#### Abstract

Following the idea of -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 -fuzzy generalized neighborhood systems. In particular, as special approximation operators based on -fuzzy generalized neighborhood systems, we will give the complete lattice structures of lower and upper approximation operators based on -fuzzy relations. Furthermore, if satisfies the double negative law, then there exists an order isomorphic mapping between upper and lower approximation operators based on -fuzzy generalized neighborhood systems; when -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 -fuzzy relations are complete lattice isomorphism.

#### 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 [310]. 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 [1114] and covering-based rough sets [1519].

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 [2031]. 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, 3238]. Furthermore, replacing the unit interval with a complete lattice as the range of the membership function [39], the more general -fuzzy rough sets further extend the theoretical framework and application range of classic rough sets [31, 4046]. 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, 4749]. 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 -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 -fuzzy generalized neighborhood systems were not studied. Following this idea, a natural problem arises: can the lattice structures of approximation operators based on -fuzzy generalized neighborhood systems be given?

In the present paper, we study the lattice structures of approximation operators based on -fuzzy generalized neighborhood systems (resp., -fuzzy relations) and give the relationship between lower and upper approximation operators based on -fuzzy generalized neighborhood systems (resp., -fuzzy relations).

The contents of this paper are organized as follows. In Section 2, we recall some notions and notations used in this paper. In Section 3, the lattice structures of approximation operators based on -fuzzy generalized neighborhood system operators are given. In Section 4, the lattice structures of lower and upper approximation operators based on -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 pair , where 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 such that for all and . is said to be integral if the top element 1 is the unique unit in the sense of for all .

In this paper, if not otherwise specified, is always assumed to be an integral, commutative quantale. Since the binary operation distributes over arbitrary joins, the function has a right adjoint given by . We collect here some basic properties of the binary operations and [5153]. is said to satisfy the double negation law if for any , , and we use to denote . If satisfies the double negative law, then (1) ; (2) ; (3) .

Proposition 1. (1).(2).(3).(4).(5).(6).(7).

Let the universe of discourse be an arbitrary nonempty set. We call a function as an -fuzzy set in . We use to denote the set of all -fuzzy sets in and call it the -fuzzy power set on . For a crisp subset , let be the characteristic function, i.e., if and if . The characteristic function of a subset can be regarded as an -fuzzy set in . We make no difference between a constant -fuzzy set and its value since no confusion will arise.

The operators on can be translated onto in a point-wise. That is, for any ,

Let be -fuzzy set in . The subsethood degree [20] of , , denoted by , is defined by . represents the semantics “ is contained in .

Let be -fuzzy set in . The intersection degree [32] of , , denoted by , is defined by . represents the semantics “ and have intersection.”

The lemma below collects some properties of subsethood degree and intersection degree. They can be found in many literatures such as [6, 2224, 30, 32, 52].

Lemma 1. Let . Then,(S1) .(S2) and .(S3) .(S4) .(S5) .(S6) .(S7) .(I1) .(I2) .(I3) .(I4) .(IS) .

Next, we recall the notions of -fuzzy generalized neighborhood systems and rough approximation operators based -fuzzy generalized neighborhood systems.

Definition 1 (see [49, 50]). Let be a set; the function is called an -fuzzy generalized neighborhood system operator on , if for any , . Usually, is called an -fuzzy generalized neighborhood system of and is interpreted as the degree of that which is a neighborhood of . The condition“ ” is a lattice-valued interpretation of “ is nonempty.” In particular,(1)An -fuzzy generalized neighborhood system operator is said to be serial (denoted by SE), if for any and , .(2)An -fuzzy generalized neighborhood system operator is said to be reflexive (denoted by RE), if for any and , .(3)An -fuzzy generalized neighborhood system operator is said to be transitive (denoted by TR), if for any and ,

Definition 2 (see [49, 50]). Let be an -fuzzy generalized neighborhood system operator. Then, for each , the lower and upper approximation operators and are defined as follows:Finally, we recall the notions of -fuzzy relations and rough approximation operator-based -fuzzy relations.

Definition 3 (see [36, 37]). An -fuzzy relation on is a mapping . For any , is interpreted as the related degree between and . Furthermore,(1) is said to be serial if for all , .(2) is said to be reflexive if for all , .(3) is said to be transitive if for all , .

Definition 4 (see [36, 37]). Let be an -fuzzy relation on . Then, for any , the lower and upper approximation operators and are defined as follows:

Lemma 2 (see [50]). Let be an -fuzzy relation on . Define an -fuzzy generalized neighborhood system operator as follows: for any and ,

Then, and for any .

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

In this section, we will study the lattice structures of lower and upper approximation operators based on -fuzzy generalized neighborhood systems.

Letbe the family of all -fuzzy generalized neighborhood system operators on , and letbe the family of all serial, reflexive, and transitive -fuzzy generalized neighborhood system operators on , respectively, where

Let and be the family of all lower and upper approximation operators on , respectively. Furthermore, letbe the family of all serial lower and upper approximation operators on , respectively. Let and be the family of all reflexive lower and upper approximation operators on , respectively. Let and be the family of all transitive lower and upper approximation operators on , respectively.

Obviously,

Theorem 1. (1)Define a relation on as follows: if and only if for any and . Then, is a poset.(2)Define a relation on as follows: if and only if for any and . Then, is a poset.(3)Define a relation on as follows: if and only if for any and . Then, is a poset.

Proof. The proof is straightforward and is omitted.

Proposition 2. Let be an index set; , we can define union of as follows:for any and ; then, . Furthermore,(1)If , then .(2)If , then .(3)If , then .

Proof. The proof of (1) and (2) are straightforward and are omitted. (3) Note that . So, if , then there must be .Thus, . Therefore, .
By Proposition 2, we obtain the following theorem.

Theorem 2. , , , and are all join-complete lattices.

Theorem 3. (1) and be an index set, we can define union of as follows:Then, is the supremum of .(2) and be an index set, we can define union of as follows:Then, is the supremum of .

Proof. Step 1. We first prove that the following equation holds: for any and ,In fact,Step 2(1)Let ; then, for each . By Step 1, we have . If is another -fuzzy generalized neighborhood system operator on such that for each , then for any . Hence,Thus, . So, is the supremum of .(2)Let ; then, for each . By Step 1, we have . If is another -fuzzy generalized neighborhood system operator on such that for each , then for any . Hence,Thus, . So, is the supremum of .
By Proposition 2 and Theorems 2 and 3, we can obtain Theorem 4 as follows.

Theorem 4. The posets , , , , , , , and are all join-complete lattices.

Furthermore, we have the following theorem.

Theorem 5. (1)If satisfies the double negative law, then the join-complete lattices and are order isomorphic.(2)If satisfies the double negative law, when -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 as follows: for any and for any , for each ; then, . Define a mapping as follows: for any and for any , for each ; then, .In fact, by Theorem 3.9 of [50], if satisfies the double negative law, then we have for any and for any . So, . Similarly, we can prove that .Step 2. and , where (resp., ) is the identity mapping on (resp., ).In fact, , for any and for any , by Step 1, we haveHence, ., for any and for any , by Step 1, we haveHence, .Step 3. The mappings and are order-preserving mappings.In fact, let and ; by Step 1 and Theorems 1 and 3, we haveSo, for any and for any , by Theorem 3, we haveIt implies that . By Theorem 1, we haveHence, the mapping is an order-preserving mapping. Similarly, we can prove that is also an order-preserving mapping.
From Steps 1 to 3, we know that the join-complete lattices and are order isomorphic.

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

We know that approximation operators based on -fuzzy relations [4, 37] can be regarded as special approximation operators based on -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 -fuzzy relations.

Theorem 6. Let be an index set; for any and ,

Proof. For any and ,Let and be the family of all lower and upper approximation operators based on -fuzzy relations, respectively.Define a relation on as follows: if and only if for any and . Then, is a poset, and is its bottom element.Define a relation on as follows: if and only if for any and . Then, is a poset, and is its bottom element.

Theorem 7. Let be an index set.(1), we can define union and intersection of as follows:whereThen, and are supremum and infimum of , respectively.(2), we can define union and intersection of as follows:where

Then, and are supremum and infimum of , respectively.

Proof. (1)We only show that is the infimum of .Note that ; for any and for any , we haveSo, . Hence, . Furthermore, for any and , by Theorem 6, we havefor each . It implies that for each .If is another -fuzzy relation on such that for each , then for any and , we have . So, . By the construction of , we can easily obtain . Thus, . Hence, is the infimum of .(2)We only show that is the infimum of .Note that ; for any and for any , we haveSo, . Hence, . Furthermore, for any and , by Theorem 6, we havefor each . It implies that for each .
If is another -fuzzy relation on such that for each , then for any and , we have . So, . By the construction of , we can easily obtain . Thus, . Hence, is the infimum of .
By Theorem 7, we know that both and are complete lattices. Furthermore, we have the following conclusion.

Theorem 8. If satisfies the double negative law, then both and are complete lattice isomorphism.

Proof. Note that both and are complete lattices. So, we only need to prove the following steps.Step 1. Define a mapping as follows: for any and for any , for each ; then, .In fact, , for any and for any , we haveHence, .Step 2. Define a mapping as follows: for any and for any , for each ; then, .In fact, , for any and for any , we haveHence, .Step 3. and , where (resp., ) is the identity mapping on (resp., ).In fact,, for any and for any , we haveHence, . , for any and for any ,Hence, .Step 4. The mapping preserves arbitrary union and arbitrary intersection.In fact, and be an index set. On the one hand, by Theorem 7 and Step 1, we have