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Some Properties of Alexandrov Topologies
Some Properties of Alexandrov Topologies
International Journal of Fuzzy Logic and Intelligent Systems. 2015. Mar, 15(1): 72-78
Copyright © 2015, 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 : July 09, 2014
  • Accepted : September 22, 2014
  • Published : March 25, 2015
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About the Authors
Yong Chan Kim
Department of Mathematics, Gangneung-Wonju National University, Gangneung 201-702, Korea
Young Sun Kim
Department of Applied Mathematics, Pai Chai University, Daejeon 302-735, Korea

Abstract
Alexandrov topologies are the topologies induced by relations. This paper addresses the properties of Alexandrov topologies as the extensions of strong topologies and strong cotopologies in complete residuated lattices. With the concepts of Zhang’s completeness, the notions are discussed as extensions of interior and closure operators in a sense as Pawlak’s the rough set theory. It is shown that interior operators are meet preserving maps and closure operators are join preserving maps in the perspective of Zhang’s definition.
Keywords
1. Introduction
Pawlak [1 , 2] introduced the rough set theory as a formal tool to deal with imprecision and uncertainty in the data analysis. Hájek [3] introduced a complete residuated lattice which is an algebraic structure for many valued logic. By using the concepts of lower and upper approximation operators, information systems and decision rules are investigated in complete residuated lattices [3 - 7] . Zhang and Fan [8] and Zhang et al. [9] introduced the fuzzy complete lattice which is defined by join and meet on fuzzy partially ordered sets. Alexandrov topologies [7 , 10 - 12] were introduced the extensions of fuzzy topology and strong topology [13] .
In this paper, we investigate the properties of Alexandrov topologies as the extensions of strong topologies and strong cotopologies in complete residuated lattices. Moreover, we study the notions as extensions of interior and closure operators. We give their examples.
Definition 1.1. [3 , 4] An algebra ( L , ∧, ∨, ⊙, →, ⊥, 𝖳) is called a complete residuated lattice if it satisfies the following conditions:
  • (C1)L= (L, ≤, ∨, ∧, ⊥, 𝖳) is a complete lattice with the greatest element 𝖳 and the least element ⊥;
  • (C2) (L, ⊙, 𝖳) is a commutative monoid;
  • (C3)x⊙y≤ziffx≤y→zforx,y,z∈L.
In this paper, we assume ( L , ∧, ∨, ⊙, →, ⊥, 𝖳) is a complete residuated lattice with a negation; i.e., x ∗∗ = x . For α L , A , 𝖳 x L X , ( α A )( x ) = α A ( x ), ( α A )( x ) = α A ( x ) and 𝖳 x ( x ) = 𝖳, 𝖳 x ( x ) = ⊥, otherwise.
Lemma 1.2. [3 , 4] For each x , y , z , xi , yi L , the following properties hold.
  • (1) Ify≤z, thenx⊙y≤x⊙z.
  • (2) Ify≤z, thenx→y≤x→zandz→x≤y→x.
  • (3)x→y= 𝖳 iffx≤y.
  • (4)x→ 𝖳 = 𝖳 and 𝖳 →x=x.
  • (5)x⊙y≤x∧y.
  • (6).
  • (7)and.
  • (8)and.
  • (9) (x→y) ⊙x≤yand (y→z) ⊙ (x→y) ≤ (x→z).
  • (10)x→y≤ (y→z) → (x→z) andx→y≤ (z→x) → (z→y).
  • (11)and.
  • (12) (x⊙y) →z=x→ (y→z) =y→ (x→z) and (x⊙y)∗=x→y∗.
  • (13)x∗→y∗=y→xand (x→y)∗=x⊙y∗.
  • (14)y→z≤x⊙y→x⊙z.
  • (15)x→y⊙z≥ (x→y) ⊙zand (x→y) →z≥x⊙ (y→z).
Definition 1.3. [7 , 10 , 12 , 13] A subset τ L X is called an Alexandrov topology if it satisfies:
  • (T1) ⊥X, 𝖳X∈τwhere 𝖳X(x) = 𝖳 and ⊥X(x) = ⊥ forx∈X.
  • (T2) IfAi∈τfori∈ Γ,.
  • (T3)α⊙A∈τfor allα∈LandA∈τ.
  • (T4)α→A∈τfor allα∈LandA∈τ.
A subset τ LX satisfying (T1), (T3) and (T4) is called a strong topology if it satisfies:
(ST) If Ai τ for i ∈ Γ,
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for each finite index Λ ⊂ Γ.
A subset τ LX satisfying (T1), (T3) and (T4) is called a strong cotopology if it satisfies:
(SC) If Ai τ for i ∈ Γ,
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for each finite index Λ ⊂ Γ.
Remark 1.4. Each Alexandrov topology is both strong topology and strong cotopology.
Definition 1.5. [8 , 9] Let X be a set. A function eX : X × X L is called:
  • (E1) reflexive ifeX(x,x) = 𝖳 for allx∈X,
  • (E2) transitive ifeX(x,y) ⊙eX(y,z) ≤eX(x,z), for allx,y,z∈X,
  • (E3) ifeX(x,y) =eX(y,x) = 𝖳, thenx=y.
If e satisfies (E1) and (E2), ( X , eX ) is a fuzzy preordered set. If e satisfies (E1), (E2) and (E3), ( X , eX ) is a fuzzy partially ordered set.
Example 1.6. (1) We define a function eLX : L X × L X L as
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. Then ( L X , eLX ) is a fuzzy partially ordered set from Lemma 1.2 (8).
(2) Let τ be an Alexandrov topology. We define a function eτ : τ × τ
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. Then ( τ , eτ ) is a fuzzy partially ordered set.
Definition 1.7. [8 , 9] Let ( X , eX ) be a fuzzy partially ordered set and A LX .
  • (1) A pointx0is called a join ofA, denoted byx0= ⊔Aif it satisfies
  • (J1)A(x) ≤eX(x,x0),
  • (J2).
  • A pointx1is called a meet ofA, denoted byx1= ⊓A, if it satisfies
  • (M1)A(x) ≤eX(x1,x),
  • (M2).
Remark 1.8. [8 , 9] Let ( X , eX ) be a fuzzy partially ordered set and A LX .
  • (1)x0is a join ofAiff
  • .
  • (2)x1is a meet ofAiff.
  • (3) Ifx0is a join ofA, then it is unique becauseeX(x0,y) =eX(y0,y) for ally∈X, puty=x0ory=y0, theneX(x0,y0) =eX(y0,x0) = 𝖳 impliesx0=y0. Similarly, if a meet ofAexist, then it is unique.
Remark 1.9. [8 , 9] Let ( L X , eLX ) be a fuzzy partially ordered and Φ ∈ LLX .
  • (1) Since
  • then.
  • (2) We havebecause
2. Some Properties of Alexandrov Topologies
Theorem 2.1. (1) A subset τ LX is an Alexandrov topology on X iff for each Φ : τ L , ⊔Φ ∈ τ and ⊓Φ ∈ τ .
(2) τ is an Alexandrov topology on X iff τ = { A LX | A τ } is an Alexandrov topology on X .
Proof. (1) (⇒) For each Φ : τ L , we define
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Since τ is an Alexandrov topology on X , (Φ( A ) ⊙ A ) ∈ τ . Thus P τ . Then P = ⊔Φ from:
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For each Φ : τ L , we define
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. Since τ is an Alexandrov topology on X , (Φ( A ) → A ) ∈ τ . Thus Q τ . Then Q = ⊓Φ from:
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(⇒) (T1) For Φ( A ) = ⊥ for all A τ ,
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and
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.
(T2) Let Φ( Ai ) = 𝖳 for all { Ai | i ∈ Γ} ⊂ τ , otherwise Φ( A ) = ⊥. We have
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(T3) Let Φ( A ) = ⊥ for A = B τ , otherwise Φ( A ) = α if A B . We have
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(2) Let A τ for A τ . Since α A = ( α A ) and α A = ( α A ) , τ is an Alexandrov topology on X .
Theorem 2.2. Let τ be an Alexandrov topology on X . Define Iτ : LX LX as follows:
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Then the following properties hold.
  • (1)eLX(A,B) ≤eLX(Iτ(A),Iτ(B)), forA,B∈LX.
  • (2)Iτ(A) ≤Afor allA∈LX.
  • (3)Iτ(Iτ(A)) =Iτ(A) for allA∈LX.
  • (4)Iτ(α→A) =α→Iτ(A) for allα∈L,A∈LX.
  • (5)for allAi∈LX.
  • (6)for each Φ :LX→Lwheredefined as.
  • (7).
  • (8) DefineτIτ= {A|A=Iτ(A)}. Thenτ=τIτ.
  • (9) There exists a fuzzy preordereX:X×X→Lsuch that
Proof . (1) By Lemma 1.2 (8,10,14), we have
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(2) Since eLX ( C , A )⊙ C A from Lemma 1.2 (9), Iτ ( A ) ≤ A .
(3) Since Iτ ( A ) ∈ τ , then
Iτ ( Iτ ( A )) ≥ eLX ( Iτ ( A ), Iτ ( A )) ⊙ Iτ ( A ) = Iτ ( A ).
By (2), Iτ ( Iτ ( A )) = Iτ ( A ).
(4) Since α Iτ ( A ) ≤ α A and α Iτ ( A ) ∈ τ ,
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(5) By (1), since Iτ ( A ) ≤ Iτ ( B ) for
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. Since
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and
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, we have
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(6) For each Φ : LX L , put
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. Since
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is a map, we have
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and
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from:
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(7)
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. Since I ( A ) ≤ A and I ( A ) ∈ τ , we have
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.
Since Iτ ( A ) ≤ A and Iτ ( A ) ∈ τ , we have I ( A ) ≥ Iτ ( A ).
(8) It follows from A τ iff Iτ ( A ) = A iff A τIτ .
(9) Since
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, by (4) and (5),
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. Put
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. Then
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Hence eX is a fuzzy preorder.
Theorem 2.3. Let τ be an Alexandrov topology on X . Define Cτ : LX LX as follows:
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Then the following properties hold.
  • (1)eLX(A,B) ≤eLX(Cτ(A),Cτ(B)), for allA,B∈LX.
  • (2)A≤Cτ(A) for allA∈LX.
  • (3)Cτ(Cτ(A)) =Cτ(A) for allA∈LX.
  • (4)Cτ(α⊙A) =α⊙Cτ(A) for allα∈L,A∈LX.
  • (5)for allAi∈LX.
  • (6)for each Φ :LX→Lwheredefined as.
  • (7).
  • (8) DefineτCτ= {A|A=Cτ(A)}. Thenτ=τCτ.
  • (9) (Cτ(A∗))∗=Iτ∗(A) for allA∈LX.
  • (10) There exists a fuzzy preordereX:X×X→Lsuch that
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Proof . (1) By Lemma 1.2 (8,10), we have
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(2) Since eLX ( A , B ) ⊙ A B iff A eLX ( A , B ) → B , then A Cτ ( A ).
(3) Since Cτ ( A ) ∈ τ , then Cτ ( Cτ ( A )) ≤ eLX ( Cτ ( A ), Cτ ( A )) → Cτ ( A ) = Cτ ( A ). By (2), Cτ ( Cτ ( A )) = Cτ ( A ).
(4) Since α A α Cτ ( A ) and α Cτ ( A ) ∈ τ ,
Cτ ( α A ) ≤ eLX ( α A , α Cτ ( A )) → α Cτ ( A ) = α Cτ ( A ).
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(5) By (1), since Cτ ( A ) ≤ Cτ ( B ) for A B ,
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. Since
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and
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we have
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(6) For each Φ : LX L , put
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. Since
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is a map, we have
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and
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from:
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(7) Put
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. Since A C ( A ) and C ( A ) ∈ τ , we have
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Since A Cτ ( A ) and Cτ ( A ) ∈ τ , we have C ( A ) ≤ Cτ ( A ).
(8) It follows from A τ iff Cτ ( A ) = A iff A τCτ .
(9)
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(10) Since
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, by (4) and (5),
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. Put eX ( x , y ) = Cτ (𝖳 x )( y ). Then
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Hence eX is a fuzzy preorder. Since
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, by Theorem 2.2(9),
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Example 2.4. Let ( L = [0, 1], ⊙, →, ) be a complete residuated lattice with a negation defined by
x y = ( x + y −1)∨0, x y = (1− x + y )∧1, x = 1− x .
Let X = { x , y , z } be a set and A 1 = (1, 0.8, 0.6), A 2 = (0.7, 1, 0.7), A 3 = (0.5, 0.7, 1).
(1) We define
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where
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(T1) For ⊥ X LX , eX (⊥ X ) = ⊥ X τ . For 𝖳 X LX , eX (𝖳 X ) = 𝖳 X τ .
(T2) For eX ( Ai ) ∈ τ for each i ∈ Γ,
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. Moreover, since eX ( A )( x ) ≥ eX ( x , x ) ⊙ A ( x ) = A ( x ) and eX ( eX ( A )) = eX ( A ),
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Hence
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.
(T3) For eX ( A ) ∈ τ , α eX ( A ) = eX ( α A ) ∈ τ .
(T4) Since α eX ( α eX ( A )) ≤ eX ( eX ( A )) = eX ( A ), we have
α eX ( A ) ≤ eX ( α eX ( A )) ≤ α eX ( A )
Hence, for eX ( A ) ∈ τ , α eX ( A ) = eX ( α eX ( A )) ∈ τ . Hence τ is an Alexandrov topology on X .
(2) For B 1 = (0.7, 0.3, 0.6), B 1 = (0.5, 0.9, 0.3), we obtain
  • Iτ(B1) = (0.5, 0.3, 0.6),Iτ(B2) = (0.5, 0.6, 0.3),
  • Cτ(B1) = (0.7, 0.5, 0.6),Cτ(B2) = (0.6, 0.9, 0.6).
Let Φ : LX L as follows
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Thus,
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.
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Thus,
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.
(3) We define
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For B 1 , B 2 and Φ in (2), we obtain
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Since ⊓Φ = (0.7, 0.4, 0.5) and
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we have
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.
Since ⊔Φ = (0.6, 0.7, 0.5) and
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then
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.
3. Conclusions
The fuzzy complete lattice is defined with join and meet operators on fuzzy partially ordered sets. Alexandrov topologies are the extensions of fuzzy topology and strong topology.
Several properties of join and meet operators induced by Alexandrov topologies in complete residuated lattices have been elicited and proved. In addition, with the concepts of Zhang’s completeness, some extensions of interior and closure operators are investigated in the sense of Pawlak’s rough set theory on complete residuated lattices. It is expected to find some interesting functorial relationships between Alexandrov topologies and two operators.
Conflict of InterestNo potential conflict of interest relevant to this article was reported.
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
Young Sun Kim received the B.S., M.S. and Ph.D. degrees in Mathematics from Yonsei University, Seoul, Korea, in 1981, 1985 and 1991, respectively. He is currently Professor of Pai Chai University. His research interests is a fuzzy topology and fuzzy logic.
E-mail: yskim@pcu.ac.kr
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