# The Lattice Structures of Approximation Operators Based on L-Fuzzy Generalized Neighborhood Systems

Authors: Qiao-Ling Song; Hu Zhao; Juan-Juan Zhang; A. A. Ramadan; Hong-Ying Zhang; Gui-Xiu Chen
Journal: Complexity
 (2021)

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## Abstract

Following the idea ofL-fuzzy generalized neighborhood systems as introduced by Zhao et al., we will give the join-complete lattice structures of lower and upper approximation operators based on L-fuzzy generalized neighborhood systems. In particular, as special approximation operators based on L-fuzzy generalized neighborhood systems, we will give the complete lattice structures of lower and upper approximation operators based on L-fuzzy relations. Furthermore, if L satisfies the double negative law, then there exists an order isomorphic mapping between upper and lower approximation operators based on L-fuzzy generalized neighborhood systems; when L-fuzzy generalized neighborhood system is serial, reflexive, and transitive, there still exists an order isomorphic mapping between upper and lower approximation operators, respectively, and both lower and upper approximation operators based on L-fuzzy relations are complete lattice isomorphism.


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## Body

## 1. Introduction

Pawlak [1, 2] defined the rough set theory to address the vagueness and granularity of information systems and data analysis. Many scholars worked on this theory and applied it to various fields [3–10]. The classical rough set theory is based on partition or equivalent relation. Thus, classical rough set theory has been extended to binary relation-based rough sets [11–14] and covering-based rough sets [15–19].Fuzzy set theory is also an important mathematical tool to study uncertainty. Nowadays, there have been many branches of fuzzy mathematics, such as fuzzy algebra, fuzzy topology, and fuzzy logic [20–31]. Particularly, fuzzy rough set theory is an important branch, which can handle more complicated uncertain problems since it has the advantages of both fuzzy set and rough set [22, 32–38]. Furthermore, replacing the unit interval 0,1 with a complete lattice L as the range of the membership function [39], the more general L-fuzzy rough sets further extend the theoretical framework and application range of classic rough sets [31, 40–46]. Fuzzy rough sets have a variety of forms due to the different approaches of fuzzification. It is easily observed that the rough sets based on L-fuzzy relations and L-fuzzy coverings are the two most well-known L-fuzzy rough sets.Recently, the notion of generalized neighborhood systems is proposed and used to define a theory of rough set, called generalized neighborhood system-based rough sets [45, 47–49]. In [49, 50], Zhao and Li gave an axiomatic characterization on generalized neighborhood system-based rough sets. Hence, it is natural to establish fuzzy general rough sets by fuzzifying them, respectively. Quite recently, by fuzzifying the notion of generalized neighborhood systems, Zhao and Li [49, 50] established a rough set model based on L-fuzzy generalized neighborhood systems. It was proved that this model brought the fuzzy relation-based rough set model, fuzzy covering-based rough set model, and generalized neighborhood system-based rough set models under a unified framework. However, the lattice structures of approximation operators based on L-fuzzy generalized neighborhood systems were not studied. Following this idea, a natural problem arises: can the lattice structures of approximation operators based on L-fuzzy generalized neighborhood systems be given?In the present paper, we study the lattice structures of approximation operators based onL-fuzzy generalized neighborhood systems (resp., L-fuzzy relations) and give the relationship between lower and upper approximation operators based on L-fuzzy generalized neighborhood systems (resp., L-fuzzy relations).The contents of this paper are organized as follows. In Section2, we recall some notions and notations used in this paper. In Section 3, the lattice structures of approximation operators based on L-fuzzy generalized neighborhood system operators are given. In Section 4, the lattice structures of lower and upper approximation operators based on L-fuzzy relations are discussed. In Section 5, we make a conclusion.


## 2. Preliminaries

In this section, we recall some basic notions and notations used in this paper.A commutative quantale is a pairL,∗, where L is a complete lattice with respect to a partial order ≤ on it, with the top (resp., bottom) element 1 (resp., 0), and ∗ is a commutative semigroup operation on L such that a∗∨j∈Jbj=∨j∈Ja∗bj for all a∈L and bj|j∈J⊆L. L,∗ is said to be integral if the top element 1 is the unique unit in the sense of 1∗a=a for all a∈L.In this paper, if not otherwise specified,L=L,∗ is always assumed to be an integral, commutative quantale. Since the binary operation ∗ distributes over arbitrary joins, the function a∗−:L⟶L has a right adjoint a⟶−:L⟶L given by a⟶b=∨c∈L:a∗c≤b. We collect here some basic properties of the binary operations ∗ and ⟶ [51–53]. L is said to satisfy the double negation law if for any a∈L, a⟶0⟶0=a, and we use ¬a to denote a⟶0. If L satisfies the double negative law, then (1) a⟶b=¬b⟶¬a; (2) a⟶b=¬a∗¬b; (3) ¬∧j∈Jaj=∨j∈J¬aj.Proposition 1.
(1)
a⟶b=1⟺a≤b.(2)
a∗a⟶b≤b.(3)
a⟶b⟶c=a∗b⟶c=b⟶a⟶c.(4)
a≤a⟶b⟶b.(5)
∨j∈Jaj⟶b=∧j∈Jaj⟶b.(6)
a⟶∧j∈Jbj=∧j∈Ja⟶bj.(7)
∀a∈L,a=∧b∈La⟶b⟶b.Let the universe of discourseX be an arbitrary nonempty set. We call a function A:X⟶L as an L-fuzzy set in X. We use LX to denote the set of all L-fuzzy sets in X and call it the L-fuzzy power set on X. For a crisp subset A⊂X, let χA be the characteristic function, i.e., χAx=1 if x∈A and χAx=0 if x∉A. The characteristic function χA of a subset A⊆X can be regarded as an L-fuzzy set in X. We make no difference between a constant L-fuzzy set and its value since no confusion will arise.The operators∨,∧,∗,⟶ on L can be translated onto LX in a point-wise. That is, for any A,B,Att∈T∈LX,(1)∨t∈TAtx=∨t∈TAtx,∧t∈TAtx=∧t∈TAtx,A∗Bx=Ax∗Bx,A⟶Bx=Ax⟶Bx,¬Ax=¬Ax.LetA,B be L-fuzzy set in X. The subsethood degree [20] of A, B, denoted by SA,B, is defined by SA,B=∧x∈XAx⟶Bx. SA,B represents the semantics “A is contained in B.”LetA,B be L-fuzzy set in X. The intersection degree [32] of A, B, denoted by IA,B, is defined by IA,B=∨x∈XAx∗Bx. IA,B represents the semantics “A and B have intersection.”The lemma below collects some properties of subsethood degree and intersection degree. They can be found in many literatures such as [6, 22–24, 30, 32, 52].Lemma 1.
LetA,B,C,D,Att∈T∈LX. Then,(S1)SA,B=1⟺A≤B.(S2)A≤B⇒SA,C≥SB,C and SC,A≤SC,B.(S3)SA,B∗SC,D≤SA∗C,B∗D.(S4)SA,B∗SB,C≤SA,C.(S5)SA,∧t∈TAt=∧t∈TSA,At.(S6)SA,α⟶B=α⟶SA,B.(S7)SA,α∗B≥α∗SA,B.(I1)IA,B=IB,A.(I2)IA,∨t∈TAt=∨t∈TIA,At.(I3)IA,α∗B=α∗IA,B.(I4)IA,α⟶B≤α⟶IA,B.(IS)IA,B⟶α=SA,B⟶α.Next, we recall the notions ofL-fuzzy generalized neighborhood systems and rough approximation operators based L-fuzzy generalized neighborhood systems.Definition 1 (see [49, 50]).
LetX be a set; the function N:X⟶LLX is called an L-fuzzy generalized neighborhood system operator on X, if for any x∈X, ∨C∈LXNxC=1. Usually, Nx is called an L-fuzzy generalized neighborhood system of x and NxC is interpreted as the degree of that C which is a neighborhood of x. The condition“ ∨C∈LXNxC=1” is a lattice-valued interpretation of “Nx is nonempty.” In particular,(1)
AnL-fuzzy generalized neighborhood system operator N is said to be serial (denoted by SE), if for any x∈X and C∈LX, NxC≤∨y∈XCy.(2)
AnL-fuzzy generalized neighborhood system operator N is said to be reflexive (denoted by RE), if for any x∈X and C∈LX, NxC≤Cx.(3)
AnL-fuzzy generalized neighborhood system operator N is said to be transitive (denoted by TR), if for any x∈X and C∈LX,(2)NxC≤∨V∈LXNxV∗∧y∈XVy⟶∨Vy∈LXNyVy∗SVy,C.Definition 2 (see [49, 50]).
LetN:X⟶LLX be an L-fuzzy generalized neighborhood system operator. Then, for each A∈LX, the lower and upper approximation operators N¯A and N¯A are defined as follows:(3)N¯Ax=∨C∈LXNxC∗SC,A,N¯Ax=∧C∈LXNxC→IC,A.
Finally, we recall the notions ofL-fuzzy relations and rough approximation operator-based L-fuzzy relations.Definition 3 (see [36, 37]).
AnL-fuzzy relation on X is a mapping R:X×X⟶L. For any x,y∈X×X, Rx,y is interpreted as the related degree between x and y. Furthermore,(1)
R is said to be serial if for all x∈X, ∨y∈XRx,y=1.(2)
R is said to be reflexive if for all x∈X, Rx,x=1.(3)
R is said to be transitive if for all x,y,z∈X, Rx,y∗Ry,z≤Rx,z.Definition 4 (see [36, 37]).
LetR:X×X⟶L be an L-fuzzy relation on X. Then, for any A∈LX, the lower and upper approximation operators R¯A and R¯A are defined as follows:(4)R¯Ax=∧y∈XRx,y⟶Ay,R¯Ax=∨y∈XRx,y∗Ay.Lemma 2 (see [50]).
LetR:X×X⟶L be an L-fuzzy relation on X. Define an L-fuzzy generalized neighborhood system operator NR as follows: for any x∈X and A∈LX,(5)NRxA=1,ifA=Rx,−;0,otherwise.Then,NR¯B=R¯B and NR¯B=R¯B for any B∈LX.


## 3. The Lattice Structures of Approximation Operators Based onL-Fuzzy Generalized Neighborhood Systems

In this section, we will study the lattice structures of lower and upper approximation operators based onL-fuzzy generalized neighborhood systems.Let(6)L−FGNSX=N:X⟶LLX|∀x∈X,∨C∈LXNxC=1,be the family of all L-fuzzy generalized neighborhood system operators on X, and let(7)SE−LFGNSX=L−FGNSX∩A,RE−LFGNSX=L−FGNSX∩B,TR−LFGNSX=L−FGNSX∩ℂ.be the family of all serial, reflexive, and transitive L-fuzzy generalized neighborhood system operators on X, respectively, where(8)A=N:X→LLX∣∀x∈X,C∈LX,NxC≤∨y∈XCy,B=N:X→LLX∣∀x∈X,C∈LX,NxC≤Cx,ℂ=N:X→LLX∣∀x∈X,C,∈LX,NxC≤∨V∈LXNxV∗∧y∈XVy→∨Vy∈LXNyVy∗SVy,C.LetLAOX=N¯:LX⟶LX|N∈L−FGNSX and UAOX=N¯:LX⟶LX|N∈L−FGNSX be the family of all lower and upper approximation operators on X, respectively. Furthermore, let(9)SE−LAOX=N¯:LX⟶LX|N∈SE−LFGNSX,SE−UAOX=N¯:LX⟶LX|N∈SE−LFGNSX,be the family of all serial lower and upper approximation operators on X, respectively. Let RE−LAOX=N¯:LX⟶LX|N∈RE−LFGNSX and RE−UAOX=N¯:LX⟶LX|N∈RE−LFGNSX be the family of all reflexive lower and upper approximation operators on X, respectively. Let TR−LAOX=N¯:LX⟶LX|N∈TR−LFGNSX and TR−UAOX=N¯:LX⟶LX|N∈TR−LFGNSX be the family of all transitive lower and upper approximation operators on X, respectively.Obviously,(10)SE−LAOX,RE−LAOX,TR−LAOX⊆LAOX,SE−UAOX,RE−UAOX,TR−UAOX⊆UAOX.Theorem 1.
(1)
Define a relation⊆ on L−FGNSX as follows: N1⊆N2 if and only if N1xC≤N2xC for any x∈X and C∈LX. Then, L−FGNSX,⊆ is a poset.(2)
Define a relation≤ on LAOX as follows: N1¯≤N2¯ if and only if N1¯Bx≤N2¯Bx for any x∈X and B∈LX. Then, LAOX,≤ is a poset.(3)
Define a relation≤ on UAOX as follows: N1¯≤N2¯ if and only if N2¯Bx≤N1¯Bx for any x∈X and B∈LX. Then, UAOX,≤ is a poset.Proof.
The proof is straightforward and is omitted.Proposition 2.
LetΛ be an index set; ∀Nii∈Λ⊆L−FGNSX,⊆, we can define union of Ni as follows:(11)∪i∈ΛNixC=∨i∈ΛNixC,for any x∈X and C∈LX; then, ∪i∈ΛNi∈L−FGNSX. Furthermore,(1)
IfNii∈Λ⊆SE−LFGNSX, then ∪i∈ΛNi∈SE−LFGNSX.(2)
IfNii∈Λ⊆RE−LFGNSX, then ∪i∈ΛNi∈RE−LFGNSX.(3)
IfNii∈Λ⊆TR−LFGNSX, then ∪i∈ΛNi∈TR−LFGNSX.Proof.
The proof of (1) and (2) are straightforward and are omitted. (3) Note thatTR−LFGNSX⊆L−FGNSX. So, if Nii∈Λ⊆TR−LFGNSX, then there must be ∪i∈ΛNi∈L−FGNSX.(12)∀x∈X,C∈LX∪i∈ΛNixC=∨i∈ΛNixC≤∨i∈Λ∨V∈LXNixV∗∧y∈XVy⟶∨Vy∈LXNiyVy∗SVy,C=∨V∈LX∨i∈ΛNixV∗∧y∈XVy⟶∨Vy∈LXNiyVy∗SVy,C≤∨V∈LX∨i,j∈Λ×ΛNixV∗∧y∈XVy⟶∨Vy∈LXNiyVy∗SVy,C=∨V∈LX∨i∈ΛNixV∗∨j∈Λ∧y∈XVy⟶∨Vy∈LXNiyVy∗SVy,C≤∨V∈LX∨i∈ΛNixV∗∧y∈XVy⟶∨i∈Λ∨Vy∈LXNiyVy∗SVy,C=∨V∈LX∨i∈ΛNixV∗∧y∈XVy⟶∨Vy∈LX∨i∈ΛNiyVy∗SVy,C=∨V∈LX∪i∈ΛNixV∗∧y∈XVy⟶∨Vy∈LX∪i∈ΛNiyVy∗SVy,C.
Thus,∪i∈ΛNi∈ℂ. Therefore, ∪i∈ΛNi∈L−FGNSX∩ℂ=TR−LFGNSX.
By Proposition2, we obtain the following theorem.Theorem 2.
L−FGNSX,⊆,∪, SE−LFGNSX,⊆,∪, RE−LFGNSX,⊆,∪, and TR−LFGNSX,⊆,∪ are all join-complete lattices.Theorem 3.
(1)
∀Ni¯i∈Λ⊆LAOX,≤ and Λ be an index set, we can define union of Ni¯ as follows:(13)∨i∈ΛNi¯=∪i∈ΛNi¯.Then,∨i∈ΛNi¯=∪i∈ΛNi¯ is the supremum of Ni¯i∈Λ.(2)
∀Ni¯i∈Λ⊆UAOX,≤ and Λ be an index set, we can define union of Ni¯ as follows:(14)∨i∈ΛNi¯=∪i∈ΛNi¯.
Then,∨i∈ΛNi¯=∪i∈ΛNi¯ is the supremum of Ni¯i∈Λ.Proof.
Step 1. We first prove that the following equation holds: for any x∈X and B∈LX,(15)∪i∈ΛNi¯Bx=∨i∈ΛNi¯Bx,∪i∈ΛNi¯Bx=∧i∈ΛNi¯Bx.In fact,(16)∪i∈ΛNi¯Bx=∨C∈LX∪i∈ΛNixC∗SC,B=∨C∈LX∨i∈ΛNixC∗SC,B=∨C∈LX∨i∈ΛNixC∗SC,B=∨i∈Λ∨C∈LXNixC∗SC,B=∨i∈ΛNi¯Bx,∪i∈ΛNi¯Bx=∧C∈LX∪i∈ΛNixC⟶IC,B=∧C∈LX∨i∈ΛNixC⟶IC,B=∧C∈LX∧i∈ΛNixC⟶IC,B=∧i∈Λ∧C∈LXNixC⟶IC,B=∧i∈ΛNi¯Bx.Step 2(1)
LetN=∪i∈ΛNi; then, Ni⊆N for each i∈Λ. By Step 1, we have Ni¯≤N¯. If N∗ is another L-fuzzy generalized neighborhood system operator on X such that Ni¯≤N∗¯ for each i∈Λ, then Ni¯Bx≤N∗¯Bx for any x,B∈LX. Hence,(17)N¯Bx=∪i∈ΛNi¯Bx=∨i∈ΛNi¯Bx≤N∗¯Bx.Thus,N¯≤N∗¯. So, ∨i∈ΛNi¯=N¯=∪i∈ΛNi¯ is the supremum of Ni¯i∈Λ.(2)
LetN=∪i∈ΛNi; then, Ni⊆N for each i∈Λ. By Step 1, we have Ni¯≤N¯. If N∗ is another L-fuzzy generalized neighborhood system operator on X such that Ni¯≤N∗¯ for each i∈Λ, then N∗¯Bx≤Ni¯Bx for any x,B∈LX. Hence,(18)N∗¯Bx≤∧i∈ΛNi¯Bx=∪i∈ΛNi¯Bx=N¯Bx.
Thus,N¯≤N∗¯. So, ∨i∈ΛNi¯=N¯=∪i∈ΛNi¯ is the supremum of Ni¯i∈Λ.
By Proposition2 and Theorems 2 and 3, we can obtain Theorem 4 as follows.Theorem 4.
The posetsLAOX,≤,∨, UAOX,≤,∨, SE−LAOX,≤,∨, SE−UAOX,≤,∨, RE−LAOX,≤,∨, RE−UAOX,≤,∨, TR−LAOX,≤,∨, and TR−UAOX,≤,∨ are all join-complete lattices.Furthermore, we have the following theorem.Theorem 5.
(1)
IfL satisfies the double negative law, then the join-complete lattices LAOX,≤,∨ and UAOX,≤,∨ are order isomorphic.(2)
IfL satisfies the double negative law, when L-fuzzy generalized neighborhood system is serial, reflexive, and transitive, there still exists an order isomorphic mapping between upper and lower approximation operators, respectively.Proof.
We only show (1).Step 1. Define a mapping f:LAOX⟶UAOX as follows: for any A∈LX and for any x∈X, fN¯Ax=¬N¯¬Ax for each N¯∈LAOX; then, fN¯∈UAOX. Define a mapping g:UAOX⟶LAOX as follows: for any A∈LX and for any x∈X, gN¯Ax=¬N¯¬Ax for each N¯∈UAOX; then, gN¯∈LAOX.In fact, by Theorem 3.9 of [50], if L satisfies the double negative law, then we have fN¯Ax=¬N¯¬Ax=N¯Ax for any A∈LX and for any x∈X. So, fN¯=N¯∈UAOX. Similarly, we can prove that gN¯∈LAOX.Step 2.f∘g=idUAOX and g∘f=idLAOX, where idUAOX (resp., idLAOX) is the identity mapping on UAOX (resp., LAOX).In fact,∀N¯∈UAOX, for any A∈LX and for any x∈X, by Step 1, we have(19)f∘gN¯Ax=fN¯Ax=N¯Ax=idUAOXN¯Ax.Hence,f∘g=idUAOX.∀N¯∈LAOX, for any A∈LX and for any x∈X, by Step 1, we have(20)g∘fN¯Ax=gN¯Ax=N¯Ax=idLAOXN¯Ax.Hence,g∘f=idLAOX.Step 3. The mappings f:LAOX⟶UAOX and f−1=g:UAOX⟶LAOX are order-preserving mappings.
In fact, letN1¯,N2¯∈LAOX and N1¯≤N2¯; by Step 1 and Theorems 1 and 3, we have(21)N2¯=fN2¯=fN1∪N2¯=N1∪N2¯=N1¯∨N2¯.
So, for anyA∈LX and for any x∈X, by Theorem 3, we have(22)N2¯Ax=N1¯∨N2¯Ax=N1¯Ax∧N2¯Ax.
It implies thatN2¯Ax≤N1¯Ax. By Theorem 1, we have(23)fN1¯=N1¯≤N2¯=fN2¯.
Hence, the mappingf:LAOX⟶UAOX is an order-preserving mapping. Similarly, we can prove that f−1=g:UAOX⟶LAOX is also an order-preserving mapping.
From Steps 1 to 3, we know that the join-complete latticesLAOX,≤,∨ and UAOX,≤,∨ are order isomorphic.


## 4. The Lattice Structures of Approximation Operators Based onL-Fuzzy Relations

We know that approximation operators based onL-fuzzy relations [4, 37] can be regarded as special approximation operators based on L-fuzzy generalized neighborhood systems (see Lemma 2.7, also see [50]). So, in this section, we will study the lattice structures of approximation operators based on L-fuzzy relations.Theorem 6.
LetΛ be an index set; for any x∈X and A∈LX,(24)1∨i∈ΛRi¯Ax=∧i∈ΛRi¯Ax,∧i∈ΛRi¯Ax≥∨i∈ΛRi¯Ax,2∨i∈ΛRi¯Ax=∨i∈ΛRi¯Ax,∧i∈ΛRi¯Ax≤∧i∈ΛRi¯Ax.Proof.
For anyx∈X and A∈LX,(25)1∨i∈ΛRi¯Ax=∧y∈X∨i∈ΛRix,y⟶Ay=∧y∈X∨i∈ΛRix,y⟶Ay=∧y∈X∧i∈ΛRix,y⟶Ay=∧i∈Λ∧y∈XRix,y⟶Ay=∧i∈ΛRi¯Ax,∧i∈ΛRi¯Ax=∧y∈X∧i∈ΛRix,y⟶Ay=∧y∈X∧i∈ΛRix,y⟶Ay≥∨i∈Λ∧y∈XRix,y⟶Ay=∨i∈ΛRi¯Ax,2∨i∈ΛRi¯Ax=∨y∈X∨i∈ΛRix,y∗Ay=∨y∈X∨i∈ΛRix,y∗Ay=∨i∈Λ∨y∈XRix,y∗Ay=∨i∈ΛRi¯Ax,∧i∈ΛRi¯Ax=∨y∈X∧i∈ΛRix,y∗Ay=∨y∈X∧i∈ΛRix,y∗Ay≤∧i∈Λ∨y∈XRix,y∗Ay=∧i∈ΛRi¯Ax.
LetRLAOX=R¯:LX⟶LX|R∈LX×X and RUAOX=R¯:LX⟶LX|R∈LX×X be the family of all lower and upper approximation operators based on L-fuzzy relations, respectively.Define a relation≤ on RLAOX as follows: R1¯≤R2¯ if and only if R2¯Bx≤R1¯Bx for any x∈X and B∈LX. Then, RLAOX,≤ is a poset, and 0X×X¯ is its bottom element.Define a relation≤ on RUAOX as follows: R1¯≤R2¯ if and only if R1¯Bx≤R2¯Bx for any x∈X and B∈LX. Then, RUAOX,≤ is a poset, and 0X×X¯ is its bottom element.Theorem 7.
LetΛ be an index set.(1)
∀Ri¯i∈Λ⊆RLAOX, we can define union and intersection of Ri¯ as follows:(26)∨i∈ΛRi¯=∨i∈ΛRi¯,∧i∈ΛRi¯=∧i∈ΛRi¯,where(27)∧i∈ΛRi=∨R∈LX×X|∀A∈LX,∀x∈X,R¯Ax≥∨i∈ΛRi¯Ax.Then,∨i∈ΛRi¯ and ∧i∈ΛRi¯ are supremum and infimum of Ri¯i∈Λ, respectively.(2)
∀Ri¯i∈Λ⊆RUAOX, we can define union and intersection of Ri¯ as follows:(28)∨i∈ΛRi¯=∨i∈ΛRi¯⋅∧i∈ΛRi¯=∧i∈ΛRi¯,where(29)∧i∈ΛRi=∨R∈LX×X|∀A∈LX,∀x∈X,R¯Ax≤∧i∈ΛRi¯Ax.Then,∨i∈ΛRi¯ and ∧i∈ΛRi¯ are supremum and infimum of Ri¯i∈Λ, respectively.Proof.
(1)
We only show that∧i∈ΛRi¯ is the infimum of Ri¯i∈Λ.Note that0X×X¯Ax=1; for any A∈LX and for any x∈X, we have(30)0X×X∈R∈LX×X|∀A∈LX,∀x∈X,R¯Ax≥∨i∈ΛRi¯Ax.So,0X×X≤∧i∈ΛRi∈LX×X. Hence, 0X×X¯≤∧i∈ΛRi¯. Furthermore, for any x∈X and B∈LX, by Theorem 6, we have(31)∧i∈ΛRi¯Bx≥∨i∈ΛRi¯Bx≥Ri¯Bx,for eachi∈Λ. It implies that 0X×X¯≤∧i∈ΛRi¯≤Ri¯ for each i∈Λ.IfR∗ is another L-fuzzy relation on X such that R∗¯≤Ri¯ for each i∈Λ, then for any x∈X and A∈LX, we have Ri¯Ax≤R∗¯Ax. So, ∨i∈ΛRi¯Ax≤R∗¯Ax. By the construction of ∧i∈ΛRi, we can easily obtain ∧i∈ΛRi≥R∗. Thus, ∧i∈ΛRi¯≥R∗¯. Hence, ∧i∈ΛRi¯=∧i∈ΛRi¯ is the infimum of Ri¯i∈Λ.(2)
We only show that∧i∈ΛRi¯ is the infimum of Ri¯i∈Λ.
Note that0X×X¯Ax=0; for any A∈LX and for any x∈X, we have(32)0X×X∈R∈LX×X|∀A∈LX,∀x∈X,R¯Ax≤∧i∈ΛRi¯Ax.
So,0X×X≤∧i∈ΛRi∈LX×X. Hence, 0X×X¯≤∧i∈ΛRi¯. Furthermore, for any x∈X and B∈LX, by Theorem 6, we have(33)∧i∈ΛRi¯Bx≤∧i∈ΛRi¯Bx≤Ri¯Bx,for each i∈Λ. It implies that 0X×X¯≤∧i∈ΛRi¯≤Ri¯ for each i∈Λ.
IfR∗ is another L-fuzzy relation on X such that R∗¯≤Ri¯ for each i∈Λ, then for any x∈X and A∈LX, we have Ri¯Ax≥R∗¯Ax. So, ∧i∈ΛRi¯Ax≥R∗¯Ax. By the construction of ∧i∈ΛRi, we can easily obtain ∧i∈ΛRi≥R∗. Thus, ∧i∈ΛRi¯≥R∗¯. Hence, ∧i∈ΛRi¯=∧i∈ΛRi¯ is the infimum of Ri¯i∈Λ.
By Theorem7, we know that both RLAOX,≤,∨,∧ and RUAOX,≤,∨,∧ are complete lattices. Furthermore, we have the following conclusion.Theorem 8.
IfL satisfies the double negative law, then both RLAOX,≤,∨,∧ and RUAOX,≤,∨,∧ are complete lattice isomorphism.Proof.
Note that bothRLAOX,≤,∨,∧ and RUAOX,≤,∨,∧ are complete lattices. So, we only need to prove the following steps.Step 1. Define a mapping F:RLAOX⟶RUAOX as follows: for any A∈LX and for any x∈X, FR¯Ax=¬R¯¬Ax for each R¯∈RLAOX; then, FR¯∈RUAOX.In fact,∀R¯∈RLAOX, for any A∈LX and for any x∈X, we have(34)FR¯Ax=¬R¯¬Ax=¬∧y∈XRx,y⟶¬Ay=∨y∈X¬Rx,y⟶¬Ay=∨y∈XRx,y∗Ay=R¯Ax.Hence,FR¯=R¯∈RUAOX.Step 2. Define a mapping G:RUAOX⟶RLAOX as follows: for any A∈LX and for any x∈X, GR¯Ax=¬R¯¬Ax for each R¯∈RUAOX; then, GR¯∈RLAOX.In fact,∀R¯∈RUAOX, for any A∈LX and for any x∈X, we have(35)GR¯Ax=¬R¯¬Ax=¬∨y∈XRx,y∗¬Ay=∧y∈X¬Rx,y∗¬Ay=∧y∈XRx,y⟶Ay=R¯Ax.Hence,GR¯=R¯∈RLAOX.Step 3. F∘G=idRUAOX and G∘F=idRLAOX, where idRUAOX (resp., idRLAOX) is the identity mapping on RUAOX (resp., RLAOX).In fact,∀R¯∈RUAOX, for any A∈LX and for any x∈X, we have(36)F∘GR¯Ax=FR¯Ax=R¯Ax=idRUAOXR¯Ax.Hence,F∘G=idRUAOX. ∀R¯∈RLAOX, for any A∈LX and for any x∈X,(37)G∘FR¯Ax=GR¯Ax=R¯Ax=idRLAOXR¯Ax.Hence,G∘F=idRLAOX.Step 4. The mapping F:RLAOX⟶RUAOX preserves arbitrary union and arbitrary intersection.
In fact,∀Ri¯i∈Λ⊆RLAOX and Λ be an index set. On the one hand, by Theorem 7 and Step 1, we have(38)F∨i∈ΛRi¯=F∨i∈ΛRi¯=∨i∈ΛRi¯=∨i∈ΛRi¯=∨i∈ΛFRi¯.
So, the mappingF:RLAOX⟶RUAOX preserves arbitrary union.
On the other hand,(39)R0∈R∈LX×X|∀A∈LX,∀x∈X,R¯Ax≤∧i∈ΛRi¯Ax⟺∀A∈LX,∀x∈X,R0¯Ax≤∧i∈ΛRi¯Ax⟺∀A∈LX,∀x∈X,R0¯¬Ax≤∧i∈ΛRi¯¬Ax⟺∀A∈LX,∀x∈X,¬R0¯¬Ax≥¬∧i∈ΛRi¯¬Ax⟺∀A∈LX,∀x∈X,R0¯Ax=¬R0¯¬Ax≥¬∧i∈ΛRi¯¬Ax=∨i∈Λ¬Ri¯¬Ax=∨i∈ΛRi¯Ax⟺∀A∈LX,∀x∈X,R0¯Ax≥∨i∈ΛRi¯Ax⟺R0∈R∈LX×X|∀A∈LX,∀x∈X,R¯Ax≥∨i∈ΛRi¯Ax.
It implies that∧i∈ΛRi=∧i∈ΛRi. Hence, by Theorem 7 and Step 1, we have(40)F∧i∈ΛRi¯=F∧i∈ΛRi¯=∧i∈ΛRi¯=∧i∈ΛRi¯=∧i∈ΛRi¯=∧i∈ΛFRi¯.
So, the mappingF:RLAOX⟶RUAOX preserves arbitrary intersection.


## 5. Conclusions

Following the idea ofL-fuzzy generalized neighborhood systems as introduced by Zhao et al., we gave the lattice structures of lower and upper approximation operators based on L-fuzzy generalized neighborhood systems. In particular,(1)
IfL satisfies the double negative law, then the join-complete lattices LAOX,≤,∨ and UAOX,≤,∨ are order isomorphic (here LAOX and UAOX are the family of all lower and upper approximation operators based on L-fuzzy generalized neighborhood systems, respectively), including when L-fuzzy generalized neighborhood system is serial, reflexive, and transitive.(2)
IfL satisfies the double negative law, then the complete lattices RLAOX,≤,∨,∧ and RUAOX,≤,∨,∧ are complete lattice isomorphism (here RLAOX and RUAOX are the family of all lower and upper approximation operators based on L-fuzzy relations, respectively).



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*Source: 5523822-2021-04-12.xml*