### The Properties of L-lower Approximation Operators

• • #### 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  introduced a complete residuated lattice which is an algebraic structure for many valued logic. Radzikowska and Kerre  developed fuzzy rough sets in complete residuated lattice. Bělohlávek  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

(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,⊤xLX,

and

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

Definition 1.4. [8, 9]

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

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

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

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

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

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:

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

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

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

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

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

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

such that

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

is an L-join meet approximation operator.

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

for ALX such that 𝜏KJ = (𝜏𝒥 )* with

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

such that

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

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

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

such that

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

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

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

for ALX such that 𝜏KHJ = (𝜏𝒥 )* with

(16) If for ALX, then

such that

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

with

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

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

for ALX such that 𝜏MHJ = (𝜏𝒥 )* with

(19) If for ALX, then

such that

(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),

(2) Since

iff , we have

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

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

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

𝒥 (∨i∈𝚪 HJ (Ai)) ≥ ∨i∈𝚪 Ai. Thus

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

(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.

Hence Hs is an L-upper approximation operator such that

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

Hence

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

Hence

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

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

Since ,

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,

Since 𝒥 (A) = A iff HJ (A) = A iff KHJ (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) 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

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

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

Hence 𝒥R is an L-lower approximation operator.

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

then

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

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

(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),

Thus, .

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

Thus, 𝜏𝒥R = (𝜏𝒥R)* with

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

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

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

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

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

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

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

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

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

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

KHJR : LX → LX

with

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

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

we have

Thus,

Hence

By (K1), such that

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

MHJR : LX → LX

is an L-meet join approximation operator as follows:

Moreover, 𝜏MHJR = (𝜏𝒥R)*.

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

for ALX such that 𝜏MHJR = (𝜏𝒥 )* with

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

Since

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

Thus,

By (M1), such that

(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

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