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

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.

KEYWORD
Complete residuated lattices , Lupper approximation operators , Alexandrov Ltopologies

1. Introduction
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 leftcontinuous tnorm ⊙,
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} ∈L^{X} ,and
Lemma 1.3. [3, 5] For eachx, y, z, x_{i}, y_{i} ∈L , we have the following properties.(1) A map
H :L^{X} →L^{X} is called anLupper approximation operator iff it satisfies the following conditions(2) A map 𝒥 :
L^{X} →L^{X} is called anLlower approximation operator iff it satisfies the following conditions(3) A map
K :L^{X} →L^{X} is called anLjoin meet approximation operator iff it satisfies the following conditions(4) A map
M :L^{X} →L^{X} is called anLmeet join approximation operator iff it satisfies the following conditionsDefinition 1.5. [6, 9] A subset 𝜏 ⊂L^{X} is called anAlexandrov Ltopology if it satisfies:(1) 𝜏 is an Alexandrov topology on
X iff 𝜏_{⁎} = {A * ∈L^{X} A ∈ 𝜏} is an Alexandrov topology onX .(2) If
H is anL upper approximation operator, then 𝜏_{H} = {A ∈L^{X} H (A ) =A } is an Alexandrov topology onX .(3) If 𝒥 is an
L lower approximation operator, then 𝜏_{𝒥} = {A ∈L^{X}  𝒥 (A ) =A } is an Alexandrov topology onX .(4) If
K is anL join meet approximation operator, then 𝜏_{K} = {A ∈L^{X} K (A ) =A *} is an Alexandrov topology onX .(5) If
M is anL meet join operator, then 𝜏_{M} = {A ∈L^{X} 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 aLfuzzy preorder .If
R satisfies (R1), (R2) and (R3),R is called aLfuzzy equivalence relation 2. The Properties of
L lower Approximation OperatorsTheorem 2.1. Let 𝒥 :L^{X} →L^{X} be anL lower approximation operator. Then the following properties hold.(1) For
A ∈L^{X} ,.(2) Define
H_{J} (B ) = ∧{A B ≤ 𝒥 (A )}. ThenH_{J} :L^{X} →L^{X} withis an
L upper approximation operator such that (H_{J} ,𝒥 )is a residuated connection;i.e.,
HJ (B) ≤ A iff B ≤ 𝒥 (A).
Moreover, 𝜏_{𝒥} = 𝜏_{HJ} .
(3) If 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈L^{X} , thenH_{J} (H_{J} (A )) =H_{J} (A ) forA ∈L^{X} such that 𝜏_{𝒥} = 𝜏_{HJ} with(4) If 𝒥 (𝒥 *(
A )) = 𝒥 *(A ) forA ∈L^{X} , then 𝒥 (𝒥 (A )) = 𝒥 (A ) such that(5) Define
H_{s} (A ) = 𝒥 (A *)*. ThenH :s L^{X} →L^{X} withis an
L upper approximation operator. Moreover, 𝜏_{Hs} = (𝜏_{𝒥} )_{*} = (𝜏_{HJ} )_{*}.(6) If 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈L^{X} , thenHs(Hs(A)) = Hs(A)
for
A ∈L^{X} such that 𝜏_{Hs} = (𝜏_{𝒥} )_{*} = (𝜏_{HJ} )_{*}. with(7) If 𝒥 (𝒥 *(
A )) = 𝒥 *(A ) forA ∈L^{X} , thensuch that
(8) Define
K_{J} (A ) = 𝒥 (A *). ThenK_{J} :L^{X} →L^{X} withis an
L join meet approximation operator.(9) If 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈L^{X} , thenfor
A ∈L^{X} such that 𝜏_{KJ} = (𝜏_{𝒥} )_{*} with(10) If 𝒥 (𝒥 *(
A )) = 𝒥 *(A ) forA ∈L^{X} , thensuch that
(11) Define
M_{J} (A ) = (𝒥 (A ))*. ThenM_{J} :L^{X} →L^{X} withis an
L meet join approximation operator. Moreover, 𝜏_{MJ} = 𝜏_{𝒥} .(12) If 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈L^{X} , then forA ∈L^{X} such that 𝜏_{MJ} = (𝜏_{𝒥} )_{*} with(13) If 𝒥 (𝒥 *(
A )) = 𝒥 *(A ) forA ∈L^{X} , thensuch that
(14) Define
K_{HJ} (A ) = (H_{J} (A ))*. ThenK_{HJ} :L^{X} →L^{X} withis an
L meet join approximation operator. Moreover, 𝜏_{KHJ} = 𝜏_{𝒥} .(15) If 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈L^{X} , thenfor
A ∈L^{X} such that 𝜏_{KHJ} = (𝜏_{𝒥} )_{*} with(16) If for
A ∈L^{X} , thensuch that
(17) Define
M_{HJ} (A ) =H_{J} (A *). ThenM_{HJ} :L^{X} →L^{X} with
is an
L join meet approximation operator. Moreover, 𝜏_{MHJ} = (𝜏_{𝒥} )_{*}.(18) If 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈L^{X} , thenfor
A ∈L^{X} such that 𝜏_{MHJ} = (𝜏_{𝒥} )_{*} with(19) If for
A ∈L^{X} , thensuch that
(20) (
K_{HJ} ,K_{J} ) is a Galois connection;i.e,A ≤ KHJ (B) iff B ≤ KJ (A).
Moreover, 𝜏_{KJ} = (𝜏_{KHJ} )_{*}.
(21) (
M_{J} ,M_{HJ} ) is a dual Galois connection;i.e,MHJ (A) ≤ B iff MJ (B) ≤ A.
Moreover, 𝜏_{MJ} = (𝜏_{MHJ} )_{*}.
Proof. (1) Since , by (J2) and (J3),
(2) Since
iff , we have
(H1) Since
H_{J} (A ) ≤H_{J} (A ) iff A ≤ 𝒥 (H_{J} (A )), we haveA ≤ 𝒥 (H_{J} (A )) ≤H_{J} (A ).(H3) By the definition of
H_{J} , sinceH_{J} (A ) ≤H_{J} (B ) forB ≤A , we haveSince 𝒥 (∨_{i∈𝚪}
H_{J} (A_{i} )) ≥ 𝒥 (H_{J} (A_{i} )) ≥A_{i} , then𝒥 (∨_{i∈𝚪}
H_{J} (A_{i} )) ≥ ∨_{i∈𝚪}A_{i} . ThusThus
H_{J} :L^{X} →L^{X} is anL upper approximation operator. By the definition ofH_{J} , we haveHJ (B) ≤ A iff B ≤ 𝒥 (A).
Since
A ≤ 𝒥 (A ) iffH_{J} (A ) ≤A , we have 𝜏_{HJ} = 𝜏_{𝒥} .(3) Let 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈L^{X} . Since 𝒥 (B ) ≥H_{J} (A ) iff 𝒥 (𝒥 (B )) = 𝒥 (B ) ≥A from the definition ofH_{J} , we have(4) Let 𝒥 *(
A ) ∈ 𝜏_{𝒥} . Since 𝒥 (𝒥 *(A )) = 𝒥 *(A ),𝒥 (𝒥 (
A )) = 𝒥 (𝒥 *(𝒥 *(A ))) = (𝒥 (𝒥 *(A )))* = 𝒥 (A ).Hence 𝒥 (
A ) ∈ 𝜏_{𝒥} ; i.e. 𝒥 *(A ) ∈ (𝜏_{𝒥} )_{*}. Thus, 𝜏_{𝒥} ⊂ (𝜏_{𝒥} )_{*}.Let
A ∈ (𝜏_{𝒥} )_{*}. ThenA * = 𝒥 (A *). Since 𝒥 (A ) = 𝒥 (𝒥 *(A *)) = 𝒥 *(A *) =A , thenA ∈ 𝜏_{𝒥} . Thus, (𝜏_{𝒥} )_{*} ⊂ 𝜏_{𝒥} .(5) (H1) Since 𝒥 (
A *) ≤A *,H_{s} (A ) = 𝒥 (A *)* ≥A .Hence
H_{s} is anL upper approximation operator such thatMoreover, 𝜏_{Hs} = (𝜏_{𝒥} )_{*} from:
A = Hs(A) iff A = 𝒥 (A*)* iff A* = 𝒥 (A*).
(6) Let 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈L^{X} . ThenHence
(7) Let 𝒥 (𝒥 *(
A )) = 𝒥 *(A ) forA ∈L^{X} . ThenHence
By a similar method in (4), 𝜏_{Hs} = (𝜏_{Hs} )_{*}.
(8) It is similarly proved as (5).
(9) If 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈L^{X} , then(10) If 𝒥 (𝒥 *(
A )) = 𝒥 *(A ) forA ∈L^{X} , thenSince ,
Hence 𝜏_{KJ} = {
K_{J} (A ) A ∈L^{X} } = (𝜏_{KJ} )_{*}.(11) , (12), (13) and (14) are similarly proved as (5), (9), (10) and (5), respectively.
(15) If 𝒥 (𝒥 (
A )) = 𝒥 (A ) forA ∈L^{X} , thenH_{J} (H_{J} (A )) =H_{J} (A ). Thus,Since 𝒥 (
A ) =A iffH_{J} (A ) =A iffK_{HJ} (A ) =A *, 𝜏_{KHJ} = (𝜏_{𝒥} )_{*} with(16) If for A ∈
L^{X} , then(17) , (18) and (19) are similarly proved as (14), (15) and (16), respectively.
(20) (
K_{HJ} ,K_{J} ) 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 * ≤K_{J} (A ) iffA ≤K_{HJ} (A *), 𝜏_{KJ} = (𝜏_{KHJ} )_{*}.(21) (
M_{J} ,M_{HJ} ) 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
M_{HJ} (A *) ≤A iffM_{J} (A ) ≤A *, 𝜏_{MJ} = (𝜏_{MHJ} )_{*}.Let
R ∈L ^{X × X} be anL fuzzy relation. Define operators as followsExample 2.2. LetR be a reflexiveL fuzzy relation. Define 𝒥_{R} :L^{X} →L^{X} as follows:(1) (J1) 𝒥_{R}(
A )(y ) ≤R (y, y ) →A (y ) =A (y ): 𝒥_{R} satisfies the conditions (J1) and (J2) from:Hence 𝒥_{R} is an
L lower approximation operator.(2) Define
H_{JR} (B ) = ∨ {A B ≤ 𝒥_{R}(A )}. Sincethen
By Theorem 2.1(2),
H_{JR} =H _{R1} is anL upper approximation operator such that (H_{JR} ,𝒥_{R}) is a residuated connection;i.e.,HJR(A) ≤ B iff A ≤ 𝒥R(B).
Moreover, 𝜏_{HJR} = 𝜏_{𝒥R}.
(3) If
R is anL fuzzy preorder, thenR ^{1} is anL fuzzy preorder. SinceBy Theorem 2.1(3),
H_{JR} (H_{JR} (A )) =H_{JR} (A ): By Theorem 2.1(3), 𝜏_{HJR} = 𝜏_{𝒥R} 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}(𝒥_{R}(
A )) = 𝒥_{R}(A ) forA ∈L^{X} .Thus, 𝜏_{𝒥R} = (𝜏_{𝒥R})_{*} with
(5) Define
H_{s} (A ) = 𝒥_{R}(A *)*. By Theorem 2.1(5),H_{s} =H_{R} is anL upper approximation operator such thatMoreover, 𝜏_{Hs} = 𝜏_{HR} = (𝜏_{HJR} )_{*}.
(6) If
R is anL fuzzy preorder, then 𝒥_{R}(𝒥_{R}(A )) = 𝒥_{R}(A ) forA ∈L^{X} . By Theorem 2.1(6), thenH_{s} (H_{s} (A )) =H_{s} (A ) forA ∈L^{X} such that 𝜏H_{s} = (𝜏_{𝒥R})_{*} = (𝜏_{HJR} )_{*} with(7) If
R is a reflexive and EuclideanL fuzzy relation, then(8) Define
K_{JR} (A ) = 𝒥_{R}(A *). ThenK_{JR} :L^{X} →L^{X} withis an
L join meet approximation operator. Moreover, 𝜏_{KJR} = (𝜏_{𝒥R})_{*}.(9)
R is anL fuzzy preorder, then 𝒥_{R}(𝒥_{R}(A )) = 𝒥_{R}(A ) forA ∈L^{X} . By Theorem 2.1(9), forA ∈L^{X} such that 𝜏_{KJR} = (𝜏_{𝒥R})_{*} with(10) If
R is a reflexive and EuclideanL fuzzy relation, then forA ∈L^{X} . By Theorem 2.1(10), such that(11) Define
M_{JR} (A ) = (𝒥_{R}(A ))*. ThenM_{JR} :L^{X} →L^{X} withis an
L join meet approximation operator. Moreover, 𝜏_{MJR} = 𝜏_{𝒥R}.(12) If
R is anL fuzzy preorder, then 𝒥_{R}(𝒥_{R}(A )) = 𝒥_{R}(A ) forA ∈L^{X} . By Theorem 2.1(12), forA ∈L^{X} such that 𝜏_{MJR} = 𝜏_{𝒥R} with(13) If
R is a reflexive and EuclideanL fuzzy relation, then forA ∈L^{X} . By Theorem 2.1(13), such that(14) Define
K_{HJR} (A ) = (H_{JR} (A ))*. ThenKHJR : LX → LX
with
is an
L join meet approximation operator. Moreover, 𝜏_{KR1} = 𝜏_{𝒥R} = 𝜏_{HR1} .(15) If
R is anL fuzzy preorder, then 𝒥_{R}(𝒥_{R}(A )) = 𝒥_{R}(A ) forA ∈L^{X} . By Theorem 2.1(15), forA ∈L^{X} such that 𝜏_{KR1} = 𝜏_{𝒥R} = 𝜏_{HR1} with(16) Let
R ^{1} be a reflexive and EuclideanL fuzzy relation. Sincewe have
Thus,
Hence
By (K1), such that
(17) Define
M_{HJR} (A ) =H_{JR} (A *). ThenMHJR : LX → LX
is an
L meet join approximation operator as follows:Moreover, 𝜏_{MHJR} = (𝜏_{𝒥R})_{*}.
(18) If
R is anL fuzzy preorder, then 𝒥_{R}(𝒥_{R}(A )) = 𝒥_{R}(A ) forA ∈L^{X} . By Theorem 2.1(18),for
A ∈L^{X} such that 𝜏_{MHJR} = (𝜏_{𝒥} )_{*} 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) (
K_{HJR} =K _{R1*} ,K_{JR} =K _{R*} ) is a Galois connection; i.e,A ≤K_{HJR} (B ) iffB ≤K_{JR} (A ): Moreover, 𝜏_{KJR} = (𝜏_{KHJR} )_{*}.(21) (
M_{JR} =M_{R} ,M_{HJR} =M _{R1} ) is a dual Galois connection; i.e,M_{HJR} (A ) ≤B iffM_{JR} (B ) ≤A . Moreover, 𝜏_{MJR} = (𝜏_{MHJR} )_{*}.3. Conclusions
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.Conflict of Interest
No potential conflict of interest relevant to this article was reported.

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