A and E τR , then E ≤ B from:
PPT Slide
Lager Image
Hence IτR = B .
(11)
PPT Slide
Lager Image
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)
PPT Slide
Lager Image
(1) ⇒ (5)
PPT Slide
Lager Image
(3) ⇒ (5)
PPT Slide
Lager Image
(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)
PPT Slide
Lager Image
PPT Slide
Lager Image
(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
PPT Slide
Lager Image
Then RX ( a , b ) ≤ RY ( f ( a ), f ( b )) for all a , b X .
PPT Slide
Lager Image
For n ≥ 2,
PPT Slide
Lager Image
and
PPT Slide
Lager Image
as follows:
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
Moreover,
PPT Slide
Lager Image
Then RτRX ( a, b ) ≤ RτRX ( f ( a ), f ( b )) for all a , b X .
(2)
PPT Slide
Lager Image
where ai L and
PPT Slide
Lager Image
For
PPT Slide
Lager Image
PPT Slide
Lager Image
For
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
where bi L and
PPT Slide
Lager Image
For
PPT Slide
Lager Image
PPT Slide
Lager Image
For
PPT Slide
Lager Image
PPT Slide
Lager Image
(3)
PPT Slide
Lager Image
where ai L and
PPT Slide
Lager Image
PPT Slide
Lager Image
where bi L and
PPT Slide
Lager Image
(4) For A = (0.2. 0.8, 0.6) ∈ LX ,
PPT Slide
Lager Image
References
Bělohlávek R. 2002 Fuzzy Relational Systems Kluwer Academic Publishers New York
De Meyer B. , De Meyer H. 2003 On the existence and construction of T-transitive closures Information Sciences 152 167 - 179    DOI : 10.1016/S0020-0255(02)00407-3
Hájek P. 1998 Metamathematices of Fuzzy Logic Kluwer Academic Publishers Dordrecht
Jinming Fang 2007 I-fuzzy Alexandrov topologies and specialization orders Fuzzy Sets and Systems 158 2359 - 2374    DOI : 10.1016/j.fss.2007.05.001
Kim Y.C. 2014 Alexandrov L-topologies International Journal of Pure and Applied Mathematics 93 (2) 165 - 179
Kim Y.C. 2014 Alexandrov L-topologies and L-join meet approximation operators International Journal of Pure and Applied Mathematics 91 (1) 113 - 129
Lai H. , Zhang D. 2006 Fuzzy preorder and fuzzy topology Fuzzy Sets and Systems 157 1865 - 1885    DOI : 10.1016/j.fss.2006.02.013
Lai H. , Zhang D. 2009 Concept lattices of fuzzy contexts: Formal concept analysis vs. rough set theory Int. J. Approx. Reasoning 50 695 - 707    DOI : 10.1016/j.ijar.2008.12.002
Pawlak Z. 1982 Rough sets Int. J. Comput. Inf. Sci. 11 341 - 356    DOI : 10.1007/BF01001956
Pawlak Z. 1984 Rough probability Bull. Pol. Acad. Sci. Math. 32 607 - 615
Radzikowska A.M. , Kerre E.E. 2002 A comparative study of fuzy rough sets Fuzzy Sets and Systems 126 137 - 155    DOI : 10.1016/S0165-0114(01)00032-X
She Y.H. , Wang G.J. 2009 An axiomatic approach of fuzzy rough sets based on residuated lattices Computers and Mathematics with Applications 58 189 - 201    DOI : 10.1016/j.camwa.2009.03.100
Ma Zhen Ming , Hu Bao Qing 2013 Topological and lattice structures of L-fuzzy rough set determined by lower and upper sets Information Sciences 218 194 - 204    DOI : 10.1016/j.ins.2012.06.029