Further study of multi-granulation fuzzy rough sets

Abstract

The multi-granulation fuzzy rough sets (MGFRS), proposed by Xu [35], is a meaningful contribution in the view of the generalization of the classical rough set model. In this paper, the main objective is to make further studies based on reference [35]. It is shown that the MGFRS do not meet the operations of intersection and union. To solve these problems, we propose the concepts of the equivalence relations with minimum (maximum) element and find that the MGFRS based on the equivalence relations with minimum (maximum) element meet the operations of intersection and union. At the same time, some algebraic properties of the MGFRS are discussed.

Keywords

Fuzzy rough set multiple granulation minimum (maximum) element operation property algebraic property

1 Introduction

Rough sets theory, initiated by Pawlak [24], is an excellent tool to handle imprecise, vague and uncertain information. The theory has been applied successfully in the fields of data mining, pattern recognition, conflict analysis and medical diagnosis [9 , 30]. However Pawlak’s rough set theory can not deal with many practical problems. In view of this fact, various generalized rough set models have been developed and their corresponding properties have been discussed.

On the one hand, rough set theory was generalized from the view of granular computing. It is well known that Pawlak’s rough set is based on a single granulation. Therefore, in order to enlarge the application scope of the single granulation rough set theory, Oian et al. initiated the multiple granulation rough set model [26]. Since Qian et al. explored the concept of multi-granulation rough set [27], many researchers have extended the multi-granulation rough set to the generalized multi-granulation rough sets [15 , 37–39]. Yang et al. studied the hierarchical structure properties of the multi-granulation rough set [40]. Lin et al. proposed the neighborhood-based multi-granulation rough set [17] and explored the topological properties of multi-granulation rough set [29]. Xu et al. developed the multi-granulation rough set in ordered information system [36].

On the other hand, rough set theory was generalized by combining with other theories that deal with uncertain knowledge. The fuzzy rough set model combining fuzzy set theory with rough set theory is one of the most important generalized rough set models [35]. It is well known that fuzzy set theory and rough set theory are complementary in terms of handling different kinds of uncertainty. Rough set theory deals with uncertainty following from ambiguity of information [24]. Fuzzy set theory, in turn, is good at dealing with possibilistic uncertainty, connected with imprecision of states, perceptions and preferences [4]. The two theories can be encountered in lots of specific problems. For this reason, rough set theory has been generalized by combining with fuzzy set theory [2 , 41–43]. Many researches discussed the fuzzy rough set model from different directions [3 , 34]. Dubois and Prade gave the concepts of rough fuzzy set and fuzzy rough set [5]. Morsi et al. discussed axiomatic of fuzzy rough set [21]. Wu et al. studied the (I, T)-fuzzy rough approximation operators [31]. Xu et al. proposed the multi-granulation fuzzy rough set model and studied the properties of MGFRS [35].

In addition, since Pawlak initiated the concept of rough sets, many researchers focussed on studying the algebraic properties of rough sets [1 , 42]. Kuroki et al. proposed concepts of upper and lower rough subgroup and studied the upper and lower approximation problem [11]. Li discussed rough algebra in the sense of quotient algebra [13]. Kong et al. proposed the optimistic covering fuzzy rough set model [10]. In reference [10], the operation properties and algebraic properties of optimistic covering fuzzy rough sets have been investigated. But the algebraic properties of MGFRS model is still an open problem. Then, the purpose of this paper is to research the operation properties and algebraic properties of the MGFRS. The paper is organized as follows. In Section 2, we briefly review some basic concepts of Pawlak’s rough set theory and fuzzy rough set theory [4]. Furthermore, we give the concepts of the equivalence relations with minimum (maximum) element. In Section 3, we discuss the properties of optimistic and pessimistic multi-granulation fuzzy approximations. In Section 4, we present that the optimistic and the pessimistic MGFRS with respect to equivalence relations do not satisfy the operations of intersection and union. Furthermore, we prove that the optimistic and the pessimistic MGFRS with respect to the equivalence relations with minimum (maximum) element meet the corresponding operation properties. In Section 5, we research the algebraic properties of the MGFRS. Finally, Section 6 concludes this study.

2 Preliminaries

In this section, we review some basic concepts and notions in the theory of fuzzy sets and multi-granulation fuzzy rough sets. More details can be seen in references [24 , 44].

$(U, ℝ)$ is referred to as an approximation space, where U = {x₁, x₂, ⋯ , x_n} is a non-empty finite set(also called the universe of discourse); $ℝ$ ={R₁, R₂, ⋯, R_s} is a set of the equivalence relations; denote [x] _R = {y| (x, y) ∈ R}, U/R = {[x] _R|x ∈ U}, then [x] _R is called the equivalence class of x and the quotient set U/R is called the equivalence class set of U.

A fuzzy set X is a mapping from U into the unit interval [0, 1], X : U → [0, 1], where each X (x) is the membership degree of x in X. The set of all the fuzzy sets defined on U is defined by F (U).

Definition 2.1. Let $(U, ℝ)$ be an approximation space, R an equivalence relation. For ∀X ∈ F (U), denote $\begin{matrix} \underline{R} (X) (x) = \land {X (y) ∣ y \in [x]_{R}} \\ \bar{R} (X) (x) = \lor {X (y) ∣ y \in [x]_{R}} \end{matrix}$ then, $\underline{R} (X)$ and $\bar{R} (X)$ are respectively called the lower and upper approximations of the fuzzy set X with respect to equivalence relation R, where “∧” means “min” and “∨” means “max”.

Definition 2.2. Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations. For ∀x ∈ U, [x] _{R
₁}, [x] _{R
₂}, ⋯ , [x] _{R
_s} are s equivalence classes of x. If there exists some [x] _{R
_i} such that [x] _{R
_i} ⊆ [x] _{R
_j}, j = 1, 2, ⋯ , s. Then R₁, R₂, ⋯ , R_s are called equivalence relations with minimum element. Moreover, [x] _{R
_i} is denoted by [x] _min. Furthermore, we denote $U / \cup_{i = 1}^{s} R_{i} = {[x]_{\min} ∣ x \in U}$

If there exists some [x] _{R
_i} such that [x] _{R
_j} ⊆ [x] _{R
_i}, j = 1, 2, ⋯ , s. Then R₁, R₂, ⋯ , R_s are called equivalence relations with maximum element. Moreover, [x] _{R
_i} is denoted by [x] _max. Furthermore, we denote $U / \cap_{i = 1}^{s} R_{i} = {[x]_{\max} ∣ x \in U}$

In the following, we give two examples to explain the Definition 2.2.

Example 2.1. Let U = {x₁, x₂, ⋯ , x₁₀}, and U/R₁ = {{x₁, x₂} , {x₃, x₄, x₉} , {x₅, x₆, x₇, x₈, x₁₀}} U/R₂ = {{x₁, x₂, x₃, x₄, x₉, x₁₀} , {x₅, x₆} , {x₇, x₈}} U/R₃ = {{x₁, x₂, x₇, x₈} , {x₃, x₄, x₅, x₆, x₉} , {x₁₀}}

Then, it is clear that R₁, R₂, R₃ are equivalence relations with minimum element. Meanwhile, we have that $U / \cup_{i = 1}^{3} R_{i} = {{x_{1}, x_{2}}, {x_{3}, x_{4}, x_{9}}, {x_{5}, x_{6}},$ {x₇, x₈} , {x₁₀}}.

Example 2.2. Let U = {x₁, x₂, ⋯ , x₁₀},and U/R₁ = {{x₁, x₂, x₃} , {x₄, x₅, x₆, x₈, x₉} , {x₇, x₁₀}} U/R₂ = {{x₁, x₂, x₃, x₇, x₁₀} , {x₄, x₅} , {x₆} , {x₈, x₉}} U/R₃ = {{x₁, x₇} , {x₂, x₃, x₁₀} , {x₄, x₆, x₈} , {x₅, x₉}}

Clearly, R₁, R₂, R₃ are equivalence relations with maximum element. Meanwhile, we have that $U / \cap_{i = 1}^{3} R_{i} = {{x_{1}, x_{2}, x_{3}, x_{7}, x_{10}}, {x_{4}, x_{5}, x_{6}, x_{8}, x_{9}}}$

Definition 2.3. [35] Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations. For any X ∈ F (U), denote $\begin{matrix} \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) (x) = \lor_{i = 1}^{s} {\land {X (y) ∣ y \in [x]_{R_{i}}}} \\ \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) (x) = \land_{i = 1}^{s} {\lor {X (y) ∣ y \in [x]_{R_{i}}}} \end{matrix}$

Then, $\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)$ and $\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)$ are respectively called the optimistic multi-granulation lower and upper fuzzy approximations of X with respect to equivalence relations R₁, R₂, ⋯ , R_s.

Denote $\begin{matrix} \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) (x) = \land_{i = 1}^{s} {\land {X (y) ∣ y \in [x]_{R_{i}}}} \\ \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) (x) = \lor_{i = 1}^{s} {\lor {X (y) ∣ y \in [x]_{R_{i}}}} \end{matrix}$

Then, $\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X)$ and $\bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X)$ are respectively called the pessimistic multi-granulation lower and upper fuzzy approximations of X with respect to equivalence relations R₁, R₂, ⋯ , R_s.

3 Properties of multi-granulation fuzzy approximations

3.1 Properties of optimistic multi-granulation fuzzy approximations

In the subsection, we will consider the properties of optimistic multi-granulation fuzzy approximations in an approximation space.

Proposition 3.1.1. [35] Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations. For any X, Y ∈ F (U), the following properties hold.

$\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (U) = U$

$\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (U) = U$

$\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (\emptyset) = \emptyset$

$\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (\emptyset) = \emptyset$

$\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \subseteq X$

$X \subseteq \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)$

$X \subseteq Y \Rightarrow \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \subseteq \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)$

$X \subseteq Y \Rightarrow \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \subseteq \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)$

$\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (\sim X) = \sim \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)$

$\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (\sim X) = \sim \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)$

Proposition 3.1.2. Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations with minimum element. For any X, Y ∈ F (U), the following properties hold.

$\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) = \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)$

$\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) = \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)$

$\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) = \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)$

$\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) = \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)$

Proof. (1) Since the number of the granulation is finite, we only prove the result is true when the approximation space has two equivalence relations with minimum element ( $R_{1}, R_{2} \subseteq ℝ$ ) for convenience. It is obvious that the proposition holds when R₁ = R₂ or X = Y. When R₁ ≠ R₂ and X ≠ Y, the proposition can be proved as follows.

For any x ∈ X, suppose that [x] _{R
₁} ⊆ [x] _{R
₂}. Then, we have that

$\begin{matrix} \underline{{OM}_{R_{1} + R_{2}}} (X) (x) \\ = {\land {X (y) ∣ y \in [x]_{R_{1}}}} \lor {\land {X (y) ∣ y \in [x]_{R_{2}}}} \\ = {\land {X (y) ∣ y \in [x]_{R_{1}}}} \end{matrix}$ (3.1)

Similarly, we also have that

$\begin{matrix} \underline{{OM}_{R_{1} + R_{2}}} (Y) (x) \\ = {\land {Y (y) ∣ y \in [x]_{R_{1}}}} \lor {\land {Y (y) ∣ y \in [x]_{R_{2}}}} \\ = {\land {Y (y) ∣ y \in [x]_{R_{1}}}} \end{matrix}$ (3.2)

By (3.1) and (3.2), it can be shown that

$\begin{matrix} (\underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y)) (x) \\ = \underline{{OM}_{R_{1} + R_{2}}} (X) (x) \land \underline{{OM}_{R_{1} + R_{2}}} (Y) (x) \\ = {\land {X (y) ∣ y \in [x]_{R_{1}}}} \land {\land {Y (y) ∣ y \in [x]_{R_{1}}}} \end{matrix}$ (3.3)

On the other hand, we have that $\begin{matrix} \underline{{OM}_{R_{1} + R_{2}}} (\underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y)) (x) \\ = {\land {(\underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y)) (y) ∣ y \in [x]_{R_{1}}}} \\ \lor {\land {(\underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y)) (y) ∣ y \in [x]_{R_{2}}}} \\ = {\land {(\underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y)) (y) ∣ y \in [x]_{R_{1}}}} \\ = {\land {\underline{{OM}_{R_{1} + R_{2}}} (X) (y) \land \underline{{OM}_{R_{1} + R_{2}}} (Y) (y) ∣ y \in [x]_{R_{1}}}} \\ = {\land {\underline{{OM}_{R_{1} + R_{2}}} (X) (y) ∣ y \in [x]_{R_{1}}}} \\ \land {\land {\underline{{OM}_{R_{1} + R_{2}}} (Y) (y) ∣ y \in [x]_{R_{1}}}} \end{matrix}$ (3.4)

For any y ∈ [x] _{R
₁}, it can be found that

$\begin{matrix} \underline{{OM}_{R_{1} + R_{2}}} (X) (y) \\ = {\land {X (z) ∣ z \in [y]_{R_{1}}}} \\ \lor {\land {X (z) ∣ z \in [y]_{R_{2}}}} \\ = {\land {X (z) ∣ z \in [y]_{R_{1}}}} \\ = {\land {X (y) ∣ y \in [x]_{R_{1}}}} \end{matrix}$ (3.5)

Similarly, for any y ∈ [x] _{R
₁}, it can be obtained that $\underline{{OM}_{R_{1} + R_{2}}} (Y) (y) = {\land {Y (y) ∣ y \in [x]_{R_{1}}}}$ (3.6)

By (3.5), (3.6) and (3.4), we can find that

$\begin{matrix} \underline{{OM}_{R_{1} + R_{2}}} (\underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y)) (x) \\ = {\land {X (y) ∣ y \in [x]_{R_{1}}}} \\ \land {\land {Y (y) ∣ y \in [x]_{R_{1}}}} \end{matrix}$ (3.7)

According to (3.3) and (3.7), we have that $\begin{matrix} \underline{{OM}_{R_{1} + R_{2}}} (\underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y)) \\ = \underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y) \end{matrix}$

Similarly, the proof of (2), (3) and (4) is similar to that of item (1).□

Proposition 3.1.3. Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations with minimum element. For any X, Y ∈ F (U) and x ∈ U, if [x] _min = {x}, then the following properties hold $\begin{matrix} (1) (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x) \\ = (\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x) \\ (2) (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x) \\ = (\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x) \end{matrix}$

For any x ∈ X, suppose that [x] _min = [x] _{R
₁} = {x} ⊆ [x] _{R
₂}. We have that $\begin{matrix} \underline{{OM}_{R_{1} + R_{2}}} (X) (x) \\ = {\land {X (y) ∣ y \in [x]_{R_{1}}}} \lor {\land {X (y) ∣ y \in [x]_{R_{2}}}} \\ = X (x) \end{matrix}$

Similarly, we also have that $\begin{matrix} \underline{{OM}_{R_{1} + R_{2}}} (Y) (x) = Y (x) \end{matrix}$

Then, it can be obtained that $\begin{matrix} (\underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y)) (x) \\ = \underline{{OM}_{R_{1} + R_{2}}} (X) (x) \land \underline{{OM}_{R_{1} + R_{2}}} (Y) (x) \\ = X (x) \land Y (x) \end{matrix}$

At the same time, we can conclude that $\begin{matrix} \bar{{OM}_{R_{1} + R_{2}}} (X) (x) \\ = {\lor {X (y) ∣ y \in [x]_{R_{1}}}} \land {\lor {X (y) ∣ y \in [x]_{R_{2}}}} \\ = X (x) \end{matrix}$

Similarly, we also have that $\begin{matrix} \bar{{OM}_{R_{1} + R_{2}}} (Y) (x) = Y (x) \end{matrix}$

Then, it can be shown that $\begin{matrix} (\bar{{OM}_{R_{1} + R_{2}}} (X) \cap \bar{{OM}_{R_{1} + R_{2}}} (Y)) (x) \\ = \bar{{OM}_{R_{1} + R_{2}}} (X) (x) \land \bar{{OM}_{R_{1} + R_{2}}} (Y) (x) \\ = X (x) \land Y (x) \end{matrix}$

Hence $\begin{matrix} (\underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y)) (x) \\ = (\bar{{OM}_{R_{1} + R_{2}}} (X) \cap \bar{{OM}_{R_{1} + R_{2}}} (Y)) (x) . \end{matrix}$

(2) The item can be proved similarly to (1).□

Proposition 3.1.4. Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations with minimum element and X, Y ∈ F (U). For any x ∈ U and any x_i, x_j ∈ [x] _min, then the following properties hold. $\begin{matrix} (1) (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{i}) \\ = (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{j}) \\ (2) (\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{i}) \\ = (\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{j}) \\ (3) (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{i}) \\ = (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{j}) \\ (4) (\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{i}) \\ = (\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{j}) \end{matrix}$

For any x ∈ X, suppose that [x] _{R
₁} ⊆ [x] _{R
₂}, we have that [x] _min = [x] _{R
₁} = [x_i] _{R
₁} ⊆ [x_i] _{R
₂}. Then, it can be obtained that $\begin{matrix} \underline{{OM}_{R_{1} + R_{2}}} (X) (x_{i}) \\ = {\land {X (y) ∣ y \in [x_{i}]_{R_{1}}}} \\ \lor {\land {X (y) ∣ y \in [x_{i}]_{R_{2}}}} \\ = {\land {X (y) ∣ y \in [x]_{\min}}} \\ \lor {\land {X (y) ∣ y \in [x_{i}]_{R_{2}}}} \\ = {\land {X (y) ∣ y \in [x]_{\min}}} \end{matrix}$

Similarly, we have that $\begin{matrix} \underline{{OM}_{R_{1} + R_{2}}} (Y) (x_{i}) & = {\land {Y (y) ∣ y \in [x]_{\min}}} \end{matrix}$

We thus conclude that $\begin{matrix} (\underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y)) (x_{i}) \\ = {\land {X (y) ∣ y \in [x]_{\min}}} \\ \land {\land {Y (y) ∣ y \in [x]_{\min}}} \end{matrix}$

Furthermore, we can obtain that $\begin{matrix} (\underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y)) (x_{j}) \\ = {\land {X (y) ∣ y \in [x]_{\min}}} \\ \land {\land {Y (y) ∣ y \in [x]_{\min}}} \end{matrix}$

Hence $\begin{matrix} (\underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y)) (x_{i}) \\ = (\underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y)) (x_{j}) \end{matrix}$

Similarly, the proof of (2), (3) and (4) is similar to that of item (1). □

3.2 Properties of pessimistic multi-granulation fuzzy approximations

In the subsection, we will investigate the properties of pessimistic multi-granulation fuzzy approximations in an approximation space.

Proposition 3.2.1. [35] Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations. For any X, Y ∈ F (U), the following properties hold.

$\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (U) = U$

$\bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (U) = U$

$\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (\emptyset) = \emptyset$

$\bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (\emptyset) = \emptyset$

$\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \subseteq X$

$X \subseteq \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X)$

$X \subseteq Y \Rightarrow \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \subseteq \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)$

$X \subseteq Y \Rightarrow \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \subseteq \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)$

$\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (\sim X) = \sim \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X)$

$\bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (\sim X) = \sim \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X)$

Proposition 3.2.2. Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations with maximum element. For any X, Y ∈ F (U), the following properties hold. $\begin{matrix} (1) \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) \\ = \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \\ (2) \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) \\ = \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \\ (3) \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (\bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) \\ = \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{MPM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \\ (4) \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (\bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) \\ = \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \end{matrix}$

Proof. The proposition can be proved similarly to Proposition 3.1.2.□

Proposition 3.2.3. Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations with maximum element. For any x ∈ U, if [x] _max = {x}, then the following properties hold. $\begin{matrix} (1) (\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x) \\ = (\bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x) \\ (2) (\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x) \\ = (\bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x) \end{matrix}$

Proof. The proposition can be proved similarly to Proposition 3.1.3. □

Proposition 3.2.4. Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations with maximum element and X, Y ∈ F (U). For any x ∈ U and any x_i, x_j ∈ [x] _max, the following properties hold. $\begin{matrix} (1) (\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{i}) \\ = (\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{j}) \\ (2) (\bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{i}) \\ = (\bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{j}) \\ (3) (\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{i}) \\ = (\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{j}) \\ (4) (\bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{i}) \\ = (\bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{j}) \end{matrix}$

Proof. The proof of this proposition is similar to that of Proposition 3.1.4.□

4 Operation properties of MGFRS

4.1 Operation properties of optimistic MGFRS

Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations. Forany X ∈ F (U), the pair $(\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X))$ is called the optimistic MGFRS of X. Obviously, it can be found that ${(\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) ∣ X \in F (U)}$ isall the optimistic MGFRS in approximation space and is denoted by $𝔹_{O}$ . ie., $𝔹_{O} = {(\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) ∣ X \in F (U)}$ .

In the following, we will propose the definitions of complement, intersection and union of the optimistic MGFRS.

Definition 4.1.1. Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations. For any X ∈ F (U), the complement of $(\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \in 𝔹_{O}$ can be defined as follows. $\begin{matrix} \sim (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \\ = (\sim \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \sim \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \end{matrix}$

Remark 4.1.1. According to Proposition 3.1.1, we have $\begin{matrix} \sim (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \\ = (\sim \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \sim \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \\ = (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (\sim X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (\sim X)) \end{matrix}$

In other words, the complement of the optimistic multi-granulation fuzzy rough set of X is equal to the optimistic multi-granulation fuzzy rough set of ∼X. Therefore, the optimistic multi-granulation fuzzy rough sets meet the complement operation.

Definition 4.1.2. Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations. For $(\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X))$ , $(\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) \in 𝔹_{O}$ , the intersection and union of them can be defined as follows. $\begin{matrix} (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \\ \cap (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) \\ = (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y), \\ \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) \\ (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \\ \cup (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) \\ = (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y), \\ \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) \end{matrix}$

Similarly, we can raise such a question: do the optimistic multi-granulation fuzzy rough sets meet the operations of intersection and union? In the following, we employ two examples to illustrate the question.

Example 4.1.1. Let $(U, ℝ)$ be an approximation space, U = {x₁, x₂, x₃} , U/R₁ = {{x₁, x₂} , {x₃}} and U/R₂ = {{x₁} , {x₂, x₃}}. For two fuzzy sets X = (0.1, 0.2, 0.3) , Y = (0.3, 0.1, 0.2) . It is obvious that there does not exist a fuzzy set V ∈ F (U) such that $\begin{matrix} \underline{{OM}_{R_{1} + R_{2}}} (V) = \underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y) \\ \bar{{OM}_{R_{1} + R_{2}}} (V) = \bar{{OM}_{R_{1} + R_{2}}} (X) \cap \bar{{OM}_{R_{1} + R_{2}}} (Y) \end{matrix}$

Similarly, according to Proposition 3.1.1, for two fuzzy sets X′ = (0.9, 0.8, 0.7), Y′ = (0.7, 0.9, 0.8). It is clear that there does not exist a fuzzy set W ∈ F (U) such that $\begin{matrix} \underline{{OM}_{R_{1} + R_{2}}} (W) = \underline{{OM}_{R_{1} + R_{2}}} (X') \cup \underline{{OM}_{R_{1} + R_{2}}} (Y') \\ \bar{{OM}_{R_{1} + R_{2}}} (W) = \bar{{OM}_{R_{1} + R_{2}}} (X') \cup \bar{{OM}_{R_{1} + R_{2}}} (Y') \end{matrix}$

Example 4.1.1 indicates that the optimistic multi-granulation fuzzy rough sets with respect to equivalence relations do not meet the operations of intersection and union.

Example 4.1.2. Let $(U, ℝ)$ be an approximation space, U = {x₁, x₂, x₃, x₄, x₅, x₆} , U/R₁ = {{x₁, x₂, x₅} , {x₃, x₆} , {x₄}} and U/R₂ = {{x₁, x₅} , {x₂} , {x₃, x₄, x₆}}. Clearly, R₁, R₂ are equivalence relations with minimum element.

For two fuzzy sets X = (0.3, 0.4, 0.5, 0.6, 0.7, 0.8) , Y = (0.9, 0.7, 0.5, 0.3, 0.2, 0.1) . Let V = (0.2, 0.4, 0.1, 0.3, 0.7, 0.5), then we have $\begin{matrix} \underline{{OM}_{R_{1} + R_{2}}} (V) = \underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y) \\ \bar{{OM}_{R_{1} + R_{2}}} (V) = \bar{{OM}_{R_{1} + R_{2}}} (X) \cap \bar{{OM}_{R_{1} + R_{2}}} (Y) \end{matrix}$

Let W = (0.3, 0.7, 0.8, 0.6, 0.9, 0.5), then we have $\begin{matrix} \underline{{OM}_{R_{1} + R_{2}}} (W) = \underline{{OM}_{R_{1} + R_{2}}} (X) \cup \underline{{OM}_{R_{1} + R_{2}}} (Y) \\ \bar{{OM}_{R_{1} + R_{2}}} (W) = \bar{{OM}_{R_{1} + R_{2}}} (X) \cup \bar{{OM}_{R_{1} + R_{2}}} (Y) \end{matrix}$

From Example 4.1.2, it can be found that the selections of Z and W are not unique, but $\underline{{OM}_{R_{1} + R_{2}}} (Z), \bar{{OM}_{R_{1} + R_{2}}} (Z), \underline{{OM}_{R_{1} + R_{2}}} (W)$ , and $\bar{{OM}_{R_{1} + R_{2}}} (W)$ are unique, respectively.

Let $(U, ℝ)$ be an approximation space, U = {x₁, x₂, ⋯ , x_n}, R₁, R₂ the equivalence relations with minimum element and X, Y ∈ F (U).

Case 1.1. If [x_i] _min = {x_i}, where x_i ∈ U, i = 1, 2, ⋯ , n. According to Proposition 3.1.3, we denote $\begin{matrix} V (x_{i}) & = (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{i}) \\ = (\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{i}); \end{matrix}$

Case 1.2. If [x_i] _min = {x_{i
₁}, x_{i
₂}, ⋯ , x_{i
_k}}, where x_{i
_u} ∈ U, i = 1, 2, ⋯ , n ; u = 1, 2, ⋯ , k. According to Proposition 3.1.4, we denote $\begin{matrix} V (x_{i_{1}}) = (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{i_{1}}) \\ V (x_{i_{2}}) = V (x_{i_{3}}) = \dots = V (x_{i_{k}}) \\ = (\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{i_{2}}); \end{matrix}$

By case 1.1 and case 1.2, denote V = (V (x₁) , V (x₂) , ⋯ , V (x_n)).

Case 2.1. If [x_j] _min = {x_j}, where x_j ∈ U, j = 1, 2, ⋯ , n. According to Proposition 3.1.3, we denote $\begin{matrix} W (x_{j}) & = (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{j}) \\ = (\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{j}) \end{matrix}$

Case 2.2. If [x_j] _min = {x_{j
₁}, x_{j
₂}, ⋯ , x_{j
_t}}, where x_{j
_v} ∈ U, j = 1, 2, ⋯ , n ; v = 1, 2, ⋯ , t. According to Proposition 3.1.4, we denote $\begin{matrix} W (x_{j_{1}}) \\ = (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{j_{1}}) \\ W (x_{j_{2}}) = W (x_{j_{3}}) = \dots = W (x_{j_{t}}) \\ = (\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{j_{2}}) \end{matrix}$

By case 2.1 and case 2.2, denote W = (W (x₁) , W (x₂) , ⋯ , W (x_n)).

According to the constructions of V, W defined above, we have the following conclusions.

Proposition 4.1.1. Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations with minimum element. For any X, Y ∈ F (U) and V, W defined above, the following propertieshold. $\begin{matrix} (1) \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \\ \subseteq V \subseteq \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \\ (2) \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \\ \subseteq W \subseteq \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \end{matrix}$

Proof. It is easy to prove by the constructions of V and W.

Proposition 4.1.2. Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations with minimum element. For any X, Y ∈ F (U) and V, W defined above, the following properties hold. $\begin{matrix} (1) \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (V) \\ = \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \\ (2) \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (V) \\ = \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \\ (3) \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (W) \\ = \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \\ (4) \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (W) \\ = \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \end{matrix}$

Proof. (1) Since the number of the granulation is finite, we only prove the result is true when the approximation space has two equivalence relations with minimum element ( $R_{1}, R_{2} \subseteq ℝ$ ) for convenience. It is obvious that the proposition holds when R₁ = R₂. When R₁ ≠ R₂, the proposition can be proved as follows.

For any x ∈ U, suppose that [x] _min = [x] _{R
₁}. It can be known that $\begin{matrix} \underline{{OM}_{R_{1} + R_{2}}} (V) (x) \\ = {\land {V (y) ∣ y \in [x]_{R_{1}}}} \\ \lor {\land {V (y) ∣ y \in [x]_{R_{2}}}} \\ = {\land {V (y) ∣ y \in [x]_{\min}}} \\ \lor {\land {V (y) ∣ y \in [x_{i}]_{R_{2}}}} \\ = {\land {V (y) ∣ y \in [x]_{\min}}} \end{matrix}$

According to the construction of V and Proposition 3.1.4, there must exist a y₁ ∈ [x] _min such that $\begin{matrix} {\land {V (y) ∣ y \in [x]_{\min}}} \\ = \underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y) (y_{1}) \\ = \underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y) (x) \end{matrix}$

Hence

$\underline{{OM}_{R_{1} + R_{2}}} (V) = \underline{{OM}_{R_{1} + R_{2}}} (X) \cap \underline{{OM}_{R_{1} + R_{2}}} (Y) .$

Similarly, the proof of (2), (3) and (4) is similar to that of item (1).□

Remark 4.1.2. From Proposition 4.1.2, we can conclude that the optimistic multi-granulation fuzzy rough sets with respect to the equivalence relations with minimum element meet the operations of intersection and union defined above.

4.2 Operation properties of pessimistic MGFRS

Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations. For any X ∈ F (U), the pair $(\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X))$ is called the pessimistic MGFRS of X. Obviously, it can be known that ${(\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X)) ∣ X \in F (U)}$ is all the pessimistic MGFRS in approximation space and is denoted by $𝔹_{P}$ . ie., $𝔹_{P} = {(\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X)) ∣ X \in F (U)}$ .

Similar to Definitions 4.1.1 and 4.1.2, we can propose the definitions of complement, intersection and union of the pessimistic MGFRS.

Clearly, it can be obtained that the complement of the pessimistic multi-granulation fuzzy rough set of X is equal to the pessimistic MGFRS of ∼X. Therefore, the pessimistic MGFRS meet the complement operation.

Meanwhile, we can raise the question: do the pessimistic MGFRS satisfy the operations of intersection and union? In the following, we still employ two examples to illustrate the question.

Example 4.2.1. Let $(U, ℝ)$ be an approximation space, U = {x₁, x₂, x₃} , U/R₁ = {{x₁, x₂} , {x₃}} and U/R₂ = {{x₁} , {x₂, x₃}}. For two fuzzy sets X = (0.1, 0.2, 0.3) , Y = (0.3, 0.2, 0.1). It is obvious that there does not exist a fuzzy set V ∈ F (U) such that $\begin{matrix} \underline{{PM}_{R_{1} + R_{2}}} (V) = \underline{{PM}_{R_{1} + R_{2}}} (X) \cap \underline{{PM}_{R_{1} + R_{2}}} (Y) \\ \bar{{PM}_{R_{1} + R_{2}}} (V) = \bar{{PM}_{R_{1} + R_{2}}} (X) \cap \bar{{PM}_{R_{1} + R_{2}}} (Y) \end{matrix}$

Similarly, according to Proposition 3.2.1, for two fuzzy sets X′ = (0.9, 0.8, 0.7), Y′ = (0.7, 0.8, 0.9). It is clear that there does not exist a fuzzy set W ∈ F (U) such that $\begin{matrix} \underline{{PM}_{R_{1} + R_{2}}} (W) = \underline{{PM}_{R_{1} + R_{2}}} (X') \cup \underline{{PM}_{R_{1} + R_{2}}} (Y') \\ \bar{{PM}_{R_{1} + R_{2}}} (W) = \bar{{PM}_{R_{1} + R_{2}}} (X') \cup \bar{{PM}_{R_{1} + R_{2}}} (Y') \end{matrix}$

Example 4.2.1 indicates that the pessimistic MGFRS with respect to equivalence relations do not meet the operations of intersection and union.

Example 4.2.2. Let $(U, ℝ)$ be an approximation space, U = {x₁, x₂, x₃, x₄, x₅, x₆} , U/R₁ = {{x₁, x₂, x₅} , {x₃, x₆} , {x₄}} and U/R₂ = {{x₁, x₅} , {x₂} , {x₃, x₄, x₆}}. Clearly, R₁, R₂ are two equivalence relations with maximum element. For two fuzzy sets X = (0.3, 0.4, 0.5, 0.6, 0.7, 0.8) , Y = (0.9, 0.7, 0.5, 0.3, 0.2, 0.1). Let V = (0.2, 0.7, 0.1, 0.5, 0.7, 0.5), then we have that $\begin{matrix} \underline{{PM}_{R_{1} + R_{2}}} (V) = \underline{{PM}_{R_{1} + R_{2}}} (X) \cap \underline{{PM}_{R_{1} + R_{2}}} (Y) \\ \bar{{PM}_{R_{1} + R_{2}}} (V) = \bar{{PM}_{R_{1} + R_{2}}} (X) \cap \bar{{PM}_{R_{1} + R_{2}}} (Y) \end{matrix}$

Let W = (0.3, 0.9, 0.5, 0.8, 0.9, 0.8), then we get that $\begin{matrix} \underline{{PM}_{R_{1} + R_{2}}} (W) = \underline{{PM}_{R_{1} + R_{2}}} (X) \cup \underline{{PM}_{R_{1} + R_{2}}} (Y) \\ \bar{{PM}_{R_{1} + R_{2}}} (W) = \bar{{PM}_{R_{1} + R_{2}}} (X) \cup \bar{{PM}_{R_{1} + R_{2}}} (Y) \end{matrix}$

From Example 4.1.2, we have known that the selections of V and W are not unique, but $\underline{{PM}_{R_{1} + R_{2}}} (Z), \bar{{PM}_{R_{1} + R_{2}}} (Z), \underline{{PM}_{R_{1} + R_{2}}} (W)$ and $\bar{{PM}_{R_{1} + R_{2}}} (W)$ are unique, respectively.

Let $(U, ℝ)$ be an approximation space, U = {x₁, x₂, ⋯ , x_n}, R₁, R₂ two equivalence relations with maximum element and X, Y ∈ F (U).

Case 1.1. If [x_i] _max = {x_i}, where x_i ∈ U, i = 1, 2, ⋯ , n. According to Proposition 3.2.3, we denote $\begin{matrix} V (x_{i}) & = (\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{i}) \\ = (\bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{i}); \end{matrix}$

Case 1.2. If [x_i] _max = {x_{i
₁}, x_{i
₂}, ⋯ , x_{i
_k}}, where x_{i
_u} ∈ U, i = 1, 2, ⋯ , n ; u = 1, 2, ⋯ , k. According to Proposition 3.2.4, we denote $\begin{matrix} V (x_{i_{1}}) \\ = (\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{i_{1}}) \\ V (x_{i_{2}}) = V (x_{i_{3}}) = \dots = V (x_{i_{k}}) \\ = (\bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{i_{2}}) \end{matrix}$

By case 1.1 and case 1.2, denote V = (V (x₁) , V (x₂) , ⋯ , V (x_n)).

Case 2.1. If [x_j] _max = {x_j}, where x_j ∈ U, j = 1, 2, ⋯ , n. According to Proposition 3.2.3, we denote $\begin{matrix} W (x_{j}) & = (\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{j}) \\ = (\bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{j}); \end{matrix}$

Case 2.2. If [x_j] _max = {x_{j
₁}, x_{j
₂}, ⋯ , x_{j
_t}}, where x_{j
_v} ∈ U, j = 1, 2, ⋯ , n ; v = 1, 2, ⋯ , t. According to Proposition 3.2.4, we denote $\begin{matrix} W (x_{j_{1}}) \\ = (\underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{j_{1}}) \\ W (x_{j_{2}}) = W (x_{j_{3}}) = \dots = W (x_{j_{t}}) \\ = (\bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) (x_{j_{2}}) \end{matrix}$

By case 2.1 and case 2.2, denote W = (W (x₁) , W (x₂) , ⋯ , W (x_n)).

According to the constructions of V, W defined above, we have the following conclusions.

Proposition 4.2.1. Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations with maximum element. For any X, Y ∈ F (U) and V, W defined above, the following properties hold. $\begin{matrix} (1) \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \\ \subseteq V \subseteq \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \\ (2) \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \\ \subseteq W \subseteq \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \end{matrix}$

Proof. It is easy to prove by the constructions of V and W.

Proposition 4.2.2. Let $(U, ℝ)$ be an approximation space, R₁, R₂, ⋯ , R_s the equivalence relations with maximum element. For any X, Y ∈ F (U) and V, W defined above, the following properties hold. $\begin{matrix} (1) \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (V) \\ = \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \\ (2) \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (V) \\ = \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \\ (3) \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (W) \\ = \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \underline{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \\ (4) \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (W) \\ = \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cup \bar{{PM}_{\sum_{i = 1}^{s} R_{i}}} (Y) \end{matrix}$

Proof. The proof of this proposition is similar to that of Proposition 4.1.2.□

Remark 4.2.2. From Proposition 4.2.2, we can conclude that all the pessimistic multi-granulation fuzzy rough sets with respect to the equivalence relations with maximum element meet the operations of intersection and union.

5 Algebraic properties of MGFRS

5.1 Algebraic properties of optimistic MGFRS

In this subsection, we will investigate the algebraic properties of optimistic multi-granulation fuzzy rough sets in an approximation space. Firstly, we review some basic definitions of algebraic theory.

Definition 5.1.1. A partially ordered set L is a lattice, if A ∩ B ∈ L and A ∪ B ∈ L, for all A, B ∈ L.

Definition 5.1.2. A lattice L is a distribute lattice, if A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) and A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C), for all A, B, C ∈ L.

Definition 5.1.3. A distribute lattice 〈L, ∪ , ∩ , ∼ 〉 is a soft algebra, if for any A, B ∈ L, the following three properties are satisfied:

A ∪ 0 = A, A ∩ 0 =0, A ∪ 1 =1, A ∩ 1 = A;

∼ (∼ A) = A;

∼ (A ∪ B) = (∼ A) ∩ (∼ B) , ∼ (A ∩ B) = (∼ A) ∪ (∼ B).

In this subsection, let (U, R) be an approximation space, R₁, R₂ · · · R₅ the equivalence relations with minimum element. Then, we have the following conclusions.

Proposition 5.1.1. $(𝔹_{O}, \cup, \cap)$ is a lattice.

Proof. It is immediate by Proposition 4.1.2, Definition 4.1.1 and Definition 5.1.1. □

Proposition 5.1.2. $(𝔹_{O}, \cup, \cap)$ is a distribute lattice.

Proof. It is immediate by Proposition 4.1.2, Definitions 4.1.1, 4.1.2 and 5.1.2. □

According to Proposition 3.1.1, let $(\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (\emptyset), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (\emptyset)) = (\emptyset, \emptyset) = 0$ and $(\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (U), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (U)) = (U, U) = 1 .$ Then, we have the following result.

Proposition 5.1.3. $(𝔹_{O}, \cup, \cap, \sim)$ is a soft algebra.

Proof. For any $(\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)), (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) \in 𝔹_{O}$ , by Proposition 3.1.1, Definitions 4.1.1 and 4.1.2, we have that $\begin{matrix} (1) (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \cup 0 \\ = (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \\ (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \cap 0 = 0 \\ (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \cup 1 = 1 \\ (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \cap 1 \\ = (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \end{matrix}$

Then, 0 and 1 are the minimal and maximal element of $(𝔹_{𝕆}, \cup, \cap, \sim)$ , respectively. $\begin{matrix} (2) \sim (\sim (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X))) \\ = \sim (\sim \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \sim \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \\ = \sim (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (\sim X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (\sim X)) \\ = (\sim \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (\sim X), \sim \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (\sim X)) \\ = (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \end{matrix}$ $\begin{matrix} (3) \sim ((\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \\ \cap (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y), \bar{{MOM}_{\sum_{i = 1}^{s} R_{i}}} (Y))) \\ = \sim (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y), \\ \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) \\ = (\sim (\bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)), \\ \sim (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X) \cap \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y))) \\ = ((\sim \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \cup (\sim \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)), \\ (\sim \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \cup (\sim \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y))) \\ = (\sim \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \sim \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \\ \cup (\sim \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y), \sim \underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y)) \\ = (\sim (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X))) \\ \cup (\sim (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y), \bar{{MOM}_{\sum_{i = 1}^{s} R_{i}}} (Y))) \end{matrix}$

Similarly, we have that $\begin{matrix} \sim ((\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X)) \\ \cup (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y))) \\ = (\sim (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (X))) \\ \cap (\sim (\underline{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y), \bar{{OM}_{\sum_{i = 1}^{s} R_{i}}} (Y))) \end{matrix}$

Hence, it can be known that $(𝔹_{O}, \cup, \cap, \sim)$ is a soft algebra.□

5.2 Algebraic properties of pessimistic MGFRS

In this subsection, we will investigate the algebraic properties of pessimistic MGFRS in an approximation space. Let (U, R) be an approximation space, R₁, R₂, · · · R₅ the equivalence relations with maximum element. Similar to the algebraic properties of optimistic MGFRS, we have the following results.

Proposition 5.2.1. $(𝔹_{P}, \cup, \cap)$ is a lattice.

Proof. This proposition can be proved similarly to Proposition 5.1.1.□

Proposition 5.2.2. $(𝔹_{P}, \cup, \cap)$ is a distribute lattice.

Proof. This proposition can be proved similarly to Proposition 5.1.2.□

Proposition 5.2.3. $(𝔹_{P}, \cup, \cap, \sim)$ is a soft algebra.

Proof. This proposition can be proved similarly to Proposition 5.1.3.□

6 Conclusion

To deal with problems of uncertainty and imprecision easily, Xu et al. proposed the multi-granulation fuzzy rough set model based on equivalence relations [35]. The model is a meaningful contribution in the view of the generalization of the classical rough set model. In this paper, the main objective is to make further studies based on reference [35]. We made conclusions that the multi-granulation fuzzy rough sets with respect to equivalence relations do not meet the operations of intersection and union. To overcome this limitation, we gave the concepts of the equivalence relations with minimum (maximum) element and proved that the multi-granulation fuzzy rough sets with respect to the equivalence relations with minimum (maximum) element meet the operations of intersection and union. Furthermore, correspondingly algebraic properties were discussed.

However, there are still several issues in multi-covering fuzzy rough set model deserving further research. In this paper, we just briefly introduced some algebraic properties of the multi-covering fuzzy rough sets. More algebraic properties need to be further studied. On the other hand, topological property of multi-covering rough sets is still an open problem. We will investigate these issues in the near future.

Footnotes

Acknowledgments

This work is supported by the Natural Science Foundation of China (Nos. 11161003, 11261006, 61472463, 61402064) and the Science-Technology Foundation of the Education Department of Fujian Province, China (No. B15111).

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