The Properties of L -lower Approximation Operators
- Author: Kim Yong Chan
- Organization: Kim Yong Chan
- Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 14, Issue1, p57~65, 25 March 2014
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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 AlexandrovL -topologies. Moreover, we give their examples as approximation operators induced by variousL -fuzzy relations.
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KEYWORD
Complete residuated lattices , L-upper approximation operators , Alexandrov L-topologies
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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 AlexandrovL -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 AlexandrovL -topologies. Moreover, we give their examples as approximation operators induced by variousL -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 iffx ≤y →z forx ,y ,z ∈L Remark 1.2. [3, 5] (1) A completely distributive latticeL = (L ,≤,∨,∧ = ⊙,→, 1, 0) is a complete residuated lattice defined by(2) The unit interval with a left-continuous t-norm ⊙,
is a complete residuated lattice defined by
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 ,and
Lemma 1.3. [3, 5] For eachx, y, z, xi, yi ∈L , we have the following properties.(1) A map
H :LX →LX is called anL-upper approximation operator iff it satisfies the following conditions(2) A map 𝒥 :
LX →LX is called anL-lower approximation operator iff it satisfies the following conditions(3) A map
K :LX →LX is called anL-join meet approximation operator iff it satisfies the following conditions(4) A map
M :LX →LX is called anL-meet join approximation operator iff it satisfies the following conditionsDefinition 1.5. [6, 9] A subset 𝜏 ⊂LX is called anAlexandrov L-topology if it satisfies:(1) 𝜏 is an Alexandrov topology on
X iff 𝜏⁎ = {A * ∈LX |A ∈ 𝜏} is an Alexandrov topology onX .(2) If
H is anL -upper approximation operator, then 𝜏H = {A ∈LX |H (A ) =A } is an Alexandrov topology onX .(3) If 𝒥 is an
L -lower approximation operator, then 𝜏𝒥 = {A ∈LX | 𝒥 (A ) =A } is an Alexandrov topology onX .(4) If
K is anL -join meet approximation operator, then 𝜏K = {A ∈LX |K (A ) =A *} is an Alexandrov topology onX .(5) If
M is anL -meet join operator, then 𝜏M = {A ∈LX |M (A ) =A *} is an Alexandrov topology onX .Definition 1.7. [8, 9] LetX be a set. A functionR :X ×X →L is called:If
R satisfies (R1) and (R3),R is called aL-fuzzy preorder .If
R satisfies (R1), (R2) and (R3),R is called aL-fuzzy equivalence relation 2. The Properties of
L -lower Approximation OperatorsTheorem 2.1. Let 𝒥 :LX →LX be anL -lower approximation operator. Then the following properties hold.(1) For
A ∈LX ,.(2) Define
HJ (B ) = ∧{A |B ≤ 𝒥 (A )}. ThenHJ :LX →LX withis an
L -upper approximation operator such that (HJ ,𝒥 )is a residuated connection;i.e.,
HJ (B) ≤ A iff B ≤ 𝒥 (A).
Moreover, 𝜏𝒥 = 𝜏
HJ .(3) If 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈LX , thenHJ (HJ (A )) =HJ (A ) forA ∈LX such that 𝜏𝒥 = 𝜏HJ with(4) If 𝒥 (𝒥 *(
A )) = 𝒥 *(A ) forA ∈LX , then 𝒥 (𝒥 (A )) = 𝒥 (A ) such that(5) Define
Hs (A ) = 𝒥 (A *)*. ThenH :s LX →LX withis an
L -upper approximation operator. Moreover, 𝜏Hs = (𝜏𝒥 )* = (𝜏HJ )*.(6) If 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈LX , thenHs(Hs(A)) = Hs(A)
for
A ∈LX such that 𝜏Hs = (𝜏𝒥 )* = (𝜏HJ )*. with(7) If 𝒥 (𝒥 *(
A )) = 𝒥 *(A ) forA ∈LX , thensuch that
(8) Define
KJ (A ) = 𝒥 (A *). ThenKJ :LX →LX withis an
L -join meet approximation operator.(9) If 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈LX , thenfor
A ∈LX such that 𝜏KJ = (𝜏𝒥 )* with(10) If 𝒥 (𝒥 *(
A )) = 𝒥 *(A ) forA ∈LX , thensuch that
(11) Define
MJ (A ) = (𝒥 (A ))*. ThenMJ :LX →LX withis an
L -meet join approximation operator. Moreover, 𝜏MJ = 𝜏𝒥 .(12) If 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈LX , then forA ∈LX such that 𝜏MJ = (𝜏𝒥 )* with(13) If 𝒥 (𝒥 *(
A )) = 𝒥 *(A ) forA ∈LX , thensuch that
(14) Define
KHJ (A ) = (HJ (A ))*. ThenKHJ :LX →LX withis an
L -meet join approximation operator. Moreover, 𝜏KHJ = 𝜏𝒥 .(15) If 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈LX , thenfor
A ∈LX such that 𝜏KHJ = (𝜏𝒥 )* with(16) If for
A ∈LX , thensuch that
(17) Define
MHJ (A ) =HJ (A *). ThenMHJ :LX →LX with
is an
L -join meet approximation operator. Moreover, 𝜏MHJ = (𝜏𝒥 )*.(18) If 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈LX , thenfor
A ∈LX such that 𝜏MHJ = (𝜏𝒥 )* with(19) If for
A ∈LX , thensuch that
(20) (
KHJ ,KJ ) is a Galois connection;i.e,A ≤ KHJ (B) iff B ≤ KJ (A).
Moreover, 𝜏
KJ = (𝜏KHJ )*.(21) (
MJ ,MHJ ) is a dual Galois connection;i.e,MHJ (A) ≤ B iff MJ (B) ≤ A.
Moreover, 𝜏
MJ = (𝜏MHJ )*.Proof. (1) Since , by (J2) and (J3),
(2) Since
iff , we have
(H1) Since
HJ (A ) ≤HJ (A ) iff A ≤ 𝒥 (HJ (A )), we haveA ≤ 𝒥 (HJ (A )) ≤HJ (A ).(H3) By the definition of
HJ , sinceHJ (A ) ≤HJ (B ) forB ≤A , we haveSince 𝒥 (∨
i ∈𝚪HJ (Ai )) ≥ 𝒥 (HJ (Ai )) ≥Ai , then𝒥 (∨
i ∈𝚪HJ (Ai )) ≥ ∨i ∈𝚪Ai . ThusThus
HJ :LX →LX is anL -upper approximation operator. By the definition ofHJ , we haveHJ (B) ≤ A iff B ≤ 𝒥 (A).
Since
A ≤ 𝒥 (A ) iffHJ (A ) ≤A , we have 𝜏HJ = 𝜏𝒥 .(3) Let 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈LX . Since 𝒥 (B ) ≥HJ (A ) iff 𝒥 (𝒥 (B )) = 𝒥 (B ) ≥A from the definition ofHJ , we have(4) Let 𝒥 *(
A ) ∈ 𝜏𝒥 . Since 𝒥 (𝒥 *(A )) = 𝒥 *(A ),𝒥 (𝒥 (
A )) = 𝒥 (𝒥 *(𝒥 *(A ))) = (𝒥 (𝒥 *(A )))* = 𝒥 (A ).Hence 𝒥 (
A ) ∈ 𝜏𝒥 ; i.e. 𝒥 *(A ) ∈ (𝜏𝒥 )*. Thus, 𝜏𝒥 ⊂ (𝜏𝒥 )*.Let
A ∈ (𝜏𝒥 )*. ThenA * = 𝒥 (A *). Since 𝒥 (A ) = 𝒥 (𝒥 *(A *)) = 𝒥 *(A *) =A , thenA ∈ 𝜏𝒥 . Thus, (𝜏𝒥 )* ⊂ 𝜏𝒥 .(5) (H1) Since 𝒥 (
A *) ≤A *,Hs (A ) = 𝒥 (A *)* ≥A .Hence
Hs is anL -upper approximation operator such thatMoreover, 𝜏
Hs = (𝜏𝒥 )* from:A = Hs(A) iff A = 𝒥 (A*)* iff A* = 𝒥 (A*).
(6) Let 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈LX . ThenHence
(7) Let 𝒥 (𝒥 *(
A )) = 𝒥 *(A ) forA ∈LX . ThenHence
By a similar method in (4), 𝜏
Hs = (𝜏Hs )*.(8) It is similarly proved as (5).
(9) If 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈LX , then(10) If 𝒥 (𝒥 *(
A )) = 𝒥 *(A ) forA ∈LX , thenSince ,
Hence 𝜏
KJ = {KJ (A ) |A ∈LX } = (𝜏KJ )*.(11) , (12), (13) and (14) are similarly proved as (5), (9), (10) and (5), respectively.
(15) If 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈LX , thenHJ (HJ (A )) =HJ (A ). Thus,Since 𝒥 (
A ) =A iffHJ (A ) =A iffKHJ (A ) =A *, 𝜏KHJ = (𝜏𝒥 )* with(16) If for A ∈
LX , then(17) , (18) and (19) are similarly proved as (14), (15) and (16), respectively.
(20) (
KHJ ,KJ ) is a Galois connection;i.e,A ≤ KHJ (B) iff A ≤ (HJ (B))*
iff HJ (B) ≤ A* iff B ≤ 𝒥 (A*) = KJ (A)
Moreover, since
A * ≤KJ (A ) iffA ≤KHJ (A *), 𝜏KJ = (𝜏KHJ )*.(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 iffMJ (A ) ≤A *, 𝜏MJ = (𝜏MHJ )*.Let
R ∈L X ×X be anL -fuzzy relation. Define operators as followsExample 2.2. LetR be a reflexiveL -fuzzy relation. Define 𝒥R :LX →LX as follows:(1) (J1) 𝒥
R (A )(y ) ≤R (y, y ) →A (y ) =A (y ): 𝒥R satisfies the conditions (J1) and (J2) from:Hence 𝒥
R is anL -lower approximation operator.(2) Define
HJR (B ) = ∨ {A |B ≤ 𝒥R (A )}. Sincethen
By Theorem 2.1(2),
HJR =H R -1 is anL -upper approximation operator such that (HJR ,𝒥R ) is a residuated connection;i.e.,HJR(A) ≤ B iff A ≤ 𝒥R(B).
Moreover, 𝜏
HJR = 𝜏𝒥R .(3) If
R is anL -fuzzy preorder, thenR -1 is anL -fuzzy preorder. SinceBy Theorem 2.1(3),
HJR (HJR (A )) =HJR (A ): By Theorem 2.1(3), 𝜏HJR = 𝜏𝒥R with(4) Let
R be a reflexive and EuclideanL -fuzzy relation. SinceR (x, z ) ⊙R (y, z ) ⊙A *(x ) ≤R (x, y ) ⊙A *(x ) iffR (x, z ) ⊙A *(x ) ≤R (y, z ) →R (x, y ) ≤A *(x ),Thus, .
By Theorem 2.1(4), 𝒥
R (𝒥R (A )) = 𝒥R (A ) forA ∈LX .Thus, 𝜏𝒥
R = (𝜏𝒥R )* with(5) Define
Hs (A ) = 𝒥R (A *)*. By Theorem 2.1(5),Hs =HR is anL -upper approximation operator such thatMoreover, 𝜏
Hs = 𝜏HR = (𝜏HJR )*.(6) If
R is anL -fuzzy preorder, then 𝒥R (𝒥R (A )) = 𝒥R (A ) forA ∈LX . By Theorem 2.1(6), thenHs (Hs (A )) =Hs (A ) forA ∈LX such that 𝜏Hs = (𝜏𝒥R )* = (𝜏HJR )* with(7) If
R is a reflexive and EuclideanL -fuzzy relation, then(8) Define
KJR (A ) = 𝒥R (A *). ThenKJR :LX →LX withis an
L -join meet approximation operator. Moreover, 𝜏KJR = (𝜏𝒥R )*.(9)
R is anL -fuzzy preorder, then 𝒥R (𝒥R (A )) = 𝒥R (A ) forA ∈LX . By Theorem 2.1(9), forA ∈LX such that 𝜏KJR = (𝜏𝒥R )* with(10) If
R is a reflexive and EuclideanL -fuzzy relation, then forA ∈LX . By Theorem 2.1(10), such that(11) Define
MJR (A ) = (𝒥R (A ))*. ThenMJR :LX →LX withis an
L -join meet approximation operator. Moreover, 𝜏MJR = 𝜏𝒥R .(12) If
R is anL -fuzzy preorder, then 𝒥R (𝒥R (A )) = 𝒥R (A ) forA ∈LX . By Theorem 2.1(12), forA ∈LX such that 𝜏MJR = 𝜏𝒥R with(13) If
R is a reflexive and EuclideanL -fuzzy relation, then forA ∈LX . By Theorem 2.1(13), such that(14) Define
KHJR (A ) = (HJR (A ))*. ThenKHJR : LX → LX
with
is an
L -join meet approximation operator. Moreover, 𝜏K R -1 = 𝜏𝒥R = 𝜏H R -1 .(15) If
R is anL -fuzzy preorder, then 𝒥R (𝒥R (A )) = 𝒥R (A ) forA ∈LX . By Theorem 2.1(15), forA ∈LX such that 𝜏K R -1 = 𝜏𝒥R = 𝜏H R -1 with(16) Let
R -1 be a reflexive and EuclideanL -fuzzy relation. Sincewe have
Thus,
Hence
By (K1), such that
(17) Define
MHJR (A ) =HJR (A *). ThenMHJR : LX → LX
is an
L -meet join approximation operator as follows:Moreover, 𝜏
MHJR = (𝜏𝒥R )*.(18) If
R is anL -fuzzy preorder, then 𝒥R (𝒥R (A )) = 𝒥R (A ) forA ∈LX . By Theorem 2.1(18),for
A ∈LX such that 𝜏MHJR = (𝜏𝒥 )* with(19) Let
R -1 be a reflexive and EuclideanL -fuzzy relation.Since
then (
R (y, x ) →A (x )) ⊙R (z, y ) ≤R (z, x ) →A (x ).Thus,
By (M1), such that
(20) (
KHJR =K R -1* ,KJR =K R * ) is a Galois connection; i.e,A ≤KHJR (B ) iffB ≤KJR (A ): Moreover, 𝜏KJR = (𝜏KHJR )*.(21) (
MJR =MR ,MHJR =M R -1 ) is a dual Galois connection; i.e,MHJR (A ) ≤B iffMJR (B ) ≤A . Moreover, 𝜏MJR = (𝜏MHJR )*.In this paper,
L -lower approximation operators induceL -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 AlexandrovL -topologies. Moreover, we give their examples as approximation operators induced by variousL -fuzzy relations.No potential conflict of interest relevant to this article was reported.
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