L -fuzzy multigranulation rough set based on residuated lattices

Abstract

This paper studies $L$ -fuzzy multigranulation rough set based on residuted lattices, which is considered as an extension of the classical $L$ -fuzzy rough set. Two models of $L$ -fuzzy optimistic multigrannulation rough set and $L$ -fuzzy pessimistic multigrannulation rough set based on residuted latteices are suggested. Then, we mainly investigate some fundamental properties of the $L$ -fuzzy multigranulation rough set model. With regards to residuated lattices’ topology, we study the topological and the lattice structure in the $L$ -fuzzy optimistic multigranulation rough set model and $L$ -fuzzy pessimistic multigranulation rough set model, respectively.

Keywords

Residuated lattice ℒ-fuzzy multigranulation rough sets optimistic-lower/upper set pessimistic-lower/upper set Topology

1 Introduction

Rough set theory was introduced by Pawlak [12, 13] to deal with insufficient and incomplete data, which is originally constructed from the basis of an indiscernibility relation (or an equivalence relation) or a partition of the universe. Using equivalence classes, an arbitrary subset can be approximated by two subsets, namely the lower approximation and the upper approximation. The lower approximation is the greatest definable set contained in the given set of objects, while the upper approximation is the smallest definable set contained the given set. Two approximation operators are basic notions in the development of rough set theory and one of main directions for the study of rough set theory is naturally the extensive definitions of rough approximation operators in various fields. Moreover, many authors have combined fuzzy set [26] and rough set to obtain fuzzy rough set or rough fuzzy set [3, 4], which is to handle fuzzy and quantitative data.

With respect to granular computing which is proposed in [27], an equivalence relation on the universe can be considered as a granulation, and a partition on the universe can be regarded as a granulation space, so it is named single granulation rough set model. In regard to the practical aspect, there are some problems of multiple information resources (give examples). Thus, it is necessary to have some restrictions for applying single granulation. There are practically some problems to apply the multiple information resources. Single granulation structure therefore limits some applications. To address this issue, Qian introduced concept of multigranulation rough set in [15] (MGRS), where the set approximations are defined by using multi equivalence relations on the universe. The multigranulation rough set model provides an effective approach to solve problems in the context of multi granulations. She and He [17] studied on the structure of the multigranulation rough set model. Yang et al. [25] discussed multigranulation rough set: from crisp to fuzzy case. Qian studied incomplete multigranulation rough set [14]. Xu et al. [24] proposed multigranulation fuzzy rough sets in a fuzzy tolerance approximation space and multigranulation fuzzy rough sets [23]. Feng [5] investigated variable precision multigranulation decision-theoretic fuzzy rough sets. Construction of multigranulation rough set was also proposed in [19 , 29]. Besides, the reader can see multigranultion rough set in various fields, such as, evidence-theory-based numerical characterization [20], intuitionistic fuzzy sets [8] and multigranulation decision-theoretic rough sets in incomplete information systems [28].

Radzikowska and Kerrein are known as pioneers of $L$ -fuzzy rough set ( $L$ -FRS) [16]. They generalized rough set theory to residuated lattice serving as a more general structure of truth values. She et al. [18] extended to the axiomatic characterization of various $L$ -FRSs later. Many authors have studied topological structure. For example, Ma et al. [11] employed the lower and upper set to study the relationship between $L$ -fuzzy rough approximation operators and $L$ -fuzzy topological spaces by the lower set and upper set. Zhao et al. [22] presented fuzzy variable precision rough sets based on generalized residuated lattices which is an extension of fuzzy rough set.

Despite the fact that MGRS theory is thought as a new direction in rough set, its connection to residuated lattice theory has not been mentioned in past studies. For this reason, the aim of this paper is to study properties of $L$ -fuzzy multigranulation rough set based on residuated lattices. To address this issue, we first examine the connection between MGRS and $L$ -fuzzy rough set based on residuated lattices and introduce definition of $L$ -fuzzy optimistic/pessimistic multigranulation rough set model based on residuated lattices. Then, we study some fundamental properties of the multigranulation rough set model. Particularly, we focus on the topological structure and the lattice structure in the $L$ -fuzzy optimistic multigranulation rough set model and $L$ -fuzzy pessimistic one by optimistic-lower/upper set and pessimistic-lower/upper set, respectively.

This paper is organized as follows. Some of concepts and properties of residuated lattices will be mentioned in Section 2. Section 3 states definitions about the $L$ -fuzzy optimistic/pessimistic multigranulation lower and upper approximation operators and their relating properties. Section 4 mainly presents the optimistic/pessimistic lower and upper sets to investigate the topological structure. The last one is conclusion.

2 Preliminaries

In this section, we firstly review definitions and properties of residuated lattice and $L$ -fuzzy rough set [11, 22].

2.1 Residuated lattice

A residuated lattice $L$ is an algebra (L, ∨ , ∧ , ⊗ , → , 0, 1), where (L, ∨ , ∧ , 0, 1) is a bounded lattice with the greatest element 1 and the smallest element 0; (L, ⊗ , 1) is a commutative monoid and (⊗ , →) is an adjoint pair on $L$ satisfying a ⊗ b ≤ c ⇔ a ≤ b → c, ∀a, b, c ∈ L.

Let a ∈ L and define an unary operator, as ¬a = a → 0, referred to as the precomplement operator. If for any a ∈ L, ¬¬ a = a, then $L$ is called a regular residuated lattice.

2.2 $L$ -sets, $L$ -relations and $L$ -topologies

Denote by $L^{X}$ the family of all $L$ -sets on X. An $L$ -set μ is constant if μ (x) = a, for all x ∈ X, written as $\hat{a}$ . An $L$ -fuzzy set μ is denoted by α_Y, if ∅ ≠ Y ∈ X and $μ (x) = {\begin{matrix} α if x \in Y \\ 0 if x \notin Y \end{matrix}$ If Y = {y}, then α_Y is denoted by α_y.

The basic and most common operations on $L^{X}$ are as follows: for all x ∈ X and $μ \in L^{X}$ , $\begin{matrix} (μ \lor ν) (x) = μ (x) \lor ν (x); \\ (μ \land ν) (x) = μ (x) \land ν (x); \\ (μ \otimes ν) (x) = μ (x) \otimes ν (x); \\ (μ \to ν) (x) = μ (x) \to ν (x); \\ (\neg μ) (x) = \neg μ (x) . \end{matrix}$

We also write u ≤ v to denote u (x) ≤ v (x) for all x ∈ X.

An $L$ -fuzzy relation (take $L$ -relation for short) [6] θ is a binary function defined on X also with truth values from $L$ , i.e., θ : X × X → L. Likewise, $L^{X \times X}$ denotes the family of all $L$ -fuzzy relations on X.

Let $θ \in L^{X \times X}$ , ∀x, y, z ∈ X. Then θ is

serial if ∨_y∈Xθ (x, y) =1;

reflexive if θ (x, x) =1;

symmetric if θ (x, y) = θ (y, x);

transitive if θ (x, y) ⊗ θ (y, z) ≤ θ (x, z);

Euclidean if θ (z, x) ⊗ θ (z, y) ≤ θ (x, y).

Note 2.1. For a transitive and Euclidean $L$ -relation, the following statements are also true.

θ is transitive ⇔ ∨_y∈X (θ (x, y) ⊗ θ (y, z)) ≤ θ (x, z), ∀x, z ∈ X.

θ is Euclidean ⇔ ∨_z∈X (θ (z, x) ⊗ θ (z, y)) ≤ θ (x, y), ∀x, y ∈ X.

2.3 $L$ -fuzzy rough sets

Definition 2.2. [16] Let (X, θ) be an $L$ -fuzzy approximation space. Define the following two mappings $\underline{θ}, \bar{θ} : L^{X} \to L^{X}$ , namely $L$ -fuzzy lower and upper approximation operators, respectively, as follows: for all x ∈ X and $μ \in L^{X}$ , $\begin{matrix} \underline{θ} (μ) (x) & = \land_{y \in X} (θ (x, y) \to μ (y)), \\ \bar{θ} (μ) (x) & = \lor_{y \in X} (θ (x, y) \otimes μ (y)) . \end{matrix}$

$θ (μ) = (\underline{θ} (μ), \bar{θ} (μ))$ is called an $L$ -fuzzy rough set ( $L$ -FRS) with respect to μ.

3 $L$ -fuzzy optimistic multigranulation rough sets and $L$ -fuzzy pessimistic multigranulation rough sets

In this section, we propose two types $L$ -fuzzy multigranulation rough sets, $L$ -fuzzy optimistic multigranulation rough sets and $L$ -fuzzy pessimistic multigranulation rough sets, as an extionsion of $L$ –fuzzy rough set, and then investigate their basic properties and their characteristics properties in residuated lattices. Set θ ={ θ₁, θ₂, . . . , θ_m } is called an $L$ -fuzzy multigranulation relations set on X if each θ_i is an $L$ -fuzzy relation on X. Then we can write $θ = {θ_{i}}_{i = \bar{1, m}}$ . An $L$ -fuzzy multigranulation relations set θ is called serial, reflexive, symmetric, transitive, Euclidean, $L$ -fuzzy preorder if and only if each θ_i, with i = 1, 2, . . . , n, is serial, reflexive, symmetric, transitive, Euclidean, $L$ -fuzzy preorder, respectively.

In the following discussion, if not being specifically stated, $L$ is always a complete residuated lattice. (X, θ) is called an m dimensional $L$ -fuzzy multigranulation approximation space if X is a finite universe and θ is an $L$ -fuzzy multigranulation relations set on X.

Let (X, θ) and (X, ϑ) be two m dimensional $L$ -fuzzy multigranulation approximation spaces, in which $θ = {θ_{i}}_{i = \bar{1, m}}$ , $ϑ = {ϑ_{i}}_{i = \bar{1, m}}$ . Then we define θ ≤ ϑ if and only if $θ_{i} \leq ϑ_{i}, \forall i = \bar{1, m}$ . Definition 3.1. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. We define two pairs of approximation operators as follows, for all x ∈ X and $μ \in L^{X}$ , $\begin{matrix} {\underline{θ}}^{O} (μ) (x) & = \underset{i = 1}{\overset{m}{\lor}} \land_{y \in X} [θ_{i} (x, y) \to μ (y)], \\ {\bar{θ}}^{O} (μ) (x) & = \underset{i = 1}{\overset{m}{\land}} \lor_{y \in X} [θ_{i} (x, y) \otimes μ (y)], \\ {\underline{θ}}^{P} (μ) (x) & = \underset{i = 1}{\overset{m}{\land}} \land_{y \in X} [θ_{i} (x, y) \to μ (y)], \\ {\bar{θ}}^{P} (μ) (x) & = \underset{i = 1}{\overset{m}{\lor}} \lor_{y \in X} [θ_{i} (x, y) \otimes μ (y)] . \end{matrix}$

The fuzzy set $θ^{O} (μ) = ({\underline{θ}}^{O} (μ), {\bar{θ}}^{O} (μ))$ is called an $L$ -fuzzy optimistic multigranulation rough set ( $L$ -FOMRS) with respect to μ and $θ^{P} (μ) = ({\underline{θ}}^{P} (μ),$ ${\bar{θ}}^{P} (μ))$ is called an $L$ -fuzzy pessimistic multigranulation rough set ( $L$ -FPMRS).

Then, $L$ -FOMRS can be described as ${\underline{θ}}^{O} (μ) (x) = \underset{i = 1}{\overset{m}{\lor}} {\underline{θ}}_{i} (μ) (x)$ and ${\bar{θ}}^{O} (μ) (x) = \underset{i = 1}{\overset{m}{\land}} {\bar{θ}}_{i} (μ) (x)$ and $L$ -FPMRS can be described as ${\underline{θ}}^{P} (μ) (x) = \underset{i = 1}{\overset{m}{\land}} {\underline{θ}}_{i} (μ) (x)$ and ${\bar{θ}}^{P} (μ) (x) = \underset{i = 1}{\overset{m}{\lor}} {\bar{θ}}_{i} (μ) (x)$ .

Note 3.2. When m = 1, the lower and upper approximation operators in Definition 3 degenerate to $L$ -fuzzy rough set in Definition 2.3.

Example 3.3. Consider the bounded lattice {0, a, b, c, 1} with the partial order ≤ defined by 0 < a < c < 1 and 0 < b < c < 1, where a and b are non-comparable. ⊗ is defined and → is computed, respectively, as follows: $\begin{array}{c} \otimes & 0 & a & b & c & 1 & \to & 0 & a & b & c & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ a & 0 & 0 & 0 & 0 & a & and & a & c & 1 & c & 1 & 1 \\ b & 0 & 0 & b & b & b & b & a & a & 1 & 1 & 1 \\ c & 0 & 0 & b & c & c & c & a & a & c & 1 & 1 \\ 1 & 0 & a & b & c & 1 & 1 & 0 & a & b & c & 1 \end{array}$ Let X = {x₁, x₂} and the $L$ -relations be defined as follows:

$θ_{1} = [\begin{matrix} 1 & b \\ a & c \end{matrix}]$ , $θ_{2} = [\begin{matrix} c & a \\ b & 1 \end{matrix}]$ and $μ = \hat{b}$ .

With the Definition 3.1, we have

${\underline{θ}}^{O} (μ) (x) = (c, c)$ and ${\bar{θ}}^{O} (μ) (x) = (b, b)$ .

${\underline{θ}}^{P} (μ) (x) = (b, b)$ and ${\bar{θ}}^{P} (μ) (x) = (b, c)$ .

Proposition 3.4. Let (X, θ) and (X, ϑ) be two m dimensional $L$ -fuzzy multigranulation approximation spaces. Then we have: for all $μ, ν, \in L^{X}$ such as μ ≤ ν,

${\underline{θ}}^{O} (\hat{1}) = {\underline{θ}}^{P} (\hat{1}) = \hat{1}$ , ${\bar{θ}}^{O} (\hat{0}) = {\bar{θ}}^{P} (\hat{0}) = \hat{0}$ .

${\underline{θ}}^{O} (μ) \leq {\underline{θ}}^{O} (ν)$ , ${\bar{θ}}^{O} (μ) \leq {\bar{θ}}^{O} (ν)$ ,

${\underline{θ}}^{P} (μ) \leq {\underline{θ}}^{P} (ν)$ , ${\bar{θ}}^{P} (μ) \leq {\bar{θ}}^{P} (ν)$ .

${\underline{θ}}^{O} (μ) \leq \neg {\bar{θ}}^{O} (\neg μ)$ , ${\bar{θ}}^{O} (μ) \leq \neg {\underline{θ}}^{O} (\neg μ)$ ,

${\underline{θ}}^{P} (μ) \leq \neg {\bar{θ}}^{P} (\neg μ)$ , ${\bar{θ}}^{P} (μ) \leq \neg {\underline{θ}}^{P} (\neg μ)$ .

If θ ≤ ϑ, then ${\underline{ϑ}}^{O} (μ) \leq {\underline{θ}}^{O} (μ)$ , ${\bar{θ}}^{O} (μ) \leq {\bar{ϑ}}^{O} (μ)$ .

${\underline{ϑ}}^{P} (μ) \leq {\underline{θ}}^{P} (μ)$ , ${\bar{θ}}^{P} (μ) \leq {\bar{ϑ}}^{P} (μ)$ .

Theorem 3.5. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space in which $L$ is a complete regular residuated lattice. Then the duality properties of two pairs of $({\underline{θ}}^{O}, {\bar{θ}}^{O})$ and $({\underline{θ}}^{P}, {\bar{θ}}^{P})$ hold, i.e., for all $μ \in L^{X}$ , $\begin{matrix} {\underline{θ}}^{O} (μ) & = \neg {\bar{θ}}^{O} (\neg μ), {\bar{θ}}^{O} (μ) = \neg {\underline{θ}}^{O} (\neg μ) . \\ {\underline{θ}}^{P} (μ) & = \neg {\bar{θ}}^{P} (\neg μ), {\bar{θ}}^{P} (μ) = \neg {\underline{θ}}^{P} (\neg μ) . \end{matrix}$

Proof. When $L$ is a complete regular residuated lattice, it can be proved from Proposition 3.4 (3). □

Proposition 3.6. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. Then we have: for all x, y ∈ X, a ∈ L and $μ \in L^{X}$ ,

$\hat{a} \otimes {\bar{θ}}^{O} (μ) \leq {\bar{θ}}^{O} (\hat{a} \otimes μ)$ ,

${\underline{θ}}^{O} (\hat{a} \to μ) \leq \hat{a} \to {\underline{θ}}^{O} (μ)$ .

$\hat{a} \otimes {\bar{θ}}^{P} (μ) = {\bar{θ}}^{P} (\hat{a} \otimes μ)$ ,

${\underline{θ}}^{P} (\hat{a} \to μ) = \hat{a} \to {\underline{θ}}^{P} (μ)$ .

${\bar{θ}}^{O} (\hat{a}) \leq \hat{a} \leq {\underline{θ}}^{O} (\hat{a})$ ,

${\bar{θ}}^{P} (\hat{a}) \leq \hat{a} \leq {\underline{θ}}^{P} (\hat{a})$ .

${\underline{θ}}^{O} (\hat{a}) = \hat{a} \Leftrightarrow {\underline{θ}}^{O} (\hat{0}) = \hat{0}$ ,

${\bar{θ}}^{O} (\hat{a}) = \hat{a} \Leftrightarrow {\bar{θ}}^{O} (\hat{1}) = \hat{1}$ .

${\underline{θ}}^{P} (\hat{a}) = \hat{a} \Leftrightarrow {\underline{θ}}^{P} (\hat{0}) = \hat{0}$ ,

${\bar{θ}}^{P} (\hat{a}) = \hat{a} \Leftrightarrow {\bar{θ}}^{P} (\hat{1}) = \hat{1}$ .

$a \otimes \underset{i = 1}{\overset{m}{\land}} θ_{i} (x, y) \leq {\bar{θ}}^{O} (1_{y} \otimes \hat{a}) (x)$ ,

${\underline{θ}}^{O} (1_{y} \to \hat{a}) (x) \leq \underset{i = 1}{\overset{m}{\lor}} θ_{i} (x, y) \to a$ ,

$a \otimes \underset{i = 1}{\overset{m}{\land}} θ_{i} (x, y) = {\bar{θ}}^{P} (1_{y} \otimes \hat{a}) (x)$ ,

${\underline{θ}}^{P} (1_{y} \to \hat{a}) (x) = \underset{i = 1}{\overset{m}{\land}} θ_{i} (x, y) \to a$ .

$\underset{i = 1}{\overset{m}{\land}} θ_{i} (x, y) = {\bar{θ}}^{O} (1_{y}) (x)$ ,

${\underline{θ}}^{O} (1_{X - {y}}) (x) = \underset{i = 1}{\overset{m}{\land}} θ_{i} (x, y) \to 0$ ,

$\underset{i = 1}{\overset{m}{\land}} θ_{i} (x, y) = {\bar{θ}}^{P} (1_{y}) (x)$ ,

${\underline{θ}}^{P} (1_{X - {y}}) (x) = \underset{i = 1}{\overset{m}{\lor}} θ_{i} (x, y) \to 0$ .

Proof.

(4) For all x, y ∈ X, we have

${\bar{θ}}^{O} (1_{y} \otimes \hat{a}) (x) = \underset{i = 1}{\overset{m}{\land}} {\bar{θ}}_{i} (1_{y} \otimes \hat{a}) (x) = \underset{i = 1}{\overset{m}{\land}} (θ_{i} (x, y) \otimes a) \geq a \otimes \underset{i = 1}{\overset{m}{\land}} θ_{i} (x, y) .$

The other one can be similarly proved.

(5) For all x, y ∈ X, by taking a = 1, it follows ${\bar{θ}}^{O} (1_{y} \otimes \hat{a}) (x) = \underset{i = 1}{\overset{m}{\land}} [θ_{i} (x, y) \otimes a] = \underset{i = 1}{\overset{m}{\land}} θ_{i} (x, y)$ .

It is similar to prove for the others. □

Proposition 3.7. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. If $θ_{1}, . . ., θ_{n} \in L^{X \times X}$ and θ₁ ≤ . . . ≤ θ_m, then the following hold

${\underline{θ}}_{m} (μ) \leq . . . \leq {\underline{θ}}_{1} (μ)$ and ${\bar{θ}}_{1} (μ) \leq . . . \leq {\bar{θ}}_{m} (μ)$ .

${\underline{θ}}^{O} (μ) = {\underline{θ}}_{m} (μ)$ and ${\bar{θ}}^{O} (μ) = {\bar{θ}}_{1} (μ)$ .

${\underline{θ}}^{P} (μ) = {\underline{θ}}_{1} (μ)$ and ${\bar{θ}}^{P} (μ) = {\bar{θ}}_{m} (μ)$ .

Proposition 3.8. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. If $μ_{i} \in L^{X}$ , then the following hold:

${\bar{θ}}^{O} (\underset{j = 1}{\overset{n}{\lor}} μ_{j}) = \underset{i = 1}{\overset{m}{\land}} \underset{j = 1}{\overset{n}{\lor}} {\bar{θ}}_{i} (μ_{j})$ ,

${\underline{θ}}^{O} (\underset{j = 1}{\overset{n}{\land}} μ_{j}) = \underset{i = 1}{\overset{m}{\lor}} \underset{j = 1}{\overset{n}{\land}} {\underline{θ}}_{i} (μ_{j})$ ,

${\bar{θ}}^{P} (\underset{j = 1}{\overset{n}{\lor}} μ_{j}) = \underset{i = 1}{\overset{m}{\land}} \underset{j = 1}{\overset{n}{\lor}} {\bar{θ}}_{i} (μ_{j})$ ,

${\underline{θ}}^{P} (\underset{j = 1}{\overset{n}{\land}} μ_{j}) = \underset{i = 1}{\overset{m}{\lor}} \underset{j = 1}{\overset{n}{\land}} {\underline{θ}}_{i} (μ_{j}) .$

${\bar{θ}}^{O} (\underset{j = 1}{\overset{n}{\lor}} μ_{j}) \leq \underset{j = 1}{\overset{n}{\lor}} {\bar{θ}}^{O} (μ_{j})$ ,

${\underline{θ}}^{O} (\underset{j = 1}{\overset{n}{\land}} μ_{j}) \leq \underset{j = 1}{\overset{n}{\land}} {\underline{θ}}^{O} (μ_{j})$ ,

${\bar{θ}}^{P} (\underset{j = 1}{\overset{n}{\lor}} μ_{j}) = \underset{j = 1}{\overset{n}{\lor}} {\bar{θ}}^{P} (μ_{j})$ ,

${\underline{θ}}^{P} (\underset{j = 1}{\overset{n}{\land}} μ_{j}) = \underset{j = 1}{\overset{n}{\land}} {\underline{θ}}^{P} (μ_{j}) .$

${\bar{θ}}^{O} (\underset{j = 1}{\overset{n}{\land}} μ_{j}) \leq \underset{j = 1}{\overset{n}{\land}} {\bar{θ}}^{O} (μ_{j})$ ,

${\underline{θ}}^{O} (\underset{j = 1}{\overset{n}{\lor}} μ_{j}) \leq \underset{j = 1}{\overset{n}{\lor}} {\underline{θ}}^{O} (μ_{j})$ ,

${\bar{θ}}^{P} (\underset{j = 1}{\overset{n}{\land}} μ_{j}) \leq \underset{j = 1}{\overset{n}{\land}} {\bar{θ}}^{P} (μ_{j})$ ,

${\underline{θ}}^{P} (\underset{j = 1}{\overset{n}{\land}} μ_{j}) \geq \underset{j = 1}{\overset{n}{\lor}} {\underline{θ}}^{P} (μ_{j}) .$

Proposition 3.9. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. Then the following statements are equivalent, for all a ∈ L:

$L$ -fuzzy multigranulation relations set is serial.

${\bar{θ}}^{O} (\hat{a}) = \hat{a}$ .

${\underline{θ}}^{O} (\hat{a}) = \hat{a}$ .

${\bar{θ}}^{P} (\hat{a}) = \hat{a}$ .

${\underline{θ}}^{P} (\hat{a}) = \hat{a}$ .

Proposition 3.10. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. Then the following statements are equivalent, for all $μ \in L^{X}$ :

$L$ -fuzzy multigranulation relations set is reflexive.

$μ \leq {\bar{θ}}^{O} (μ)$ .

${\underline{θ}}^{O} (μ) \leq μ$ .

$μ \leq {\bar{θ}}^{P} (μ)$ .

${\underline{θ}}^{P} (μ) \leq μ$ .

Proof. (1) ⇒ (2) For all $μ \in L^{X}$ and x ∈ X, we have

${\bar{θ}}^{O} (μ) (x) = \underset{i = 1}{\overset{m}{\land}} \lor_{y \in X} [θ_{i} (x, y) \otimes μ (y)] \geq \underset{i = 1}{\overset{m}{\land}} {θ_{i} (x, x) \otimes μ (x)} = μ (x)$ .

Hence, it holds that $μ \leq {\bar{θ}}^{O} (μ)$ .

(1) ⇒ (3) can be proven by a similar way.

(2) ⇒ (1) For any x ∈ X, let μ = 1_x. As we know ${\bar{θ}}^{0} (1_{x}) (x) = \underset{i = 1}{\overset{m}{\land}} θ_{i} (x, x)$ and ${\bar{θ}}^{O} (1_{x}) (x) \geq 1_{x} (x) = 1$ at the same time, it follows that $\underset{i = 1}{\overset{m}{\land}} θ_{i} (x, x) = 1$ , i,e, θ_i (x, x) =1.

The others can be proven in a similar way. □

Proposition 3.11. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. For all x, y ∈ X and a ∈ L, then the following statements are equivalent:

$L$ -fuzzy multigranulation relations set is symmetric.

${\bar{θ}}^{O} (1_{y}) (x) = {\bar{θ}}^{O} (1_{x}) (y)$ .

${\bar{θ}}^{O} (1_{y} \otimes \hat{a}) (x) = {\bar{θ}}^{O} (1_{x} \otimes \hat{a}) (y)$ .

${\underline{θ}}^{O} (1_{y} \to \hat{a}) (x) = {\underline{θ}}^{O} (1_{x} \to \hat{a}) (y)$ .

${\bar{θ}}^{P} (1_{y}) (x) = {\bar{θ}}^{P} (1_{x}) (y)$ .

${\bar{θ}}^{P} (1_{y} \otimes \hat{a}) (x) = {\bar{θ}}^{P} (1_{x} \otimes \hat{a}) (y)$ .

${\underline{θ}}^{P} (1_{y} \to \hat{a}) (x) = {\underline{θ}}^{P} (1_{x} \to \hat{a}) (y)$ .

Remark 3.12. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space in which $L$ is a complete regular residuated lattice. Then $L$ -fuzzy multigranulation relations set is symmetric, for all x, y ∈ X

$\Leftrightarrow {\underline{θ}}^{O} (1_{X - {x}}) (y) = {\underline{θ}}^{O} (1_{X - {y}}) (x)$

$\Leftrightarrow {\underline{θ}}^{P} (1_{X - {x}}) (y) = {\underline{θ}}^{P} (1_{X - {y}}) (x)$ .

Proposition 3.13. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. For all $μ \in L^{X}$ , Then the following statements are equivalent:

$L$ -fuzzy multigranulation relations set is symmetric.

${\bar{θ}}^{O} ({\underline{θ}}^{O} (μ)) \leq μ$ .

$μ \leq {\underline{θ}}^{O} ({\bar{θ}}^{O} (μ))$ .

${\bar{θ}}^{P} ({\underline{θ}}^{P} (μ)) \leq μ$ .

$μ \leq {\underline{θ}}^{P} ({\bar{θ}}^{P} (μ))$ .

Proof.

As $L$ -fuzzy multigranulation relations set is symmetric, it follows that ${\bar{θ}}^{O} ({\underline{θ}}^{O} (μ)) \leq μ$ .

${\bar{θ}}^{O} ({\underline{θ}}^{O} (μ)) (x) = = \underset{i = 1}{\overset{m}{\land}} \lor_{y \in X} {({\underline{θ}}^{O} (μ)) (y) \otimes θ_{i} (x, y)} = \underset{i = 1}{\overset{m}{\land}} \lor_{y \in X} {\underset{i = 1}{\overset{m}{\lor}} \land_{z \in X} (θ_{i} (y, z) \to μ (z)) \otimes θ_{i} (x, y)} \leq \underset{i = 1}{\overset{m}{\land}} \lor_{y \in X} {\underset{i = 1}{\overset{m}{\lor}} (θ_{i} (y, x) \to μ (x)) \otimes θ_{i} (x, y)} = \underset{i = 1}{\overset{m}{\land}} \lor_{y \in X} \underset{i = 1}{\overset{m}{\lor}} {(θ_{i} (x, y) \to μ (x)) \otimes θ_{i} (x, y)} = μ (x) .$

${\bar{θ}}^{O} ({\underline{θ}}^{O} (μ)) \leq μ$ implies $L$ -fuzzy multigranulation relations set is symmetric.

Assume that $L$ -fuzzy multigranulation relations set is not symmetric, then there exist x₀, y₀ ∈ X such that θ (x₀, y₀) ≠ θ (y₀, x₀). Consider the following three cases:

Assume that θ (x₀, y₀) ≤ θ (y₀, x₀), thenθ (y₀, x₀) ≰ θ (x₀, y₀) holds, so θ (y₀, x₀)→θ (x₀, y₀) <1. Let μ (y) = θ_i (x₀, y). Then

$θ^{O} ({\underline{θ}}^{O} (μ)) (y_{0}) \to μ (y_{0}) = \underset{i = 1}{\overset{m}{\land}} \lor_{x \in X} {\underset{i = 1}{\overset{m}{\lor}} \land_{z \in X} (θ_{i} (y, z) \to μ (z)) \otimes θ_{i} (y_{0}, x)} \to μ (y_{0}) \leq \underset{i = 1}{\overset{m}{\land}} \lor_{x \in X} \underset{i = 1}{\overset{m}{\lor}} \land_{z \in X} {(θ_{i} (y, z) \to μ (z)) \otimes θ_{i} (y_{0}, x)} \to μ (y_{0}) = \underset{i = 1}{\overset{m}{\land}} \underset{i = 1}{\overset{m}{\lor}} \lor_{x \in X} \land_{z \in X} {(θ_{i} (y, z) \to μ (z)) \otimes θ_{i} (y_{0}, x)} \to μ (y_{0}) \leq \underset{i = 1}{\overset{m}{\land}} \underset{i = 1}{\overset{m}{\lor}} \land_{z \in X} {(θ_{i} (x_{0}, z) \to θ_{i} (x_{0}, z) \otimes θ_{i} (y_{0}, x_{0})} \to θ_{i} (x_{0}, y_{0}) < 1$ .

We obtain ${\bar{θ}}^{O} ({\underline{θ}}^{O} (μ)) (y_{0}) ≰ μ (y_{0})$ , which implies a contradiction.

Assume that θ (y₀, x₀) ≤ θ (x₀, y₀) and μ (y) = θ (y₀, y), then it can be proved in a similar way as Case 1 that ${\bar{θ}}^{O} ({\underline{θ}}^{O} (μ)) (y_{0}) \leq μ (y_{0})$ does not hold.

Assume that θ (y₀, x₀) and θ (x₀, y₀) are incomparable, then in a similar way as Case 1, we have ${\bar{θ}}^{O} ({\underline{θ}}^{O} (μ)) (y_{0}) \leq μ (y_{0})$ does not hold.

In accordance with the above methods, it be proved that an $L$ -fuzzy multigranulation relations set is symmetric ⇔ ${\bar{θ}}^{O} ({\underline{θ}}^{O} (μ)) \leq μ$ .

The others can be proved in a similar way. □

Proposition 3.14. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. Then the following statements are equivalent, for all $μ \in L^{X}$ :

$L$ -fuzzy multigranulation relations set is transitive.

${\bar{θ}}^{O} ({\bar{θ}}^{O} (μ)) \leq {\bar{θ}}^{O} (μ)$ .

${\underline{θ}}^{O} (μ) \leq {\underline{θ}}^{O} ({\underline{θ}}^{O} (μ))$ .

${\bar{θ}}^{P} ({\bar{θ}}^{P} (μ)) \leq {\bar{θ}}^{P} (μ)$ .

${\underline{θ}}^{P} (μ) \leq {\underline{θ}}^{P} ({\underline{θ}}^{P} (μ))$ .

Proof.

(1) ⇒ (2) For all x ∈ X, we have ${\bar{θ}}^{O} ({\bar{θ}}^{O} (μ))$ $(x) = = \lor_{y \in X} [θ_{i} (x, y) \otimes {\bar{θ}}^{O} (μ) (y)] = \underset{i = 1}{\overset{m}{\land}} \lor_{y \in X} {θ_{i} (x, y) \otimes \underset{i =}{\overset{m}{\land}} \lor_{z \in X} [θ_{i} (y, z) \otimes μ (z)]} \leq \underset{i = 1}{\overset{m}{\land}} \lor_{y \in X} \underset{i = 1}{\overset{m}{\land}} \lor_{z \in X} {θ_{i} (x, y) \otimes [θ (y, z) \otimes μ (z)]} = \underset{i = 1}{\overset{m}{\land}} \lor_{y \in X} \lor_{z \in X} {[θ_{i} (x, y) \otimes θ_{i} (y, z)] \otimes μ (z)} \leq \underset{i = 1}{\overset{m}{\land}} \lor_{y \in X} \lor_{z \in X} {θ_{i} (x, z) \otimes μ (z)} = \underset{i = 1}{\overset{m}{\land}} \lor_{z \in X} [θ_{i} (x, z) \otimes μ (z)] = {\bar{θ}}^{O} (μ) (x),$

i.e., ${\bar{θ}}^{O} ({\bar{θ}}^{O} (μ)) \leq {\bar{θ}}^{O} (μ)$ .

(2) ⇒ (1) For any x, y ∈ X, set μ = 1_y, it then follows from the assertion in (2) that

${\bar{θ}}^{0} ({\bar{θ}}^{0} (1_{y})) (x) = = \underset{i = 1}{\overset{m}{\land}} \lor_{z \in X} {θ_{i} (x, z) \otimes [\underset{i = 1}{\overset{m}{\land}} \lor_{w \in X} (θ_{i} (z, w) \otimes 1_{y} (w))]} = \underset{i = 1}{\overset{m}{\land}} \lor_{z \in X} (θ (x, z) \otimes \underset{i = 1}{\overset{m}{\land}} θ_{i} (z, y)) \leq \underset{i = 1}{\overset{m}{\land}} \lor_{z \in X} \underset{i = 1}{\overset{m}{\land}} (θ (x, z) \otimes θ_{i} (z, y)) \leq \underset{i = 1}{\overset{m}{\land}} \lor_{z \in X} (θ (x, z) \otimes θ_{i} (z, y)) \leq \land {\bar{θ}}^{0} (1_{y}) (x) = θ (x, y) .$

Thus, $L$ -fuzzy multigranulation relations set is transitive.

(1) ⇔ (3) can be proved in a similar way as item (1) and (2).

The others can be proved in a similar way. □

Corollary 3.15. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space and $L$ -fuzzy multigranulation relations set be reflexive. Then $L$ -fuzzy multigranulation relations set is transitive $\Leftrightarrow {\bar{θ}}^{O} ({\bar{θ}}^{O} (μ)) = {\bar{θ}}^{O} (μ) \Leftrightarrow {\underline{θ}}^{O} ({\underline{θ}}^{O} (μ)) = {\underline{θ}}^{O} (μ)$ $\Leftrightarrow {\bar{θ}}^{P} ({\bar{θ}}^{α} (μ)) = {\bar{θ}}^{P} (μ) \Leftrightarrow {\underline{θ}}^{P} ({\underline{θ}}^{P} (μ)) = {\underline{θ}}^{P} (μ)$ , for all $μ \in L^{X}$ .

Proposition 3.16. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. Then the following statements are equivalent, for all $μ \in L^{X}$ :

$L$ -fuzzy multigranulation relations set is Euclidean.

${\bar{θ}}^{O} (μ) \leq {\underline{θ}}^{O} ({\bar{θ}}^{O} (μ))$ .

${\underline{θ}}^{O} (μ) \leq {\bar{θ}}^{O} ({\bar{θ}}^{O} (μ))$ .

${\bar{θ}}^{P} (μ) \leq {\underline{θ}}^{P} ({\bar{θ}}^{P} (μ))$ .

${\underline{θ}}^{P} (μ) \leq {\bar{θ}}^{P} ({\bar{θ}}^{P} (μ))$ .

Proof. (1) ⇒ (2) ∀x ∈ X, we have

${\underline{θ}}^{O} ({\bar{θ}}^{O} (μ)) (x) = = \underset{i = 1}{\overset{m}{\lor}} \land_{y \in X} [θ_{i} (x, y) \to {\bar{θ}}^{O} (μ) (y)] = \underset{i = 1}{\overset{m}{\lor}} \land_{y \in X} {θ_{i} (x, y) \to \underset{i = 1}{\overset{m}{\land}} \lor_{z \in X} [θ_{i} (y, z) \otimes μ (z)]} \geq \underset{i = 1}{\overset{m}{\lor}} \land_{y \in X} {θ_{i} (x, y) \to \underset{i = 1}{\overset{m}{\land}} \lor_{z \in X} [θ_{i} (x, y) \otimes θ_{i} (x, z) \otimes μ (z)]} = \underset{i = 1}{\overset{m}{\lor}} \land_{y \in X} {θ_{i} (x, y) \to \underset{i = 1}{\overset{m}{\land}} [θ_{i} (x, y) \otimes (\lor_{z \in X} θ_{i} (x, z) \otimes μ (z))]} = \underset{i = 1}{\overset{m}{\lor}} \land_{y \in X} \underset{i = 1}{\overset{m}{\land}} \lor_{z \in X} θ_{i} (x, z) \otimes μ (z) = {\bar{θ}}^{O} (μ) (x),$

i.e. ${\bar{θ}}^{O} (μ)) \leq {\underline{θ}}^{O} ({\bar{θ}}^{O} (μ))$ .

(2) ⇒ (1) For the arbitrary x, y ∈ X, let μ = 1_y, then we have

${\underline{θ}}^{O} ({\bar{θ}}^{O} (1_{y})) (x) = = \underset{i = 1}{\overset{m}{\lor}} \land_{z \in X} {θ_{i} (x, z) \to [\underset{i = 1}{\overset{m}{\land}} \lor_{w \in X} (θ_{i} (z, w) \otimes 1_{y} (w))]} = \underset{i = 1}{\overset{m}{\lor}} \land_{z \in X} (θ_{i} (x, z) \to \underset{i = 1}{\overset{m}{\land}} θ_{i} (z, y)) = \underset{i = 1}{\overset{m}{\lor}} \land_{z \in X} \underset{i = 1}{\overset{m}{\land}} (θ_{i} (x, z) \to θ_{i} (z, y)) = \underset{i = 1}{\overset{m}{\land}} \land_{z \in X} (θ_{i} (x, z) \to θ_{i} (z, y)) \geq {\bar{θ}}^{0} (1_{y}) (x) = \underset{i = 1}{\overset{m}{\land}} θ_{i} (x, y) .$

Thus for all x, y, z ∈ X, we have θ_i (x, z) → θ_i (z, y) ≥ θ_i (x, y) ⇔ θ_i (x, z) ⊗ θ_i (x, y) ≤ θ_i (z, y). Consequently, θ is Euclidean.

The others can be proved in a similar way. □

Proposition 3.17. Let $L$ be a residuated lattice and (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. Then the following hold: for all a ∈ L and $μ \in L^{X}$ ,

${\underline{θ}}^{O} (μ) \leq \land_{a \in L} ({\bar{θ}}^{O} (μ \to \hat{a}) \to \hat{a})$ ,

${\underline{θ}}^{P} (μ) = \land_{a \in L} ({\bar{θ}}^{P} (μ \to \hat{a}) \to \hat{a})$ .

${\bar{θ}}^{O} (μ) = \land_{a \in L} ({\underline{θ}}^{O} (μ \to \hat{a}) \to \hat{a})$ ,

${\bar{θ}}^{P} (μ) = \land_{a \in L} ({\underline{θ}}^{P} (μ \to \hat{a}) \to \hat{a})$ .

${\underline{θ}}^{O} (μ \to \hat{a}) \leq {\bar{θ}}^{O} (μ) \to \hat{a}$ ,

${\underline{θ}}^{P} (μ \to \hat{a}) = {\bar{θ}}^{P} (μ) \to \hat{a}$ .

${\bar{θ}}^{O} (μ \to \hat{a}) \leq {\underline{θ}}^{O} (μ) \to \hat{a}$ ,

${\bar{θ}}^{P} (μ \to \hat{a}) \leq {\underline{θ}}^{P} (μ) \to \hat{a}$ .

Höhle [7] described definition of subsethood as a the degree of μ. Let $μ, ν \in L^{X}$ . Then the degree of μ being the subset of ν is μ ↦ ν = ∧ _x∈X (μ (x) → ν (x)).

Proposition 3.18. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. Then we have: for all $μ, ν \in L^{X}$ , $\begin{matrix} μ \mapsto {\underline{θ}}^{O} (ν) & \geq {\bar{θ^{- 1}}}^{O} (μ) \mapsto ν, \\ μ \mapsto {\underline{θ}}^{P} (ν) & = {\bar{θ^{- 1}}}^{P} (μ) \mapsto ν . \end{matrix}$

Proof. For all $μ, ν \in L^{X}$ , we have

$μ \mapsto {\underline{θ}}^{O} (ν) = = \land_{x \in X} [μ (x) \to {\underline{θ}}^{O} (ν) (x)] = \land_{x \in X} {μ (x) \to \underset{i = 1}{\overset{m}{\lor}} \land_{y \in X} [θ (x, y) \to ν (y)]} \geq \land_{x \in X} \underset{i = 1}{\overset{m}{\lor}} \land_{y \in X} {μ (x) \to [θ (x, y) \to ν (y)]} = \land_{x \in X} \underset{i = 1}{\overset{m}{\lor}} \land_{y \in X} {[μ (x) \otimes θ (x, y)] \to ν (y)} \geq \land_{y \in X} {\underset{i = 1}{\overset{m}{\land}} \lor_{x \in X} [μ (x) \otimes θ^{- 1} (y, x)] \to ν (y)} = \land_{y \in X} {{\bar{θ^{- 1}}}^{O} (μ) (y) \to ν (y)} = {\bar{θ^{- 1}}}^{O} (μ) \mapsto ν .$ □

Corollary 3.19. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. If $L$ -fuzzy multigranulation relations set is symmetric, then we have: for all $μ, ν \in L^{X}$ , $\begin{matrix} μ \mapsto {\underline{θ}}^{O} (ν) & \geq {\bar{θ}}^{O} (μ) \mapsto ν, \\ μ \mapsto {\underline{θ}}^{P} (ν) & = {\bar{θ}}^{P} (μ) \mapsto ν . \end{matrix}$

Theorem 3.20. Let (X, θ) be an $L$ -fuzzy approximation space and $L$ -fuzzy multigranulation relations set be symmetric. Then for all $μ \in L^{X}$ , $μ \leq {\underline{θ}}^{P} (ν) \Leftrightarrow {\bar{θ}}^{P} (μ) \leq ν .$

4 The topological structure of $L$ -fuzzy multigranulation rough sets

Lai et al. [10] presents the definition of lower sets and upper sets in fuzzy preorder sets and studies the relationship between fuzzy preordered set and fuzzy topologies. This notion can be seen in [1 , 11]. In this paper, we investigate optimistic/pessimistic-lower and optimistic/pessimistic-upper sets and their topology structures in $L$ -fuzzy approximation spaces.

4.1 Optimistic/pessimistic-lower and optimistic/pessimistic-upper sets in $L$ -fuzzyapproximation spaces

Definition 4.1. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. An $L$ -set μ is said to be an optimistic-lower set if $\underset{i = 1}{\overset{m}{\land}} {μ (y) \otimes θ_{i} (x, y)} \leq μ (x)$ , ∀x, y ∈ X and optimistic-upper set if $\underset{i = 1}{\overset{m}{\land}} {μ (x) \otimes θ_{i} (x, y)} \leq μ (y)$ , ∀x, y ∈ X. Dually, μ is said to be a pessimistic-lower set if $\underset{i = 1}{\overset{m}{\lor}} {μ (y) \otimes θ_{i} (x, y)} \leq μ (x)$ , ∀x, y ∈ X and a pessimistic-upper set if $\underset{i = 1}{\overset{m}{\lor}} {μ (x) \otimes θ_{i} (x, y)} \leq μ (y)$ , ∀x, y ∈ X.

Theorem 4.2. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. Then the following hold: for all $μ \in L^{X}$ ,

μ is an optimistic-lower set ⇒ ${\bar{θ}}^{O} (μ) \leq μ$ .

μ is an optimistic-upper set ⇒ $μ \leq {\underline{θ}}^{O} (μ)$ .

μ is a pessimistic-lower set ⇔ ${\bar{θ}}^{P} (μ) \leq μ$ .

μ is a pessimistic-upper set ⇔ $μ \leq {\underline{θ}}^{P} (μ)$ .

Proof. (1) For all $μ \in L$ ,

μ is an optimistic-upper set

⇔ $\underset{i = 1}{\overset{m}{\land}} {μ (x) \otimes θ_{i} (x, y)} \leq μ (y)$

⇔ ${μ (x) \otimes \underset{i = 1}{\overset{m}{\land}} θ_{i} (x, y)} \leq \underset{i = 1}{\overset{m}{\land}} {μ (x) \otimes θ_{i} (x, y)} \leq μ (y)$ ⇔ $μ (x) \leq \lor_{y \in L} {\underset{i = 1}{\overset{m}{\land}} θ_{i} (x, y) \to μ (y)} = \lor_{y \in L} \underset{i = 1}{\overset{m}{\land}} {θ_{i} (x, y) \to μ (y)} \leq \underset{i = 1}{\overset{m}{\land}} \lor_{y \in L} {θ_{i} (x, y) \to μ (y)}$

⇔ $μ (x) \leq {\underline{θ}}^{O} (μ) (x), \forall y \in X$ .

The another item can be proven in a similar way as (1). □

Corollary 4.3. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space and θ be reflexive. Then we have: for all $μ \in L^{X}$ ,

μ is an optimistic-lower set ⇒ ${\bar{θ}}^{O} (μ) = μ$ .

μ is an optimistic-upper set ⇒ $μ = {\underline{θ}}^{O} (μ)$ .

μ is a pessimistic-lower set ⇔ ${\bar{θ}}^{P} (μ) = μ$ .

μ is a pessimistic-upper set ⇔ $μ = {\underline{θ}}^{P} (μ)$ .

Theorem 4.4. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. For all $μ \in L^{X}$ , then we have:

$L$ -fuzzy multigranulation relations set is transitive ⇔ ${\bar{θ}}^{O} (μ)$ is an optimistic-lower set ⇔ ${\underline{θ}}^{O} (μ)$ is an optimistic-upper set.

$L$ -fuzzy multigranulation relations set is transitive ⇔ ${\bar{θ}}^{P} (μ)$ is a pessimistic-lower set ⇔ ${\underline{θ}}^{P} (μ)$ is a pessimistic-upper set.

Proof.

For all $x \in X, μ \in L^{X} .$

$L$ -fuzzy multigranulation relations set is transitive

$\Leftrightarrow {\bar{θ}}^{O} ({\bar{θ}}^{O} (μ)) \leq {\bar{θ}}^{O} (μ), \Leftrightarrow \underset{i = 1}{\overset{m}{\land}} \lor_{y \in X} [θ_{i} (x, y) \otimes ({\bar{θ}}^{O} (μ) (y))] \leq {\bar{θ}}^{O} (μ) (x), \Leftrightarrow \underset{i = 1}{\overset{m}{\land}} ({\bar{θ}}^{O} (μ) (y)) \otimes θ_{i} (x, y) \leq {\bar{θ}}^{O} (μ) (x), \Leftrightarrow {\bar{θ}}^{O} (μ)$ is an optimistic-lower set.

The others can be proved similarly. □

Theorem 4.5. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. For all $μ \in L^{X}$ , then we have:

$L$ -fuzzy multigranulation relations set is Euclidean ⇔ ${\bar{θ}}^{O} (μ)$ is an optimistic-lower set ⇔ ${\underline{θ}}^{O} (μ)$ is an optimistic-upper set.

$L$ -fuzzy multigranulation relations set is Euclidean ⇔ ${\bar{θ}}^{P} (μ)$ is a pessimistic-lower set ⇔ ${\underline{θ}}^{P} (μ)$ is a pessimistic-upper set.

4.2 $L$ -topology generated by $L$ -relation

In this subsection, we consider that $L$ -fuzzy multigranulation relations set is reflexive.

Definition 4.6. A subset $τ \subseteq L^{X}$ is an $L$ -topology if it satisfies:

$\hat{0}, \hat{1} \in τ$ ,

{μ_i} _i∈Λ ⊆ τ implies ∪_i∈Λμ_i ∈ τ,

μ, ν ∈ τ implies μ ∩ ν ∈ τ.

Definition 4.7. A subset $τ \subseteq L^{X}$ is an Alexandrov $L$ -topology if it satisfies:

τ is an $L$ -topology,

$α \in L$ and μ ∈ τ implies α ⊗ μ ∈ τ,

$α \in L$ and μ ∈ τ implies α → μ ∈ τ.

Definition 4.8. A subset $τ \subseteq L^{X}$ is a clopen $L$ -topology if it satisfies:

τ is an $L$ -topology,

$μ \in L \Leftrightarrow \neg μ \in L$ .

Definition 4.9. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. We put

${\underline{H}}^{O} = {μ \in L^{X} ∣ {\underline{θ}}^{O} (μ) = μ}$ ,

${\bar{H}}^{O} = {μ \in L^{X} ∣ {\bar{θ}}^{O} (μ) = μ}$ ,

$H^{O} = {μ \in L^{X} ∣ {\underline{θ}}^{O} (μ) = μ = {\bar{θ}}^{O} (μ)}$ .

Remark 4.10.

μ is an optimistic-lower set $\Rightarrow μ \in {\bar{H}}^{O}$ .

μ is an optimistic-upper set $\Rightarrow μ \in {\underline{H}}^{O}$ .

Proof. It follows directly from Corollary 4.3. □

Theorem 4.11. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. For all $μ \in L^{X}$ , then the following hold:

$0, 1 \in {\underline{H}}^{O}$ .

For all i, $μ_{i} \in {\underline{H}}^{O}, i = 1, . ., n$ , $\underset{i = 1}{\overset{m}{\lor}} μ_{i} \in {\underline{H}}^{O}$ .

If θ_i ∈ {θ₁, . . . , θ_m} then $μ \in {\underline{H}}^{O}$ implies $θ_{i} (μ) \in {\underline{H}}^{O}$ when $L$ -fuzzy multigranulation relations set is transitive.

Proof.

It can be proved from Proposition 3.4 and that $L$ -fuzzy multigranulation relations set is reflexive.

From Proposition 3.8(3), we have ${\underline{θ}}^{O} (\underset{j = 1}{\overset{n}{\lor}} μ_{j}) \leq \underset{j = 1}{\overset{n}{\lor}} {\underline{θ}}^{O} (μ_{j}) = \underset{j = 1}{\overset{n}{\lor}} μ_{j}$ .

When θ is a reflexive, we have $\underset{j = 1}{\overset{n}{\lor}} μ_{j} \leq {\underline{θ}}^{O}$ $(\underset{j = 1}{\overset{n}{\lor}} μ_{j})$ . So ${\underline{θ}}^{O} (\underset{j = 1}{\overset{n}{\lor}} μ_{j}) = \underset{j = 1}{\overset{n}{\lor}} μ_{j}$ .

It can be proved from Corollary 3.15. □

Remark 4.12. ${\underline{H}}^{O}$ is not closed under finite intersection of sets.

We have ${\underline{θ}}^{O} (\underset{j = 1}{\overset{n}{\land}} μ_{j}) \geq \underset{j = 1}{\overset{n}{\land}} {\underline{θ}}^{O} (μ_{j}) .$

Theorem 4.13.

$0, 1 \in {\bar{H}}^{O}$ .

For all i, $μ_{i} \in {\bar{H}}^{O}, i = 1, . ., n$ , $\underset{i = 1}{\overset{m}{\land}} μ_{i} \in {\bar{H}}^{O}$ .

$\forall μ \in L^{X}$ , θ_i ∈ {θ₁, . . . , θ_m}, $μ \in {\bar{H}}^{O}$ implies $θ_{i} (μ) \in {\bar{H}}^{O}$ when $L$ -fuzzy multigranulation relations set is transitive.

$\forall μ \in L^{X}$ , $μ \in {\bar{H}}^{O}$ iff $\neg μ \in {\underline{H}}^{O}$ when $L$ is a complete regular residuated lattice.

Remark 4.14. ${\bar{H}}^{O}$ is not closed under finite union of sets.

Theorem 4.15. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space. For all $μ \in L^{X}$ , then the following hold:

0, 1 ∈ H^O.

$\forall μ \in L^{X}$ , θ_i ∈ {θ₁, . . . , θ_m}, it follows μ ∈ H^O implies θ_i (μ) ∈ H^O when $L$ -fuzzy multigranulation relations set is transitive.

$\forall μ \in L^{X}$ , μ ∈ H^O iff ¬μ ∈ H^O when $L$ is a complete regular residuated lattice.

Remark 4.16. In general,

H^O is not closed under finite intersection of sets.

H^O is not closed under finite union of sets.

Theorem 4.17. Let (X, θ) be an m dimensional $L$ -fuzzy multigranulation approximation space, in which $θ = {θ_{i}}_{i = \bar{1, m}}$ such as θ₁ ≤ . . . ≤ θ_m and $L$ -fuzzy multigranulation relations set be a transitive, then ${\underline{H}}^{O}$ forms a topology on $L^{X}$ .

Proof. Write ${\underline{H}}^{O}$ as τ. For $μ \in L^{X}$ ,

$\lor_{μ \in τ} μ (x) = \lor_{μ \in τ} {\underline{θ}}_{m}^{O} (μ) (x) \leq {\underline{θ}}_{m}^{O} (\lor_{μ \in τ} μ) (x) \leq \lor_{μ \in τ} μ (x)$ . So ${\underline{θ}}^{O} (\lor_{μ \in τ} μ) (x) = \lor_{μ \in τ} μ (x)$ ,i.e, ∨_μ∈τμ ∈ τ.

Besides, it follows from Theorem 4.11. Then τ forms a topology. □

Theorem 4.18. ${\underline{H}}^{O}$ determines a topology on U if and only if ${\bar{H}}^{O}$ determines a topology on U.

Theorem 4.19. Let $L$ be a finite partially ordered set. Then, ${\underline{H}}^{O} ({\bar{H}}^{O})$ is a lattice.

Theorem 4.20. ${\bar{H}}^{O}$ is dually isomorphic to ${\underline{H}}^{O}$ .

Proof. Let $f : {\bar{H}}^{O} \to {\underline{H}}^{O}$ be a mapping such that $\forall μ \in {\underline{H}}^{O}$ , f (μ) = ¬ μ and f be definable on Theorem 4.13. Then f is bijective. For $μ, ν \in {\underline{H}}^{O}$ , f (μ ∨ ν) = ¬ (μ ∨ ν) = μ ∧ ν = f (μ) ∧ f (ν) and $f (μ \land ν) = f ({\underline{θ}}^{O} (μ \land ν)) = \neg {\underline{θ}}^{O} (μ \land ν) = {\bar{θ}}^{O} [\neg (μ \land ν)] = \neg (μ \land ν) = \neg μ \lor \neg ν = f (μ) \lor f (ν)$ . □

Definition 4.21. Let (X, θ) be an m dimensional $L$ -fuzzy approximation space. We put

${\underline{H}}^{P} = {μ \in L^{X} ∣ {\underline{θ}}^{P} (μ) = μ}$ ,

${\bar{H}}^{P} = {μ \in L^{X} ∣ {\bar{θ}}^{P} (μ) = μ}$ ,

$H^{P} = {μ \in L^{X} ∣ {\underline{θ}}^{P} (μ) = μ = {\bar{θ}}^{O} (μ)}$ .

Theorem 4.22.

$\bar{0}, \bar{1} \in {\underline{H}}^{P}$ .

$\forall μ_{j} \in {\underline{H}}^{P}, 1 \leq j \leq n$ implies $\underset{j = 1}{\overset{n}{\lor}} μ_{j} \in {\underline{H}}^{P}$ .

$\forall μ_{j} \in {\underline{H}}^{P}, 1 \leq j \leq n$ implies $\underset{j = 1}{\overset{n}{\land}} μ_{j} \in {\underline{H}}^{P}$ .

$μ_{j} \in {\underline{H}}^{P}$ if and only if $\neg μ_{j} \in {\bar{H}}^{P}$ .

$μ_{j} \in {\bar{H}}^{P}$ if and only if $\neg μ_{j} \in {\underline{H}}^{P}$ .

Theorem 4.23.

${\underline{H}}^{P} = {μ ∣ μ is a pessimistic - upper set}$ .

${\bar{H}}^{P} = {μ ∣ μ is a pessimistic - lower set}$ .

It follows directly from Corollary 4.1.

From Theorem 4.2, we can conclude the following theorem.

Theorem 4.24. ${\underline{H}}^{P} = {\bar{H}}^{P} = H^{P}$ .

Theorem 4.25. ${\underline{H}}^{P}$ forms an Alexandrov $L$ -topology U.

Proof.

${\underline{H}}^{P}$ forms an $L$ -topology U. It can be proved from Theorem 4.22.

For all $μ \in {\underline{H}}^{P}$ , by Theorem 4.23, μ is a pessimistic-upper set, i.e. for all x, y ∈ X, we have $\underset{i = 1}{\overset{m}{\lor}} {μ (x) \otimes θ_{i} (x, y)} \leq μ (y)$ . And hence for all $α \in L$ , $α \otimes \underset{i = 1}{\overset{m}{\lor}} {μ (x) \otimes θ_{i} (x, y)} \leq α \otimes μ (y)$ ⇔ $\underset{i = 1}{\overset{m}{\lor}} {(\hat{α} \otimes μ) (x) \otimes θ_{i} (x, y)} \leq (\hat{α} \otimes μ) (y)$ . Applying Theorem 4.23 again, we have $(\hat{α} \otimes μ) \in τ$ .

It follows immediately from Proposition 3.6(3). □

Theorem 4.26. ${\underline{H}}^{P}$ forms a clopen topology on U.

5 Conclusion

This paper focuses on research on some kinds of essential properties and topological structures on $L$ -fuzzy pessimistic multigranulation rough set model and $L$ -fuzzy optimistic multigranulation rough set model. Besides, we see that $L$ -fuzzy pessimistic multigranulation rough set model satisfies some kinds of properties in "Axiomatic characterizations of generalized $L$ -fuzzy rough approximation operators”, so we do not introduce it in this paper.

Acknowledgments

This research was supported by the National Nature Science Foundation of China (Grant Nos. 11571010, 61179038).

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L -fuzzy multigranulation rough set based on residuated lattices

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Residuated lattice

2.2 L -sets, L -relations and L -topologies

2.3 L -fuzzy rough sets

3 L -fuzzy optimistic multigranulation rough sets and L -fuzzy pessimistic multigranulation rough sets

4 The topological structure of L -fuzzy multigranulation rough sets

4.1 Optimistic/pessimistic-lower and optimistic/pessimistic-upper sets in L -fuzzyapproximation spaces

4.2 L -topology generated by L -relation

5 Conclusion

Acknowledgments

References

2.2 $L$ -sets, $L$ -relations and $L$ -topologies

2.3 $L$ -fuzzy rough sets

3 $L$ -fuzzy optimistic multigranulation rough sets and $L$ -fuzzy pessimistic multigranulation rough sets

4 The topological structure of $L$ -fuzzy multigranulation rough sets

4.1 Optimistic/pessimistic-lower and optimistic/pessimistic-upper sets in $L$ -fuzzyapproximation spaces

4.2 $L$ -topology generated by $L$ -relation