The Properties of L-lower Approximation Operators

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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 Alexandrov L-topologies. Moreover, we give their examples as approximation operators induced by various L-fuzzy relations.


  • KEYWORD

    Complete residuated lattices , L-upper approximation operators , Alexandrov L-topologies

  • 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 Alexandrov L-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 Alexandrov L-topologies. Moreover, we give their examples as approximation operators induced by various L-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) xyz iff xyz for x, y, zL

    Remark 1.2. [3, 5] (1) A completely distributive lattice L = (L,≤,∨,∧ = ⊙,→, 1, 0) is a complete residuated lattice defined by

    image

    (2) The unit interval with a left-continuous t-norm ⊙,

    image

    is a complete residuated lattice defined by

    image

    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,⊤xLX,

    image

    and

    image

    Lemma 1.3. [3, 5] For each x, y, z, xi, yiL, we have the following properties.

    image

    Definition 1.4. [8, 9]

    (1) A map H : LXLX is called an L-upper approximation operator iff it satisfies the following conditions

    image

    (2) A map 𝒥 : LXLX is called an L-lower approximation operator iff it satisfies the following conditions

    image

    (3) A map K : LXLX is called an L-join meet approximation operator iff it satisfies the following conditions

    image

    (4) A map M : LXLX is called an L-meet join approximation operator iff it satisfies the following conditions

    image

    Definition 1.5. [6, 9] A subset 𝜏 ⊂ LX is called an Alexandrov L-topology if it satisfies:

    image

    Theorem 1.6. [8, 9]

    (1) 𝜏 is an Alexandrov topology on X iff 𝜏 = {A* ∈ LX | A ∈ 𝜏} is an Alexandrov topology on X.

    (2) If H is an L-upper approximation operator, then 𝜏H = {ALX | H(A) = A} is an Alexandrov topology on X.

    (3) If 𝒥 is an L-lower approximation operator, then 𝜏𝒥 = {ALX | 𝒥 (A) = A} is an Alexandrov topology on X.

    (4) If K is an L-join meet approximation operator, then 𝜏K = {ALX | K(A) = A*} is an Alexandrov topology on X.

    (5) If M is an L-meet join operator, then 𝜏M = {ALX | M(A) = A*} is an Alexandrov topology on X.

    Definition 1.7. [8, 9] Let X be a set. A function R : X × XL is called:

    image

    If R satisfies (R1) and (R3), R is called a L-fuzzy preorder.

    If R satisfies (R1), (R2) and (R3), R is called a L-fuzzy equivalence relation

    2. The Properties of L-lower Approximation Operators

    Theorem 2.1. Let 𝒥 : LXLX be an L-lower approximation operator. Then the following properties hold.

    (1) For ALX,.

    (2) Define HJ (B) = ∧{A | B ≤ 𝒥 (A)}. Then HJ : LXLX with

    image

    is an L-upper approximation operator such that (HJ ,𝒥 )

    is a residuated connection;i.e.,

    HJ (B) ≤ A iff B ≤ 𝒥 (A).

    Moreover, 𝜏𝒥 = 𝜏HJ .

    (3) If 𝒥 (𝒥 (A)) = 𝒥 (A) for ALX, then HJ (HJ (A)) = HJ (A) for ALX such that 𝜏𝒥 = 𝜏HJ with

    image

    (4) If 𝒥 (𝒥 *(A)) = 𝒥 *(A) for ALX, then 𝒥 (𝒥 (A)) = 𝒥 (A) such that

    image

    (5) Define Hs(A) = 𝒥 (A*)*. Then Hs : LXLX with

    image

    is an L-upper approximation operator. Moreover, 𝜏Hs = (𝜏𝒥 )* = (𝜏HJ )*.

    (6) If 𝒥 (𝒥 (A)) = 𝒥 (A) for ALX, then

    Hs(Hs(A)) = Hs(A)

    for ALX such that 𝜏Hs = (𝜏𝒥 )* = (𝜏HJ )*. with

    image

    (7) If 𝒥 (𝒥 *(A)) = 𝒥 *(A) for ALX, then

    image

    such that

    image

    (8) Define KJ (A) = 𝒥 (A*). Then KJ : LXLX with

    image

    is an L-join meet approximation operator.

    (9) If 𝒥 (𝒥 (A)) = 𝒥 (A) for ALX, then

    image

    for ALX such that 𝜏KJ = (𝜏𝒥 )* with

    image

    (10) If 𝒥 (𝒥 *(A)) = 𝒥 *(A) for ALX, then

    image

    such that

    image

    (11) Define MJ (A) = (𝒥 (A))*. Then MJ : LXLX with

    image

    is an L-meet join approximation operator. Moreover, 𝜏MJ = 𝜏𝒥 .

    (12) If 𝒥 (𝒥 (A)) = 𝒥 (A) for ALX, then for ALX such that 𝜏MJ = (𝜏𝒥 )* with

    image

    (13) If 𝒥 (𝒥 *(A)) = 𝒥 *(A) for ALX, then

    image

    such that

    image

    (14) Define KHJ (A) = (HJ (A))*. Then KHJ : LXLX with

    image

    is an L-meet join approximation operator. Moreover, 𝜏KHJ = 𝜏𝒥 .

    (15) If 𝒥 (𝒥 (A)) = 𝒥 (A) for ALX, then

    image

    for ALX such that 𝜏KHJ = (𝜏𝒥 )* with

    image

    (16) If for ALX, then

    image

    such that

    image

    (17) Define MHJ (A) = HJ (A*). Then MHJ : LXLX

    with

    image

    is an L-join meet approximation operator. Moreover, 𝜏MHJ = (𝜏𝒥 )*.

    (18) If 𝒥 (𝒥 (A)) = 𝒥 (A) for ALX, then

    image

    for ALX such that 𝜏MHJ = (𝜏𝒥 )* with

    image

    (19) If for ALX, then

    image

    such that

    image

    (20) (KHJ ,KJ ) is a Galois connection;i.e,

    A ≤ KHJ (B) iff B ≤ KJ (A).

    Moreover, 𝜏KJ = (𝜏KHJ )*.

    (21) (MJ ,MHJ ) is a dual Galois connection;i.e,

    MHJ (A) ≤ B iff MJ (B) ≤ A.

    Moreover, 𝜏MJ = (𝜏MHJ )*.

    Proof.

    (1) Since , by (J2) and (J3),

    image

    (2) Since

    iff , we have

    image

    (H1) Since HJ (A) ≤ HJ (A) iff A ≤ 𝒥 (HJ (A)), we have A ≤ 𝒥 (HJ (A)) ≤ HJ (A).

    image

    (H3) By the definition of HJ , since HJ (A) ≤ HJ (B) for BA, we have

    image

    Since 𝒥 (∨i∈𝚪 HJ (Ai)) ≥ 𝒥 (HJ (Ai)) ≥ Ai, then

    𝒥 (∨i∈𝚪 HJ (Ai)) ≥ ∨i∈𝚪 Ai. Thus

    image

    Thus HJ : LXLX is an L-upper approximation operator. By the definition of HJ , we have

    HJ (B) ≤ A iff B ≤ 𝒥 (A).

    Since A ≤ 𝒥 (A) iff HJ (A) ≤ A, we have 𝜏HJ = 𝜏𝒥 .

    (3) Let 𝒥 (𝒥 (A)) = 𝒥 (A) for ALX. Since 𝒥 (B) ≥ HJ (A) iff 𝒥 (𝒥 (B)) = 𝒥 (B) ≥ A from the definition of HJ , we have

    image

    (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 )*. Then A* = 𝒥 (A*). Since 𝒥 (A) = 𝒥 (𝒥 *(A*)) = 𝒥 *(A*) = A, then A ∈ 𝜏𝒥 . Thus, (𝜏𝒥 )* ⊂ 𝜏𝒥 .

    (5) (H1) Since 𝒥 (A*) ≤ A*, Hs(A) = 𝒥 (A*)* ≥ A.

    image
    image

    Hence Hs is an L-upper approximation operator such that

    image

    Moreover, 𝜏Hs = (𝜏𝒥 )* from:

    A = Hs(A) iff A = 𝒥 (A*)* iff A* = 𝒥 (A*).

    (6) Let 𝒥 (𝒥 (A)) = 𝒥 (A) for ALX. Then

    image

    Hence

    (7) Let 𝒥 (𝒥 *(A)) = 𝒥 *(A) for ALX. Then

    image

    Hence

    image

    By a similar method in (4), 𝜏Hs = (𝜏Hs )*.

    (8) It is similarly proved as (5).

    (9) If 𝒥 (𝒥 (A)) = 𝒥 (A) for ALX, then

    image

    (10) If 𝒥 (𝒥 *(A)) = 𝒥 *(A) for ALX, then

    image

    Since ,

    image

    Hence 𝜏KJ = {KJ (A) | ALX} = (𝜏KJ )*.

    (11) , (12), (13) and (14) are similarly proved as (5), (9), (10) and (5), respectively.

    (15) If 𝒥 (𝒥 (A)) = 𝒥 (A) for ALX, then HJ (HJ (A)) = HJ (A). Thus,

    image

    Since 𝒥 (A) = A iff HJ (A) = A iff KHJ (A) = A*, 𝜏KHJ = (𝜏𝒥 )* with

    image

    (16) If for A ∈ LX, then

    image

    (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) iff AKHJ (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 iff MJ (A) ≤ A*, 𝜏MJ = (𝜏MHJ )*.

    Let R LX × X be an L-fuzzy relation. Define operators as follows

    image

    Example 2.2. Let R be a reflexive L-fuzzy relation. Define 𝒥R : LXLX as follows:

    image

    (1) (J1) 𝒥R(A)(y) ≤ R(y, y) → A(y) = A(y): 𝒥R satisfies the conditions (J1) and (J2) from:

    image

    Hence 𝒥R is an L-lower approximation operator.

    (2) Define HJR(B) = ∨ {A | B ≤ 𝒥R(A)}. Since

    image

    then

    image

    By Theorem 2.1(2), HJR = HR-1 is an L-upper approximation operator such that (HJR,𝒥R) is a residuated connection;i.e.,

    HJR(A) ≤ B iff A ≤ 𝒥R(B).

    Moreover, 𝜏HJR = 𝜏𝒥R.

    (3) If R is an L-fuzzy preorder, then R-1 is an L-fuzzy preorder. Since

    image

    By Theorem 2.1(3), HJR(HJR(A)) = HJR(A): By Theorem 2.1(3), 𝜏HJR = 𝜏𝒥R with

    image

    (4) Let R be a reflexive and Euclidean L-fuzzy relation. Since R(x, z) ⊙ R(y, z) ⊙ A*(x) ≤ R(x, y) ⊙ A*(x) iff R(x, z) ⊙ A*(x) ≤ R(y, z) → R(x, y) ≤ A*(x),

    image

    Thus, .

    By Theorem 2.1(4), 𝒥R(𝒥R(A)) = 𝒥R(A) for ALX.

    Thus, 𝜏𝒥R = (𝜏𝒥R)* with

    image

    (5) Define Hs(A) = 𝒥R(A*)*. By Theorem 2.1(5), Hs = HR is an L-upper approximation operator such that

    image

    Moreover, 𝜏Hs = 𝜏HR = (𝜏HJR )*.

    (6) If R is an L-fuzzy preorder, then 𝒥R(𝒥R(A)) = 𝒥R(A) for ALX. By Theorem 2.1(6), then Hs(Hs(A)) = Hs(A) for ALX such that 𝜏Hs = (𝜏𝒥R)* = (𝜏HJR )* with

    image

    (7) If R is a reflexive and Euclidean L-fuzzy relation, then

    image
    image
    image

    (8) Define KJR(A) = 𝒥R(A*). Then KJR : LXLX with

    image

    is an L-join meet approximation operator. Moreover, 𝜏KJR = (𝜏𝒥R)*.

    (9) R is an L-fuzzy preorder, then 𝒥R(𝒥R(A)) = 𝒥R(A) for ALX. By Theorem 2.1(9), for ALX such that 𝜏KJR = (𝜏𝒥R)* with

    image

    (10) If R is a reflexive and Euclidean L-fuzzy relation, then for ALX. By Theorem 2.1(10), such that

    image

    (11) Define MJR(A) = (𝒥R(A))*. Then MJR : LXLX with

    image

    is an L-join meet approximation operator. Moreover, 𝜏MJR = 𝜏𝒥R.

    (12) If R is an L-fuzzy preorder, then 𝒥R(𝒥R(A)) = 𝒥R(A) for ALX. By Theorem 2.1(12), for ALX such that 𝜏MJR = 𝜏𝒥R with

    image

    (13) If R is a reflexive and Euclidean L-fuzzy relation, then for ALX. By Theorem 2.1(13), such that

    image

    (14) Define KHJR (A) = (HJR(A))*. Then

    KHJR : LX → LX

    with

    image

    is an L-join meet approximation operator. Moreover, 𝜏KR-1 = 𝜏𝒥R = 𝜏HR-1 .

    (15) If R is an L-fuzzy preorder, then 𝒥R(𝒥R(A)) = 𝒥R(A) for ALX. By Theorem 2.1(15), for ALX such that 𝜏KR-1 = 𝜏𝒥R = 𝜏HR-1 with

    image

    (16) Let R-1 be a reflexive and Euclidean L-fuzzy relation. Since

    image

    we have

    image

    Thus,

    image

    Hence

    image

    By (K1), such that

    image

    (17) Define MHJR (A) = HJR(A*). Then

    MHJR : LX → LX

    is an L-meet join approximation operator as follows:

    image

    Moreover, 𝜏MHJR = (𝜏𝒥R)*.

    (18) If R is an L-fuzzy preorder, then 𝒥R(𝒥R(A)) = 𝒥R(A) for ALX. By Theorem 2.1(18),

    image

    for ALX such that 𝜏MHJR = (𝜏𝒥 )* with

    image

    (19) Let R-1 be a reflexive and Euclidean L-fuzzy relation.

    Since

    image

    then (R(y, x) → A(x)) ⊙ R(z, y) ≤ R(z, x) → A(x).

    Thus,

    image

    By (M1), such that

    image

    (20) (KHJR = KR-1* ,KJR = KR* ) is a Galois connection; i.e, AKHJR (B) iff BKJR(A): Moreover, 𝜏KJR = (𝜏KHJR )*.

    (21) (MJR = MR,MHJR = MR-1 ) is a dual Galois connection; i.e, MHJR (A) ≤ B iff MJR(B) ≤ A. Moreover, 𝜏MJR = (𝜏MHJR )*.

    3. Conclusions

    In this paper, L-lower approximation operators induce L-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 Alexandrov L-topologies. Moreover, we give their examples as approximation operators induced by various L-fuzzy relations.

    Conflict of Interest

    No potential conflict of interest relevant to this article was reported.

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