STRUCTURES INDUCED BY ALEXANDROV FUZZY TOPOLOGIES
STRUCTURES INDUCED BY ALEXANDROV FUZZY TOPOLOGIES
The Pure and Applied Mathematics. 2014. Aug, 21(3): 183-194
• Received : March 26, 2014
• Accepted : June 03, 2014
• Published : August 31, 2014
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YONG CHAN, KIM

Abstract
In this paper, we investigate the properties of Alexandrov fuzzy topologies and meet-join approximation operators. We study fuzzy preorder, Alexandrov topologies and meet-join approximation operators induced by Alexandrov fuzzy topologies. We give their examples.
Keywords
1. Introduction
Hájek [2] introduced a complete residuated lattice which is an algebraic structure for many valued logic. Höhle [3] introduced L -fuzzy topologies and L -fuzzy interior operators on complete residuated lattices. Pawlak [8 , 9] introduced rough set theory as a formal tool to deal with imprecision and uncertainty in data analysis. Radzikowska [10] developed fuzzy rough sets in complete residuated lattice. Bělohlávek [1] investigated information systems and decision rules in complete residuated lattices. Zhang [6 , 7] introduced Alexandrov L -topologies induced by fuzzy rough sets. Kim [5] investigated the properties of Alexandrov topologies in complete residuated lattices.
In this paper, we investigate the properties of Alexandrov fuzzy topologies and meet-join approximation operators in a sense as Höhle [3] . We study fuzzy preorder, Alexandrov topologies and meet-join approximation operators induced by Alexandrov fuzzy topologies. We give their examples.
2. Preliminaries
Definition 2.1 ( [1 - 3] ). A structure ( L ,∨,∧,⊙, →, ⊥,⊤) is called a complete residuated lattice iff it satisfies the following properties:
• (L1) (L,∨,∧,⊥,⊤) is a complete lattice where ⊥ is the bottom element and ⊤ is the top element;
• (L2) (L, ⊙, ⊤) is a monoid;
• (L3) It has an adjointness,i.e.
• x ≤ y → ziffx⊙y ≤ z.
An operator * : L L defined by a * = a → ⊥ is called strong negations if a ** = a .
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In this paper, we assume that ( L , ∨, ∧, ⊙, →, *, ⊥, ⊤) be a complete residuated lattice with a strong negation *.
Definition 2.2 ( [6 , 7] ). Let X be a set. A function eX : X × X → L is called a fuzzy preorder if it satisfies the following conditions
• (E1) reflexive ifeX(x, x) = 1 for allx∈X,
• (E2) transitive ifeX(x, y) ⊙eX(y, z) ≤eX(x, z), for allx, y, z∈ X’
Example 2.3. (1) We define a function eL : L × L → L as eL ( x, y ) = x → y. Then eL is a fuzzy preorder on L .
• (2) We define a functioneLX:LX× LX→LasTheneLXis a fuzzy preorder from Lemma 2.4 (9).
Lemma 2.4 ( [1 , 2] ). Let ( L ,∨,∧,⊙, →,*, ⊥,⊤) be a complete residuated lattice with a strong 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)x→y= ⊤iff x≤y.
• (4)x→ ⊤ = ⊤ and ⊤ →x = x.
• (5)x⊙y≤x∧y.
• (6)and.
• (7)and.
• (8)and.
• (9) (x→y) ⊙x≤y and(y→ z) ⊙ (x → y) ≤ (x →z).
• (10)x→ y ≤ (y→z) → (x→z)and x→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→x and(x→y)* =x⊙y*.
• (14)y→z≤x⊙y→x⊙z.
Definition 2.5 ( [5] ). A map
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: LX → LY is called an meet-join approximation operator if it satisfies the following conditions, for all A, Ai ∈ LX , and α ∈ L ,
• (M1)where (α → A)(x) =α →A(x) for eachx ∈ X,
• (M2)
• (M3)A* ≤(A),
• (M4)(*(A))≤(A).
Definition 2.6 ( [4] ). An operator T : LX → L is called an Alexandrov fuzzy topology on X iff it satisfies the following conditions, for all A, Ai LX , and α L ,
• (T1)T(αX) = ⊤, whereαX(x) =αfor eachx∈X,
• (T2)T(Ai) ≥T(Ai) andT(Ai) ≥T(Ai),
• (T3)T(α⊙A) ≥T(A), where (α⊙A)(x) =α⊙A(x) for eachx∈X,
• (T4)T(α→A) ≥T(A).
Definition 2.7 ( [5] ). A subset τ LX is called an Alexandrov topology if it satisfies satisfies the following conditions.
• (O1)αX∈τ.
• (O2) IfAi∈τfori∈ Γ,Ai,Ai∈τ.
• (O3)α⊙A∈τfor allα∈LandA∈τ.
• (O4)α→A∈τfor allα∈LandA∈τ.
Remark 2.8. (1) If T : LX → L is an Alexandrov fuzzy topology. Define T *( A ) = T ( A *). Then T * is an Alexandrov fuzzy topology.
• (2) IfTbe an Alexandrov fuzzy topology onX,= {A∈LX|T(A) ≥r} is an Alexandrov topology onXandfors≤r∈L.
3. Structures Induced by Alexandrov Fuzzy Topologies
Theorem 3.1 . If
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is a meet-join approximation operator, then
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= {A ∈ LX |
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( A ) = A *} is an Alexandrov topology on X.
Proof. (O1) Since ⊤ X
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(⊥ X ) and
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(⊤ X ) =
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(⊥ X → A) = ⊥ X
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( A ) = ⊥, ⊥ X =
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(⊤ X ) and ⊤ X =
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(⊤ X ). Then ⊥ X ;⊤ X
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.
• (O2) ForAi∈for eachi∈ Γ , by (M2),So,Ai∈. SinceThus,Ai∈
• (O3) ForA∈, sinceα⊙(α⊙A) =(α→ (α⊙A)) ≥(A),(α⊙A) ≥α→(A) = (α⊙A)*. Thenα⊙A∈.
• (O4) ForA∈, by (M4),(α→A) =α⊙(A) =α⊙A*. Henceα→A∈.
Theorem 3.2. Let T be an Alexandorv fuzzy topology on X. Define
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We have the following properties .
• (1)is a fuzzy preorder with≤for each s≤r.
• (2)is a fuzzy preorder with≤for each s≤r and= (x, y) =*(x, y)
• (3)Defineas follows
Then
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is a meet-join approximation operator on X with
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for each s r.
• (4)
• (5)is a meet-join approximation operator on X such that
• (6)
• (7)for all A∈LXand r∈L.
• Moreover,,for each x, y∈X.
• (8)for all A∈LXand r∈L.
• Moreover, for each x, y∈X.
• (9)If=B for all i∈ Γ≠,thenwith s=ri.
• (10)Iffor all i∈ Γ≠,thenwith s=ri.
Proof. (1) Since T ( B ) ≥ r * iff
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then
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Since
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and
Hence
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is a fuzzy preorder.
• Fors≤r, sinceT(B) ≥s* ≥r*, we have≤
• (2) By a similar method as (1),is a fuzzy preorder. Moreover,
• (3) (M1)
• (M2)
• (M3)
• (M4)
• Fors≤r, since≤r, since, then
• (4) Since; i.e.T(A) ≥r*,=⊙A*(x) ≤ (A*(x) →A*(y)) ⊙A*(x) ≤A*(y), by M(3),So,ThusLet; i.e. LetThen
Since
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and
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we have
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. Hence
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* .
• (5) It is similarly proved as (4).
• (6) Letsince A ∈*,
Hence
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( A ) = A *; i.e.
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. Thus
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• Let; i.e.Then
Since
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and
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we have
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Hence
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• (7) For eachA∈LXwithA* ≤Ai,T(Ai) ≥r*, sincethen
So,
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Since A
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Since
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and
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So, ,
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. Hence
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=
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for all A LX and r L .
• (8) It is proved in a similar way as (7).
• (9) Let=Bfor alli∈ Γ ≠. Since
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then
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where
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Since
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then
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So,
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Thus
Since s ri ,
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Thus
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⧠
Theorem 3.3. Let T be an Alexandorv fuzzy topology on X. We have the following properties.
• (1) Define:LX→Las
Then
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= T * is an Alexandrov fuzzy topology on X .
• (2) Define:LX→Las
Then
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= T * is an Alexandrov fuzzy topology on X .
• (3)for all A, B∈LX.
• (4)There exists an Alexandrov fuzzy topology Trsuch that
If r ≤ s, then T r T s f or all A LX .
• (5)There exists an Alexandrov fuzzy topology T*rsuch that
• T*r(A) =eLX((A) ,A*).
Moreover, T * r ( A ) = T r ( A *) for all A LX . If r s , then T * r T * s for all A LX .
• (6) DefineTM:LX→L as
Then T M = T * = T MT is an Alexandrov fuzzy topology on X.
• (7) DefineTM*:LX→L as
Then T M* = T = T MT* is an Alexandrov fuzzy topology on X.
Proof. (1) We only show that T MT = T *. Let
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= A *. Then
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form Theorem 3.3 (6). So T * ( A ) = T ( A *) =
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Thus,
Since T *( A ) ≥ ( T ( A ))* then
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with s = T( A ). Thus,
Hence T MT = T *.
• (4) (T1) Since
• (T2) Sincewe have
• (T3) SincethenThus
• (T4)
Hence T r is an Alexandrov fuzzy topology. Since
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for r ≤ s , T s ( A ) =
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= T r ( A ).
• (5) From a similar method as (4),T*ris an Alexandrov fuzzy topology. By (3),Tr(A*) ==T*r(A) for allA∈LX.
• (6) SinceTr(A) =iffby (9),
• (2) and (7) are similarly proved as (1) and (6), respectively.                           ⧠
Example 3.4. Let ( L = [0, 1], ⊙, →, * ) be a complete residuated lattice with a strong negation.
(1) Let X = { x, y, z } be a set. Define a map T : [0, 1] X → [0, 1] as
• T(A) =A(x) →A(z).
Trivially, T ( αX ) = 1
Since α A ( x ) → α A ( z ) ≥ A ( x ) → A ( z ) from Lemma 2.4 (14), T ( α A ) ≥ T ( A ). Since ( α A ( x )) → ( α A ( z )) ≥ A ( x ) → A ( z ) from Lemma 2.4 (10), T ( α A ) ≥ T(A). By Lemma 2.4 (8), T(
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Ai ) ≥
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T ( Ai ) and T (
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Ai ) ≥
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T ( Ai ). Hence T is an Alexandrov fuzzy topology.
If T ( A ) = A ( x ) → A ( z ) ≥ r*, then A ( z ) ≥ A ( x ) ⊙ r*. Put A ( x ) = 1, A ( y ) = 0. So,
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and
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similarly, we can obtain
By Theorem 3.2(3), we obtain
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such that
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If A *( x ) ⊙ r * ≤ A *( z ), then
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Thus
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. Moreover, since T *( A ) = A *( x )→ A *( z ) ≥ r * iff A*(z) ≥ A *( x ) ⊙ r *,
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iff
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. So,
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From Theorem 3.3(1), we have
Moreover, we obtain
Hence T M = T MT = T *.
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Since B ( x ) = 1 and T ( B ) = 1 → B ( z ) = B( z ) ≥ r *, then
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Since B ( z ) = 1 and T ( B ) = B ( x ) → 1= 1, then
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Then
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(2) By (1), we obtain a map T* : [0, 1] Y → [0, 1] as
• T*(A) =A*(x) →A*(z) =A(z) →A(x).
Since T *( A ) = A ( z ) → A ( x ) ≥ r *, then A ( x ) ≥ A ( z ) ⊙ r *. Put A ( z ) = 1, A ( y ) = 0. So,
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and
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Moreover,
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for all x, y X .
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If A *( z ) ⊙ r * ≤ A *( x ), then
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If
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, then A *( z ) ⊙ r* ≤ A *( x ). Moreover, since T ( A ) = A ( x ) → A ( z ) ≥ r * iff A *( z ) ⊙ r * ≤ A *( z ),
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iff
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Thus
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Moreover, we obtain
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Hence T M* = T MT* = T .
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Since B ( x ) = 1 and T *( B ) = B ( z ) →1 = 1, then
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Since B ( z ) = 1 and T *( B ) = 1 → B ( x ) = B ( x ) ≥ r *, then B ( x ) ≥ r *. We have
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Then
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(3) Let ( L = [0, 1], ⊙, →, * ) be a complete residuated lattice with a strong negation defined by, for each n N ,
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By (1) and (2), we obtain
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Since
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we have
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