Covering-based multi-granulation fuzzy rough sets

Abstract

As a new and meaningful extension of the Pawlak rough set, multi-granulation rough sets (MGRSs) have attracted much attention and fruitful achievements have been reported in different aspects. By combining with fuzzy rough set, the paper introduces multi-granulation fuzzy rough sets in the covering approximation space, namely, covering-based multi-granulation fuzzy rough sets (CMFRS), which form the extension of fuzzy rough sets. We first investigate several important properties of lower and upper approximations of concepts in covering-based multi-granulation fuzzy rough sets and elaborate on the differences between the proposed models and the existing ones in literature. By employing the notions of reduct and exclusion of a covering, the paper studies the necessary and sufficient conditions for two CMFRS to generate identical lower and upper approximations of a target concept in the given covering approximation space. Finally, the relationships between the new models are explored in the paper.

Keywords

Covering exclusion fuzzy rough sets multi-granulation reduct

1 Introduction

As a tool for data analysis and processing, rough set theory has been successfully applied to many fields, such as such as pattern recognition, machine learning, knowledge acquisition and data mining [1 –14].

As Pawlak [15] introduced the rough set approximation operators with respect to equivalence relations, this may induced some restrictions in practical applications. To enhance the scope of usage of Pawlak’s rough sets, many meaningful extensions have been proposed, for example, the graded rough sets [16], tolerance or similarity relations based rough sets [18 –20], rough fuzzy sets and fuzzy rough sets [21], etc. As one of the extension models, covering-based rough sets, which was first proposed by Zakowski [22], has attracted much attention and induced interesting results. Pomykala [23] introduced several pairs of dual approximation operators of covering-based rough sets. Bonikowski et al. [24] studied the covering-based rough approximation operators based on the mutual correspondence of the concepts of extension and intension. Tsang et al. [25] proposed a new upper approximation operator of covering-based rough sets, which may help to extract more efficient rules. Mordeson [26] applied covering-based rough sets to ideal theory and discussed basic properties of the upper approximation operator and showed how it can be used to give algebraic structural properties of certain subsets. Zhu and Wang [27, 28] proposed three types of covering-based rough sets and introduced the notions of reduct and exclusion of a covering. Grzymala-Busse [63 –66] has done a series of studies on the missing attribute values, which also an important aspect of covering-based rough approximation operators. Liu and Sai gave a comparison of two types of rough sets induced by coverings [67]. Qin et al. discussed some properties of five pairs of dual covering approximation operators, and presented conditions with which these covering approximation operators are identical [68]. Zhang and Luo changed five pairs of covering approximation operators into the same pair of relation approximation operators [69]. Restrepo et al. investigated partial order relation for approximation operators in covering based rough sets [70]. Restrepo et al. used the concepts of duality, conjugacy and adjointness to establish relationships between the most commonly used covering approximation operators [71]. Zhu et al [29, 30] discussed some measurements of uncertainty in covering-based rough sets. Ma [31] introduced some types of neighborhood-related covering rough sets. Zhang et al. [32] have shown the closeness of union and intersection operations of rough approximation pairs. As a guideline and directional research of covering-based rough sets, Yao et al. [33] proposed a framework for the study of covering based rough set approximations. They summarized and classified the existing approximation operators into element-based, granule-based and subsystem-based definitions, which enables us to reproduce many existing approximation operators and introduce some new approximationoperators.

From the viewpoint of Granular Computing [34 –37], in Pawlak’s rough set model and its various extended rough set models, a target concept is characterized by a pair of approximations under a single granulation, which cannot be applied in some practical multi-granulation circumstances, and therefore, they can be named as the single-granulation rough sets. To enable rough set theory to fit the multi-granulation circumstances, Qian et al. [38, 39] introduced the concept of multi-granulation rough sets (MGRS), where the approximations of a target concept are characterized under multi-equivalence relations. In MGRS, due to the different requirements, a target concept can be described by two different strategies, i.e., “seeking common reserving difference” and “seeking common rejecting difference”. Following these two strategies, the optimistic and pessimistic MGRS approximations have been proposed, respectively. Up to now, many researchers have focused on the development of the MGRS. Those studies can be divided into three categories. The first category is the construction of approximation operators. For instance, Qian et al. [40] extended MGRS to incomplete information systems with respect to multiple tolerance relations. Yang et al. [41] constructed multi-granulation rough sets based on the fuzzy approximation space. Xu et al. [42 –44] constructed three kinds of multi-granulation rough sets based on the tolerance, ordered and generalized relations, respectively. Lin et al. [45] discussed two kinds of neighborhood-based multi-granulation rough sets and three kinds of covering based multigranulation rough sets [46]. Liu et al. [47 –49] studied the covering-based multi-granulation rough sets by employing the minimal and maximum descriptions of target concepts. Yang et al. [50] discussed the updating multigranulation rough approximations with increasing of granular structures. Xu et al. introduced [51] multi-granulation fuzzy rough sets and Huang et al. [52] proposed intuitionistic fuzzy multigranulation rough sets. Qian et al. [53] developed the multigranulation decision-theoretic rough set and proved that many existing multigranulation rough set models can be derived from the multigranulation decision-theoretic rough set framework. The second category is the topology properties of multi-granulation rough sets. For example, Yang et al. [54] discussed the hierarchical structures of multi-granulation rough sets. She et al. [55] and He et al. [56] investigated the topological and lattice-theoretic properties of multi-granulation rough sets. Lin et al. [57] explored multigranulationtopological rough space and proposed topological granularity and topological entropy to characterize the uncertainty of the multigranulation topological rough space. Geetha et al. [58] discussed the algebraic properties of multigranulation rough set on two universes. The third category is feature selection. For example, Liang et al. [59] introduced an efficient rough feature selection algorithm from a multi-granulation view. Lin et al. [60] discussed feature selection via neighborhood multi-granulation fusion. Liu et al. [61] proposed a rule-extraction framework under multigranulation rough sets.

The main objective of this paper is to establish a CMFRS model by using different approximation strategies according to different applied backgrounds. As it is known that the objective concepts in most existing MGRS models are suppose to be distinct. But in reality, people often lack an explicit relation to granulate the universe due to imprecise human knowledge. That is to say, many real problems involve the situation of vague concepts, such as symbols of diseases, mechanical defects, and so on. Most existing MGRS models based on distinct concepts have a limitation for this issue. Therefore,it is necessary to employ the idea of fuzzy rough sets to the domain. To provide a theoretical framework for dealing with real problems involved vague concepts, in this paper we propose six kinds of covering-based multi-granulation rough set models, in which a concept is described by multiple binary relations instead of a single binary relation. This idea is also accordant with decision making and information fusion in practice. It is may more credible and reasonable to make the decision by combining the diagnoses from all specialists rather than from only one specialist when diagnosing whether or not a patient has contracted some diseases in a medical diagnosis problem. In another aspect, the six types of covering rough sets proposed in this paper are different from the types of covering-based fuzzy rough sets available in the literature, since we discuss the issues of covering-based fuzzy rough sets in the multi-granulation environment, and little attention has been given to the covering-based multi-granulation fuzzy rough sets. Moreover, we use the maximal and minimal descriptors of objects to approximate a target concept, which is different from the current methods. And our models are based on the multi-covering, whereas, the other types of covering-based fuzzy rough sets introduced at various stages are based on only one covering, thus, they could be considered as special cases of our model or our models are extensions of them. Furthermore, from the viewpoint of granular computing, many existing MGRS models can be derived from our new framework of covering-based multi-granulation fuzzy rough sets by setting specific binary relations or granular structures.

The paper is organized as follows. The next section deals with some preliminary concepts and properties regarding the Pawlak’s rough sets, covering-based rough sets, fuzzy rough sets and multi-granulation roughsets. In section 3, we introduce six kinds of constructing methods of rough approximation operations of CMFRS and discuss their basic properties and applications. Firstly, to compare with MGRS, we detail the similarities and different points between MGRS and CMFRS. Secondly, based on the concept of reduction of a covering, we obtain the sufficient conditions for two same type CMFRS to generate identical lower approximation operator or upper approximation operator and find that the necessary conditions may not satisfied. In section 4, we present some results concerning the interrelationship among six kinds of CMFRS proposed in this paper. The conditions under which six kinds of approximation operators are identical are also discussed. The last section concludes the present research and brings some remarks for future works.

2 Basic concepts and properties

2.1 Pawlak’s rough sets

Let U denote a nonempty and finite set called the universe and R ⊆ U × U be an equivalence relation on U. The pair apr = (U, R) is called an approximation space. The equivalence relation R partitions the universe U into disjoint subsets, each subset is called equivalence class or equivalence granule. Given an arbitrary set X ⊆ U, it may not be described precisely by the present information in (U, R). One may characterize X by a pair of lower and upper approximations defined asfollows.

$\underline{R} (X) = {x \in U | [x]_{R} \subseteq X}$ , $\bar{R} (X) = {x \in U | [x]_{R}$ ∩X ≠ ∅}.

The pair $(\underline{R} (X), \bar{R} (X))$ is referred to as rough set approximation of X.

Let ∼X = U - X, we have the following basic properties of Pawlak’s rough sets.

$(L 1) \underline{R} (X) \subseteq X$	$(H 1) X \subseteq \bar{R} (X)$
$(L 2) \underline{R} (\emptyset) = \emptyset$	$(H 2) \bar{R} (\emptyset) = \emptyset$
$(L 3) \underline{R} (U) = U$	$(H 3) \bar{R} (U) = U$
$(L 4) \underline{R} (X \cap Y) = \underline{R} (X) \cap \underline{R} (Y)$	$(H 4) \bar{R} (X \cup Y) = \bar{R} (X) \cup \bar{R} (Y)$
$(L 5) X \subseteq Y \Rightarrow \underline{R} (X) \subseteq \underline{R} (Y)$	$(H 5) X \subseteq Y \Rightarrow \bar{R} (X) \subseteq \bar{R} (Y)$
$(L 6) \underline{R} (X \cup Y) \supseteq \underline{R} (X) \cup \underline{R} (Y)$	$(H 6) \bar{R} (X \cap Y) \subseteq \bar{R} (X) \cap \bar{R} (Y)$
$(L 7) \underline{R} (\sim X) = \sim \bar{R} (X)$	$(H 7) \bar{R} (\sim X) = \sim \underline{R} (X)$
$(L 8) \underline{R} (\underline{R} (X)) = \underline{R} (X)$	$(H 8) \bar{R} (\bar{R} (X)) = \bar{R} (X)$
$(L 9) \forall K \in U / R \Rightarrow \underline{R} (K) = K$	$(H 9) \forall K \in U / R \Rightarrow \bar{R} (K) = K$

2.2 Covering-based rough sets

We review some fundamental concepts about covering-based rough sets such as the minimal descriptor, neighborhoods of an object and approximation operators of a given concept in this subsection.

Definition 1. [22, 33] Let U be a universe of discourse and C a family of nonempty subsets of U. If ∪C = U, then C is called a covering of U. The ordered pair 〈U, C〉 is called a covering approximation space.

Definition 2. [22, 33] Let 〈U, C〉 be a covering approximation space, x ∈ U, then

md_C (x) ={K ∈ C_x| ∀ S ∈ C_x (S ⊆ K ⇒ K = S)},

MD_C (x) ={K ∈ C_x| ∀ S ∈ C_x (S ⊇ K ⇒ K = S)} are called the minimal description and the maximal description of x, respectively, where C_x ={K ∈ C|x ∈ K}.

Definition 3. [33] Let 〈U, C〉 be a covering approximation space, then, neighborhood operators on U can be defined as follows.

n₁ (x) = ∩{K|K ∈ C_x}, n₂ (x) = ∪{K|K ∈ C_x}

n₃ (x) = ∩{K|K ∈ md_C (x)}, n₄ (x) = ∪{K|K ∈ MD_C (x)}

There are many kinds of approximation operators have been proposed in covering-based rough sets, but as Yao et al. [33] have pointed out that it seems more reasonable the operators as a pair meet some relationships. Thus, Yao et al. introduced several pairs of approximation operators which satisfy the property of duality.

Definition 4. [33] For a covering C of U, two pairs of approximation operators $({\underline{apr}}_{C}^{1} (A), {\bar{apr}}_{C}^{1} (A))$ and $({\underline{apr}}_{C}^{2} (A), {\bar{apr}}_{C}^{2} (A))$ are defined as:

${\underline{apr}}_{C}^{1} (A) = \cup {K | K \in C, K \subseteq A} = {x | x \in U, \exists K \in C [x \in K, K \subseteq A]}$

${\bar{apr}}_{C}^{1} (A) = \sim {\underline{apr}}_{C}^{1} (\sim A) = {x | x \in U, \forall K \in C [x \in K \Rightarrow K \cap A \neq \emptyset]}$

${\underline{apr}}_{C}^{2} (A) = \sim {\bar{apr}}_{C}^{2} (\sim A) = {x | x \in U, \forall K \in C [x \in K \Rightarrow K \subseteq A]}$

${\bar{apr}}_{C}^{2} (A) = \cup {K | K \in C, K \cap A \neq \emptyset} = {x | x \in U, \exists K \in C [x \in K, K \cap A \neq \emptyset]}$

Theorem 1. [27] Let C be a covering of U and K ∈ C. If K is a union of some sets in C - {K}, we say K is a reducible element of C, otherwise K is an irreducible elements of C. If all the reducible elements have been discarded, then the remainder of C is called the reduction of C on U, denoted by reduct (C). For any x ∈ U, C and reduct (C) have the same md (x).

2.3 Fundamentals of MGRS

In recent years, Qian et al. [38, 39] have proposed a new rough set model named multi-granulation rough sets, where a target concept is approximated by multiple binary relations. In this subsection, we will briefly outline two definitions of multi-granulation rough set models, i.e., optimistic and pessimistic multi-granulation rough sets, respectively. The detailed descriptions could be found in [38, 39].

Definition 5. Let K = (U, R) be a knowledge base, where R is a family of equivalence relations on the universe U. Let A₁, A₂, . . . , A_m ∈ R, where m is a natural number. For any X ⊆ U, its optimistic lower and upper approximations with respect to A₁, A₂ . . . , A_m are defined as follows. $\begin{matrix} \underline{\sum_{i = 1}^{m} A_{i} (X)} = {x \in U | [x]_{A_{1}} \subseteq X or [x]_{A_{2}} \\ \subseteq X or \dots or [x]_{A_{m}} \subseteq X}; \end{matrix}$ $\bar{\sum_{i = 1}^{m} A_{i} (X)} = \sim \underline{\sum_{i = 1}^{m} A_{i} (\sim X)} .$ $(\underline{\sum_{i = 1}^{m} A_{i} (X)}, \bar{\sum_{i = 1}^{m} A_{i} (X)})$ is called the optimistic multi-granulation rough sets of X. Here, the word “optimistic” means that only a single granular structure is needed to satisfy the inclusion condition between an equivalence class and a target concept when multiple independent granular structures are available in the problem.

Definition 6. Let K = (U, R) be a knowledge base, where R is a family of equivalence relations on the universe U. Let A₁, A₂, . . . , A_m ∈ R, where m is a natural number. For any X ⊆ U, its pessimistic lower and upper approximations with respect to A₁, A₂, . . . , A_m are defined as follows. $\begin{matrix} \underline{\sum_{i = 1}^{m} A_{i} (X)} = {x \in U | [x]_{A_{1}} \subseteq X and [x]_{A_{2}} \\ \subseteq X and \dots and [x]_{A_{m}} \subseteq X}; \end{matrix}$ $\bar{\sum_{i = 1}^{m} A_{i} (X)} = \sim \underline{\sum_{i = 1}^{m} A_{i} (\sim X)} .$ $(\underline{\sum_{i = 1}^{m} A_{i} (X)}, \bar{\sum_{i = 1}^{m} A_{i} (X)})$ is called the pessimistic multi-granulation rough sets of X. Here, the word“pessimistic” means that all granular structures are needed to satisfy the inclusion condition between an equivalence class and a target concept when multiple independent granular structures are available.

3 Covering-based Multi-granulation Fuzzy Rough Sets(CMFRS)

In this section, we address the problems of the rough approximations of a fuzzy set based on multiple covering relations, that is, we extend fuzzy rough sets to a multi-granulation covering approximation space. Being different from the definition of covering approximation space introduced by Zakowski [22], we define the pair 〈U, C〉 as a multi-granulation covering approximation space in this paper, where U is a universe of discourse and C is a family of coverings on the universe U. First, we investigate some basic properties of covering-based multi-granulation fuzzy rough sets after introducing the concepts of CMFRS, then discuss the similarities and differences of CMFRS and MGRS. Second, for the same type CMFRS, we explore the conditions under which two distinct CMFRS models produce identical lower and upper approximations of a target concept. Finally, the relationships of the proposed CMFRS are given.

For the purposes of clearness and easy understanding, we only take the case of two coverings into consideration in this paper. In what follows, F (U) denotes all the fuzzy sets defined on U.

Definition 7. (Type-1 CMFRS) Let 〈U, C〉 be a covering approximation space and C₁, C₂, ⋯ , C_m ∈ C, where m is a natural number. For any X ∈ F (U), its type-1 covering-based lower and upper approximations with respect to C₁, C₂, ⋯ , C_m are defined as follows. For any x ∈ U,

$\underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) (x) = \lor_{i = 1}^{m} inf {X (y) | y \in \cap {K | K \in m d_{C_{i}} (x)}}$

$\bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) (x) = \land_{i = 1}^{m} sup {X (y) | y \in \cap {K | K \in m d_{C_{i}} (x)}}$

If $\underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) \neq \bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (X)$ , then X is called a type-1 covering-based multi-granulation fuzzy rough set with respect to C₁, C₂, ⋯ , C_m, else X is called a fuzzy definable set.

Definition 8. (Type-2 CMFRS) Let 〈U, C〉 be a covering approximation space and C₁, C₂, ⋯ , C_m ∈ C, where m is a natural number. For any X ∈ F (U), its type-2 covering-based lower and upper approximations with respect to C₁, C₂, ⋯ , C_m are defined as follows. For any x ∈ U,

$\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) (x) = \lor_{i = 1}^{m} inf {X (y) | y \in \cup {K | K \in m d_{C_{i}} (x)}}$

$\bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) (x) = \land_{i = 1}^{m} sup {X (y) | y \in \cup {K | K \in m d_{C_{i}} (x)}}$

If $\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) \neq \bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X)$ , then X is called a type-2 covering-based multi-granulation fuzzy rough set with respect to C₁, C₂, ⋯ , C_m, else X is called a fuzzy definable set.

Definition 9. (Type-3 CMFRS) Let 〈U, C〉 be a covering approximation space and C₁, C₂, ⋯ , C_m ∈ C, where m is a natural number. For any X ∈ F (U), its type-3 covering-based lower and upper approximations with respect to C₁, C₂, ⋯ , C_m are defined as follows. For any x ∈ U,

$\underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) (x) = \lor_{i = 1}^{m} inf {X (y) | y \in \cap {K | K \in M D_{C_{i}} (x)}}$

$\bar{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) (x) = \land_{i = 1}^{m} sup {X (y) | y \in \cap {K | K \in M D_{C_{i}} (x)}}$

If $\underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) \neq \bar{M 3_{\sum_{i = 1}^{m} C_{i}}} (X)$ , then X is called a type-3 covering-based multi-granulation fuzzy rough set with respect to C₁, C₂, ⋯ , C_m, else X is called a fuzzy definable set.

Definition 10. (Type-4 CMFRS) Let 〈U, C〉 be a covering approximation space and C₁, C₂, ⋯ , C_m ∈ C, where m is a natural number. For any X ∈ F (U), its type-4 covering-based lower and upper approximations with respect to C₁, C₂, ⋯ , C_m are defined as follows. For any x ∈ U,

$\underline{M 4_{\sum_{i = 1}^{m} C_{i}}} (X) (x) = \lor_{i = 1}^{m} inf {X (y) | y \in \cup {K | K \in M D_{C_{i}} (x)}}$

$\bar{M 4_{\sum_{i = 1}^{m} C_{i}}} (X) (x) = \land_{i = 1}^{m} sup {X (y) | y \in \cup {K | K \in M D_{C_{i}} (x)}}$

If $\underline{M 4_{\sum_{i = 1}^{m} C_{i}}} (X) \neq \bar{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ , then X is called a type-4 covering-based multi-granulation fuzzy rough set with respect to C₁, C₂, ⋯ , C_m, else X is called a fuzzy definable set.

Definition 11. (Type-5 CMFRS) Let 〈U, C〉 be a covering approximation space and C₁, C₂, ⋯ , C_m ∈ C, where m is a natural number. For any X ∈ F (U), its type-5 covering-based lower and upper approximations with respect to C₁, C₂, ⋯ , C_m are defined as follows. For any x ∈ U,

$\underline{M 5_{\sum_{i = 1}^{m} C_{i}}} (X) (x) = \lor_{i = 1}^{m} sup_{K \in m d_{C_{i}} (x)} {inf_{y \in K} {X (y)}}$

$\bar{M 5_{\sum_{i = 1}^{m} C_{i}}} (X) (x) = \land_{i = 1}^{m} inf_{K \in m d_{C_{i}} (x)} {sup_{y \in K} {X (y)}}$

If $\underline{M 5_{\sum_{i = 1}^{m} C_{i}}} (X) \neq \bar{M 5_{\sum_{i = 1}^{m} C_{i}}} (X)$ , then X is called a type-5 covering-based multi-granulation fuzzy rough set with respect to C₁, C₂, ⋯ , C_m, else X is called a fuzzy definable set.

Definition 12. (Type-6 CMFRS) Let 〈U, C〉 be a covering approximation space and C₁, C₂, ⋯ , C_m ∈ C, where m is a natural number. For any X ∈ F (U), its type-6 covering-based lower and upper approximations with respect to C₁, C₂, ⋯ , C_m are defined as follows. For any x ∈ U,

$\underline{M 6_{\sum_{i = 1}^{m} C_{i}}} (X) (x) = \lor_{i = 1}^{m} sup_{K \in M D_{C_{i}} (x)} {inf_{y \in K} {X (y)}}$

$\bar{M 6_{\sum_{i = 1}^{m} C_{i}}} (X) (x) = \land_{i = 1}^{m} inf_{K \in M D_{C_{i}} (x)} {sup_{y \in K} {X (y)}}$

If $\underline{M 6_{\sum_{i = 1}^{m} C_{i}}} (X) \neq \bar{M 6_{\sum_{i = 1}^{m} C_{i}}} (X)$ , then X is called a type-6 covering-based multi-granulation fuzzy rough set with respect to C₁, C₂, ⋯ , C_m, else X is called a fuzzy definable set.

Next, an example is given to show the differences among the CMFRS models defined above.

Example 1. Let 〈U, C〉 be a covering approximation space. C a family of coverings on U and U = {x₁, x₂, x₃, x₄}. C₁, C₂ ∈ C are two coverings on U such that C₁ = {{x₁, x₂} , {x₂, x₃, x₄} , {x₃, x₄}}, C₂ = {{x₁, x₂, x₃} , {x₂, x₄} , {x₁, x₂, x₄}}.

Suppose $X = {\frac{0.7}{x_{1}}, \frac{0.5}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.2}{x_{4}}}$ is a fuzzy set defined in U. According to the above definitions, we have the following results.

$\underline{M 1_{C_{1} + C_{2}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.2}{x_{4}}}$ ,

$\bar{M 1_{C_{1} + C_{2}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.3}{x_{4}}}$ ;

$\underline{M 2_{C_{1} + C_{2}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.2}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.2}{x_{4}}}$ ,

$\bar{M 2_{C_{1} + C_{2}}} (X) = {\frac{0.7}{x_{1}}, \frac{0.7}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.3}{x_{4}}}$ ;

$\underline{M 3_{C_{1} + C_{2}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.2}{x_{4}}}$ ,

$\bar{M 3_{C_{1} + C_{2}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{2}}, \frac{0.5}{x_{3}}, \frac{0.5}{x_{4}}}$ ;

$\underline{M 4_{C_{1} + C_{2}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.2}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.2}{x_{4}}}$ ,

$\bar{M 4_{C_{1} + C_{2}}} (X) = {\frac{0.7}{x_{1}}, \frac{0.7}{x_{2}}, \frac{0.5}{x_{3}}, \frac{0.5}{x_{4}}}$ ;

$\underline{M 5_{C_{1} + C_{2}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.2}{x_{4}}}$ ,

$\bar{M 5_{C_{1} + C_{2}}} (X) = {\frac{0.7}{x_{1}}, \frac{0.5}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.3}{x_{4}}}$ ;

$\underline{M 6_{C_{1} + C_{2}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.2}{x_{4}}}$ ,

$\bar{M 6_{C_{1} + C_{2}}} (X) = {\frac{0.7}{x_{1}}, \frac{0.7}{x_{2}}, \frac{0.5}{x_{3}}, \frac{0.5}{x_{4}}}$ .

From Example 1, we found that for a given fuzzy subset $X = {\frac{0.7}{x_{1}}, \frac{0.5}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.2}{x_{4}}}$ of U, the six types of approximations of X are different from each other, which means that the above six types of CMFRS models are also different from each other.

Yao [17] and many others [62] emphasized that the duality is one of the most important properties of lower and upper approximations and a number of results on the duality of approximation operators have been reported.

Proposition 1 gives the duality of the six types of approximations defined in the paper.

Proposition 1. Let 〈U, C〉 be a covering approximation space and C₁, C₂, ⋯ , C_m ∈ C, where m is a natural number. For any X ∈ F (U) and any x ∈ U, we have that

(1) $\underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (\sim X) = \sim \bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (\sim X) = \sim \underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (X)$ ;

(2) $\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (\sim X) = \sim \bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (\sim X) = \sim \underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X)$ ;

(3) $\underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (\sim X) = \sim \bar{M 3_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\bar{M 3_{\sum_{i = 1}^{m} C_{i}}} (\sim X) = \sim \underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X)$ ;

(4) $\underline{M 4_{\sum_{i = 1}^{m} C_{i}}} (\sim X) = \sim \bar{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\bar{M 4_{\sum_{i = 1}^{m} C_{i}}} (\sim X) = \sim \underline{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ ;

(5) $\underline{M 5_{\sum_{i = 1}^{m} C_{i}}} (\sim X) = \sim \bar{M 5_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\bar{M 5_{\sum_{i = 1}^{m} C_{i}}} (\sim X) = \sim \underline{M 5_{\sum_{i = 1}^{m} C_{i}}} (X)$ ;

(6) $\underline{M 6_{\sum_{i = 1}^{m} C_{i}}} (\sim X) = \sim \bar{M 6_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\bar{M 6_{\sum_{i = 1}^{m} C_{i}}} (\sim X) = \sim \underline{M 6_{\sum_{i = 1}^{m} C_{i}}} (X)$ .

Proof. we only prove (1), other properties can be proved in a similar way.

According to Definition 7, for any x ∈ U

$\underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (\sim X) (x) = \lor_{i = 1}^{m} inf {\sim X (y) | y \in \cap {K | K \in m d_{C_{i}} (x)}}$

$= \lor_{i = 1}^{m} inf {(1 - X (y)) | y \in \cap {K | K \in m d_{C_{i}} (x)}}$

$= 1 - \land_{i = 1}^{m} sup {X (y) | y \in \cap {K | K \in m d_{C_{i}} (x)}}$

$= \sim \bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) (x)$

Analogously,

$\bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (\sim X) (x) = \land_{i = 1}^{m} sup {\sim X (y) | y \in \cap {K | K \in m d_{C_{i}} (x)}}$

$= \land_{i = 1}^{m} sup {(1 - X (y)) | y \in \cap {K | K \in m d_{C_{i}} (x)}}$

$= 1 - \lor_{i = 1}^{m} inf {X (y) | y \in \cap {K | K \in m d_{C_{i}} (x)}}$

$= \sim \underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) (x)$

This completes the proof of Proposition 1. □

In the following, we will use symbol ◊ to denote M1, M2, . . . , M6.

Proposition 2. Let 〈U, C〉 be a covering approximation space and C₁, C₂, ⋯ , C_m ∈ C, where m is a natural number. Then, for any X, Y ∈ F (U), the following properties hold.

If X ⊆ Y, then $\underline{◊_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \underline{◊_{\sum_{i = 1}^{m} C_{i}}} (Y)$ ;

If X ⊆ Y, then $\bar{◊_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \bar{◊_{\sum_{i = 1}^{m} C_{i}}} (Y)$ ;

$\underline{◊_{\sum_{i = 1}^{m} C_{i}}} (X \cap Y) \subseteq \underline{◊_{\sum_{i = 1}^{m} C_{i}}} (X) \cap \underline{◊_{\sum_{i = 1}^{m} C_{i}}} (Y)$ ;

$\bar{◊_{\sum_{i = 1}^{m} C_{i}}} (X \cup Y) \supseteq \bar{◊_{\sum_{i = 1}^{m} C_{i}}} (X) \cup \bar{◊_{\sum_{i = 1}^{m} C_{i}}} (Y)$ ;

$\underline{◊_{\sum_{i = 1}^{m} C_{i}}} (X \cup Y) \supseteq \underline{T_{1} + T_{2}} (X) \cup \underline{◊_{\sum_{i = 1}^{m} C_{i}}} (Y)$ ;

$\bar{◊_{\sum_{i = 1}^{m} C_{i}}} (X \cap Y) \subseteq \bar{◊_{\sum_{i = 1}^{m} C_{i}}} (X) \cap \bar{◊_{\sum_{i = 1}^{m} C_{i}}} (Y)$ .

Proof: we only prove (1) and (2), others can be proved by employing (1) or (2).

(1) According to Definition 7, we have $\underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) (x) = \lor_{i = 1}^{m} inf {X (y) | y \in \cap {K | K \in m d_{C_{i}} (x)}}$ . If X ⊆ Y, then X (y) ≤ Y (y), therefore, $\underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (Y)$ ;

One can prove the remaining five cases in a similar manner, the proof is omitted here.

(2) It can be proved in a similar way like (1).

This completes the proof of Proposition 2. □

In Proposition 2, (1) and (2) show the monotonicity of covering rough approximations, w.r.t., the variety of target or concept, and each of (3)-(6) expresses the relationship between the rough approximation of X ∩ Y (or X ∪ Y) with the intersection (or union) of the two rough approximations of X and Y under the multi-covering environment.

Proposition 3. Let 〈U, C〉 be a covering approximation space, and C₁, C₂, ⋯ , C_m ∈ C, where C_i = {C_i1, C_i2, ⋯ , C_in}, 1 ≤ i ≤ m and m, n are two natural numbers. For any K_j ∈ C_i, where j ∈ {i1, i2, ⋯ , in}, we have $\underline{◊_{\sum_{i = 1}^{m} C_{i}}} (K_{j}) \subseteq K_{j}$ . In particular, if C₁, C₂, ⋯ , C_m ∈ C are all partitions of U, then for any K_j ∈ C_i, we have $\underline{◊_{\sum_{i = 1}^{m} C_{i}}} (K_{j}) = K_{j}$ .

From Proposition 3, one may see that for any element in the coverings which construct a given CMFRS model, its lower approximation in that model is not larger than itself, if the element is in partitions then its lower approximation in that model is itself.

Proposition 4. Let 〈U, C〉 be a covering approximation space, and C₁, C₂, ⋯ , C_m ∈ C, where C_i = {C_i1, C_i2, ⋯ , C_in}, 1 ≤ i ≤ m and m, n are two natural numbers. For any K_j ∈ C_i, where j ∈ {i1, i2, ⋯ , in}, we have $\bar{◊_{\sum_{i = 1}^{m} C_{i}}} (K_{j}) \supseteq K_{j}$ . Specially, if C₁, C₂, ⋯ , C_m ∈ C are all partitions of U, then for any K_j ∈ C_i, we have $\bar{◊_{\sum_{i = 1}^{m} C_{i}}} (K_{j}) = K_{j}$ . From Proposition 4, one may find that for any element in the coverings which construct a given CMFRS model, its upper approximation in that model is not smaller than itself, if the element is in partitions then its lower approximation in that model is itself.

Definition 13. [49] Let 〈U, C〉 be a coveringapproximation space, C₁, C₂ ∈ C and reduct (C₁) = {C₁₁, C₁₂, …, C_1p}, reduct (C₂) = {C₂₁, C₂₂, …, C_2q}. For any x ∈ U, x ∈ C_1i (1 ≤ i ≤ p) and x ∈ C_2j (1 ≤ j ≤ q), there is C_1i ⊆ C_2j holds, that is reduct (C₁) ⊆ reduct (C₂), then we say that C₁ is finer than C₂, denoted by C₁ ⪯ C₂.

From Definition 13, if C₁ ⪯ C₂, then for any x ∈ U, we have md_{C
₁} (x) ⊆ md_{C
₂} (x) and MD_{C
₁} (x) ⊆ MD_{C
₂} (x).

Theorem 2. Let 〈U, C〉 be a covering approximation space, and C₁, C₂, ⋯ , C_m ∈ C. For any X ⊆ U, we have

(1) If C₁ ⪯ C₂ ⪯ ⋯ ⪯ C_m, then $\underline{◊_{\sum_{i = 1}^{m} C_{i}}} (X) = \underline{◊_{C_{1}}} (X)$ ;

(2) if C₁ ⪯ C₂ ⪯ ⋯ ⪯ C_m, then $\bar{◊_{\sum_{i = 1}^{m} C_{i}}} (X) = \bar{◊_{C_{1}}} (X)$ .

Proof. Here, we only give the proofs for the first type CMFRS, others can be proved in the same manner.

(1) If C₁ ⪯ C₂ ⪯ ⋯ ⪯ C_m, by Definition 13, we have md_{C
₁} (x) ⊆ md_{C
₂} (x) ⊆ ·· · ⊆ md_{C
_m} (x).

Then for any X ∈ F (U) and x ∈ U, we have $\begin{matrix} \underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) (x) \\ = \lor_{i = 1}^{m} inf {X (y) | y \in \cap {K | K \in m d_{C_{i}} (x)}} \\ = inf {X (y) | y \in \cap {K | K \in m d_{C_{1}} (x)}} \\ = \underline{M 1_{C_{1}}} (X) (x); \end{matrix}$

(2) If C₁ ⪯ C₂ ⪯ ⋯ ⪯ C_m, by Definition 13, we have md_{C
₁} (x) ⊆ md_{C
₂} (x) ⊆ ·· · ⊆ md_{C
_m} (x).

Then for any X ∈ F (U) and x ∈ U, we have $\begin{matrix} \bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) (x) \\ = \land_{i = 1}^{m} sup {X (y) | y \in \cap {K | K \in m d_{C_{i}} (x)}} \end{matrix}$ $\begin{matrix} = sup {X (y) | y \in \cap {K | K \in m d_{C_{1}} (x)}} \\ = \bar{M 1_{C_{1}}} (X) (x) . \end{matrix}$

This completes the proof of Theorem 2. □

A reduct of a covering is a minimal subcovering that retains each object subset’s approximation. The relationship between approximations produced by C and corresponding reduct (C) in same covering approximation space is an interesting topic. Next we shall discuss the relationships of approximations produced by C₁, C₂, ⋯ C_m and their corresponding reduct (C₁), reduct (C₂), ⋯, reduct (C_m).

Theorem 3. Let 〈U, C〉 be a covering approximation space, C₁, C₂, ⋯ C_m ∈ C and reduct (C₁), reduct (C₂), ⋯, reduct (C_m) are the reducts of C₁, C₂, ⋯ , C_m respectively. For any X ⊆ U, $M 1_{\sum_{i = 1}^{m} C_{i}}$ and $M 1_{\sum_{i = 1}^{m} reduct (C_{i})}$ , $M 2_{\sum_{i = 1}^{m} C_{i}}$ and $M 2_{\sum_{i = 1}^{m} reduct (C_{i})}$ , $M 5_{\sum_{i = 1}^{m} C_{i}}$ and $M 5_{\sum_{i = 1}^{m} reduct (C_{i})}$ cultivate identical CMFRS lower and upper approximations.

Proof. By Theorem 1, for any x ∈ U, C and reduct (C) share the same md (x), according to Definition 7, 8 and 11, we prove Theorem 3. □

Remark 1 For any X ⊆ U, $M 3_{\sum_{i = 1}^{m} C_{i}}$ and $M 3_{\sum_{i = 1}^{m} reduct (C_{i})}$ , $M 4_{\sum_{i = 1}^{m} C_{i}}$ and $M 4_{\sum_{i = 1}^{m} reduct (C_{i})}$ , $M 6_{\sum_{i = 1}^{m} C_{i}}$ and $M 6_{\sum_{i = 1}^{m} reduct (C_{i})}$ may not cultivate identical CMFRS lower and upper approximations.

Proof. For any x ∈ U, C and reduct (C) may not share the same MD (x), therefore, $M 3_{\sum_{i = 1}^{m} C_{i}}$ and $M 3_{\sum_{i = 1}^{m} reduct (C_{i})}$ , $M 4_{\sum_{i = 1}^{m} C_{i}}$ and $M 4_{\sum_{i = 1}^{m} reduct (C_{i})}$ , $M 6_{\sum_{i = 1}^{m} C_{i}}$ and $M 6_{\sum_{i = 1}^{m} reduct (C_{i})}$ may not cultivate identical CMFRS lower and upper approximations. □

Definition 14. [27] Let C be a covering of U and K an element of C. If there exists another element of C such that , we say that K is an immured element of covering C.

Definition 15. [27] Let C be a covering of U. When we remove all immured elements from C, the set of all remaining elements is still a covering of U, and this new covering has no immured element. We call this new covering an exclusion of C, and it is denoted by exclusion (C).

Theorem 4. Let 〈U, C〉 be a covering approximation space, C₁, C₂, C₃, C₄ ∈ C. reduct (C₁), reduct (C₂), reduct (C₃), reduct (C₄) are the reducts of C₁, C₂, C₃ and C₄ respectively. If (1) or (2) is satisfied, then for any X ⊆ U, ◊_C₁+C₂ and ◊_C₃+C₄ produce the identical lower CMFRS approximations.

reduct (C₁) = reduct (C₃), reduct (C₂) = reduct (C₄) and at the same time, exclusion (C₁) = exclusion (C₃), exclusion (C₂) = exclusion (C₄);

reduct (C₁) = reduct (C₄), reduct (C₂) = reduct (C₃) and at the same time, exclusion (C₁) = exclusion (C₄), exclusion (C₂) = exclusion (C₃).

Proof. We only prove that if (1) is satisfied, then ◊_C₁+C₂ and ◊_C₃+C₄ produce the identical lower approximation operators of CMFRS, another case can be proved in a same manner.

According to Theorem 1, for any x ∈ U, C and its corresponding reduct (C) yield the same md (x). If reduct (C₁) = reduct (C₃) and reduct (C₂) = reduct (C₄), then C₁ and C₃ produce the identical md (x), and C₂ and C₄ share the other md (x). According to Definition 7, 8 and 11, M1_C₁+C₂ and M1_C₃+C₄, M2_C₁+C₂ and M2_C₃+C₄, M5_C₁+C₂ and M5_C₃+C₄ produce the identical lower approximation operators of CMFRS.

For any x ∈ U, C and corresponding exclusion (C) yield the identical MD (x). If exclusion (C₁) = exclusion (C₃) and exclusion (C₂) = exclusion (C₄), then C₁ and C₃ produce the same MD (x), and C₂ and C₄ share the other MD (x). According to Definition 9, 10 and 12, M3_C₁+C₂ and M3_C₃+C₄, M4_C₁+C₂ and M4_C₃+C₄, M6_C₁+C₂ and M6_C₃+C₄ produce the identical lower approximation operators of CMFRS.

This completes the proof of Theorem 4. □

In fact, the above theorem offers sufficient conditions for which different CMFRS models produce the identical lower approximation operators.

Remark 2. The reverse of Theorem 4 is not hold, i.e., ◊_C₁+C₂ and ◊_C₃+C₄ may generate identical covering multi-granulation fuzzy rough lower approximation operator, but their reducts or exclusions are not the same.

Remark 2 can be illustrated by using the following example.

Example 2. (Continued from Example 1) Let C₃ = {{x₁, x₂} , {x₂, x₃, x₄}}, C₄ = {{x₁, x₂, x₃} , {x₂, x₄}}, then we have the following outputs.

According to Theorem 1, we have

reduct (C₁) = {{x₁, x₂} , {x₂, x₃, x₄} , {x₃, x₄}},

reduct (C₂) = {{x₁, x₂, x₃} , {x₂, x₄} , {x₁, x₂, x₄}};

reduct (C₃) = {{x₁, x₂} , {x₂, x₃, x₄}},

reduct (C₄) = {{x₁, x₂, x₃} , {x₂, x₄}}.

Hence,

reduct (C₁) ≠ reduct (C₃) , reduct (C₂) ≠ reduct (C₄)

But, we have

$\underline{M 1_{C_{1} + C_{2}}} (X) = \underline{M 1_{C_{3} + C_{4}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.2}{x_{4}}}$ ,

$\underline{M 2_{C_{1} + C_{2}}} (X) = \underline{M 2_{C_{3} + C_{4}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.2}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.2}{x_{4}}}$ ,

$\underline{M 3_{C_{1} + C_{2}}} (X) = \underline{M 3_{C_{3} + C_{4}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.2}{x_{4}}}$ ,

$\underline{M 4_{C_{1} + C_{2}}} (X) = \underline{M 4_{C_{3} + C_{4}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.2}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.2}{x_{4}}}$ .

Theorem 5. Let 〈U, C〉 be a covering approximation space, C₁, C₂, C₃, C₄ ∈ C and reduct (C₁), reduct (C₂), reduct (C₃), reduct (C₄) are the reducts of C₁, C₂, C₃ and C₄ respectively. For any X ⊆ U, M1_C₁+C₂ and M1_C₃+C₄, M2_C₁+C₂ and M2_C₃+C₄, M5_C₁+C₂ and M5_C₃+C₄ produce the identical covering multi-granulation fuzzy rough lower and upper approximations.

(1) reduct (C₁) = reduct (C₃) and reduct (C₂) = reduct (C₄)

(2) reduct (C₁) = reduct (C₄) and reduct (C₂) = reduct (C₃)

Proof: We only prove (1) here, the case (2) can be proved in the similar way.

According to Theorem 1, for any x ∈ U, C and reduct (C) generate the same md (x). If reduct (C₁) = reduct (C₃) and reduct (C₂) = reduct (C₄), then C₁ and C₃ own the same md (x), and at the same time, C₂ and C₄ share another md (x). According to Definition 7, 8 and 11, M1_C₁+C₂ and M1_C₃+C₄, M2_C₁+C₂ and M2_C₃+C₄, M5_C₁+C₂ and M5_C₃+C₄ produce the identical covering multi-granulation fuzzy rough lower and upper approximations.

This completes the proof of Theorem 5. □

Remark 3. The reverse of Theorem 5 does not hold, that is M1_C₁+C₂ and M1_C₃+C₄, M2_C₁+C₂ and M2_C₃+C₄, M5_C₁+C₂ and M5_C₃+C₄ may produce identical covering multi-granulation fuzzy rough upper approximation operator, but their reducts do not equal to each other.

Example 4 gives a proof to Remark 3.

Example 4. Let 〈U, C〉 be a covering approximationspace and C₁, C₂, C₃, C₄ ∈ C, where, U = {x₁, x₂, x₃}, C₁ = {{x₁} , {x₂} , {x₃}}, C₂ = {{x₁, x₂},{x₁, x₂, x₃} , {x₁, x₃}}, C₃ = {{x₁, x₂} , {x₂, x₃} , {x₁, x₃}}, C₄ = {{x₁, x₃} , {x₁, x₂, x₃}}.

From Theorem 1, we can calculate that

reduct (C₁) = {{x₁} , {x₂} , {x₃}},

reduct (C₂) = {{x₁, x₂} , {x₁, x₃}},

reduct (C₃) = {{x₁, x₂} , {x₂, x₃} , {x₁, x₃}},

reduct (C₄) = {{x₁, x₃} , {x₁, x₂, x₃}}.

Therefore,

reduct (C₁) ≠ reduct (C₃), reduct (C₁) ≠ reduct (C₄)

reduct (C₂) ≠ reduct (C₃), reduct (C₂) ≠ reduct (C₄)

But, given $X = {\frac{0.6}{x_{1}} + \frac{0.3}{x_{3}}}$ , $Y = {\frac{0.6}{x_{1}} + \frac{0.1}{x_{2}}$ $+ \frac{0.3}{x_{3}}}$ , we have

$\underline{M 1_{C_{1} + C_{2}}} (X) = {\frac{0.6}{x_{1}} + \frac{0.3}{x_{3}}} = \underline{M 1_{C_{3} + C_{4}}} (X)$ ,

$\bar{M 1_{C_{1} + C_{2}}} (X) = {\frac{0.6}{x_{1}} + \frac{0.3}{x_{3}}} = \bar{M 1_{C_{3} + C_{4}}} (X)$ ;

$\underline{M 2_{C_{1} + C_{2}}} (Y) = {\frac{0.6}{x_{1}} + \frac{0.1}{x_{2}} + \frac{0.3}{x_{3}}} = \underline{M 2_{C_{3} + C_{4}}} (Y)$ ,

$\bar{M 2_{C_{1} + C_{2}}} (Y) = {\frac{0.6}{x_{1}} + \frac{0.1}{x_{2}} + \frac{0.3}{x_{3}}} = \bar{M 2_{C_{3} + C_{4}}} (Y)$ ;

$\underline{M 5_{C_{1} + C_{2}}} (Y) = {\frac{0.6}{x_{1}} + \frac{0.1}{x_{2}} + \frac{0.3}{x_{3}}} = \underline{M 5_{C_{3} + C_{4}}} (Y)$ ,

$\bar{M 5_{C_{1} + C_{2}}} (Y) = {\frac{0.6}{x_{1}} + \frac{0.1}{x_{2}} + \frac{0.3}{x_{3}}} = \bar{M 5_{C_{3} + C_{4}}} (Y)$ .

Theorem 6 Let 〈U, C〉 be a covering approximation space, C₁, C₂, C₃, C₄ ∈ C and reduct (C₁), reduct (C₂), reduct (C₃) and reduct (C₄) are the reducts of C₁, C₂, C₃ and C₄ respectively. M3_C₁+C₂ and M3_C₃+C₄, M4_C₁+C₂ and M4_C₃+C₄, M6_C₁+C₂ and M6_C₃+C₄ produce the identical upper approximation operators of CMFRS, if at least one of the following conditions is satisfied.

exclusion (C₁) = exclusion (C₃) and exclusion (C₂) = exclusion (C₄) ;

exclusion (C₁) = exclusion (C₄) and exclusion (C₂) = exclusion (C₃).

Proof. It can be analogously proved as Theorem 5. □

Theorem 6 gives the sufficient conditions for which two different Type-3 or Type-4 or Type-6 CMFRS models produce the identical upper approximation operators.

Remark 4. The reverse of Theorem 6 is not true, i.e., M3_C₁+C₂ and M3_C₃+C₄, M4_C₁+C₂ and M4_C₃+C₄, M6_C₁+C₂ and M6_C₃+C₄ may produce identical upper approximation operators, but their exclusions may differ from each other.

Remark 4 can be proved by the following example.

Example 5. Let 〈U, C〉 be a covering approximationspace and C₁, C₂, C₃, C₄ ∈ C, where U = {x₁, x₂, x₃, x₄}, C₁ = {{x₁, x₂} , {x₃} , {x₃, x₄} , {x₁, x₂, x₃} , {x₁, x₂, x₄}}, C₂ = {{x₁, x₂} , {x₄} , {x₃, x₄}}, C₃ = {{x₁, x₂, x₃} , {x₁, x₂, x₃, x₄} , {x₂, x₄}}, C₄ = {{x₁, x₂} , {x₁, x₂, x₄} , {x₃, x₄}}.

From Definition 15, we can obtain

exclusion (C₁) = {{x₃, x₄} , {x₁, x₂, x₃} , {x₁, x₂, x₄}},

exclusion (C₂) = {{x₁, x₂} , {x₃, x₄}},

exclusion (C₃) = {{x₁, x₂, x₃, x₄}},

exclusion (C₄) = {{x₁, x₂, x₄} , {x₃, x₄}}.Therefore,

exclusion (C₁) ≠ exclusion (C₃) and

exclusion (C₂)≠ exclusion (C₄) ;

exclusion (C₁) ≠ exclusion (C₄) and

exclusion (C₂) ≠ exclusion (C₃). But, let $X = {\frac{0.7}{x_{1}}, \frac{0.5}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.2}{x_{4}}}$ , we have $\begin{matrix} \bar{M 3_{C_{1} + C_{2}}} (X) = {\frac{0.7}{x_{1}}, \frac{0.7}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.2}{x_{4}}} = \bar{M 3_{C_{3} + C_{4}}} (X), \\ \bar{M 6_{C_{1} + C_{2}}} (X) = {\frac{0.7}{x_{1}}, \frac{0.7}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.3}{x_{4}}} = \bar{M 6_{C_{3} + C_{4}}} (X) . \end{matrix}$

4 Relationships Among Types of CMFRS

We investigate the relationships among the six types of covering-based multi-granulation fuzzy rough sets proposed in the previous section. In addition, some interesting properties are also discussed here.

According to the definitions of CMFRS, the following theorem can be obtained.

Theorem 7. Let 〈U, C〉 be a covering approximation space and C₁, C₂, ⋯ , C_m ∈ C. For any X ∈ F (U), we have

$\underline{M 4_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \underline{M 6_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \underline{M 5_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq X$ ;

$\underline{M 4_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \underline{M 6_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \underline{M 5_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq X$ ;

$\bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \bar{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \bar{M 5_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \bar{M 6_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \bar{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ ;

$\bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \bar{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ ;

$X \subseteq \bar{M 5_{\sum_{i = 1}^{m} C_{i}}} (X) \bar{M 6_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \bar{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ .

Theorem 7 can be easily proved by employing the definitions of CMFRS, we omit them here. For the readers’ convenience, the relationships among the six types of CMFRS models are described by Fig. 1. In Fig. 1, each node denotes an approximation or a concept, and each line connects two approximations, where the lower node is a subset of the upper node. Unfortunately, for any X ⊆ U, the set of all the six types of lower and upper approximations equipped with the binary relation ⊆ of inclusion, can not construct a lattice.

From Fig. 1, the following conclusions are obtained.

First, from Fig. 1, one could find that among those six types of lower approximations of CMFRS, the first type lower approximation $\underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (X)$ is the best one to describe X, since in Fig. 1 the first type lower approximation is closer to X than other types of lower approximations. While for the case of upper approximation, we may not decide which type of upper approximations is the best one to approximate the concept X. Moreover, the fourth type of lower approximation $(\underline{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ is the worst to describe X, since in Fig. 1 the fourth type of lower approximation has the furthest distance with X with respect to other types of lower approximations. And we can not figure out which is the worst to describe X for the case of upper approximations.

Second, from Fig. 1, one could also find that $\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X)$ and $\underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X)$ are generally not equal, and one does not contain another, that is, neither $\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X)$ nor $\underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X)$ , that is to say, $\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X)$ and $\underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X)$ are independent. So do $\bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X)$ and $\bar{M 3_{\sum_{i = 1}^{m} C_{i}}} (X)$ , $\bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X)$ and $\bar{M 5_{\sum_{i = 1}^{m} C_{i}}} (X)$ , $\bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X)$ and $\bar{M 6_{\sum_{i = 1}^{m} C_{i}}} (X)$ .

Finally, from Fig. 1, one could obtain the following theorem, which contains some important conclusions about the relationships among the six pairs of lower and upper approximations.

Theorem 8. Let 〈U, C〉 be a covering approximation space and C₁, C₂, ⋯ , C_m ∈ C. For any X ∈ F (U), we have

$\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋃ \underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \underline{M 5_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋂ \underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) \supseteq \underline{M 6_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋃ \underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋂ \underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) \supseteq \underline{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋂ \bar{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) \supseteq \bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋂ \bar{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \bar{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ .

Proof. It can be easily proved by employing Theorem 7.

□

Corollary 1. Let 〈U, C〉 be a covering approximation space and C₁, C₂, ⋯ , C_m ∈ C. If C₁, C₂, ⋯ , C_m are all partitions of U, then for any X ∈ F (U), we have

$\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋃ \underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) = \underline{M 5_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋂ \underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) = \underline{M 6_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋃ \underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) = \underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋂ \underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) = \underline{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋂ \bar{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) = \bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋂ \bar{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) = \bar{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ .

From Theorem 8 and Corollary 1, one could see that there exist some meaningful relationships among the six pairs of lower and upper approximations. For instance, from (1) of Theorem 8, one could find that for any X ⊆ F (U), the type-5 lower approximation of X is a superset of the union of the type-2 and type-3 lower approximations of X. Specially, from (1) of Corollary 1, if C₁, C₂, ⋯ , C_m are all partitions of U, then the type-5 lower approximation of X is equal to the union of the type-2 and type-3 lower approximations of X. Moreover, from (2)-(6) of Theorem 8 and (2)-(6) of Corollary 1, some similar conclusions could be obtained.

Example 6 shows the above results about the six pairs of lower and upper approximations given in Fig. 1.

Example 6. Let 〈U, C〉 be a covering approximationspace, where U = {x₁, x₂, x₃, x₄} and C₁, C₂ ∈ C are two coverings of U such that C₁ = {{x₁, x₂} , {x₂, x₃, x₄} , {x₃, x₄}}, C₂ = {{x₁, x₂, x₃} , {x₂, x₄} , {x₁, x₂, x₄}}. Let $X = {\frac{0.4}{x_{1}}, \frac{0.3}{x_{2}}, \frac{0.7}{x_{3}}, \frac{0.9}{x_{4}}}$ , then the following outputs can be obtained.

$\underline{M 1_{C_{1} + C_{2}}} (X) = {\frac{0.3}{x_{1}}, \frac{0.3}{x_{2}}, \frac{0.7}{x_{3}}, \frac{0.7}{x_{4}}}$ ,

$\bar{M 1_{C_{1} + C_{2}}} (X) = {\frac{0.4}{x_{1}}, \frac{0.3}{x_{2}}, \frac{0.7}{x_{3}}, \frac{0.9}{x_{4}}}$ ;

$\underline{M 2_{C_{1} + C_{2}}} (X) = {\frac{0.3}{x_{1}}, \frac{0.3}{x_{2}}, \frac{0.7}{x_{3}}, \frac{0.7}{x_{4}}}$ ,

$\bar{M 2_{C_{1} + C_{2}}} (X) = {\frac{0.4}{x_{1}}, \frac{0.9}{x_{2}}, \frac{0.7}{x_{3}}, \frac{0.9}{x_{4}}}$ ;

$\underline{M 3_{C_{1} + C_{2}}} (X) = {\frac{0.3}{x_{1}}, \frac{0.3}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.3}{x_{4}}}$ ,

$\bar{M 3_{C_{1} + C_{2}}} (X) = {\frac{0.4}{x_{1}}, \frac{0.3}{x_{2}}, \frac{0.7}{x_{3}}, \frac{0.9}{x_{4}}}$ ;

$\underline{M 4_{C_{1} + C_{2}}} (X) = {\frac{0.3}{x_{1}}, \frac{0.3}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.3}{x_{4}}}$ ,

$\bar{M 4_{C_{1} + C_{2}}} (X) = {\frac{0.4}{x_{1}}, \frac{0.9}{x_{2}}, \frac{0.7}{x_{3}}, \frac{0.9}{x_{4}}}$ ;

$\underline{M 5_{C_{1} + C_{2}}} (X) = {\frac{0.3}{x_{1}}, \frac{0.3}{x_{2}}, \frac{0.7}{x_{3}}, \frac{0.7}{x_{4}}}$ ,

$\bar{M 5_{C_{1} + C_{2}}} (X) = {\frac{0.4}{x_{1}}, \frac{0.4}{x_{2}}, \frac{0.7}{x_{3}}, \frac{0.9}{x_{4}}}$ ;

$\underline{M 6_{C_{1} + C_{2}}} (X) = {\frac{0.3}{x_{1}}, \frac{0.3}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.3}{x_{4}}}$ ,

$\bar{M 6_{C_{1} + C_{2}}} (X) = {\frac{0.4}{x_{1}}, \frac{0.4}{x_{2}}, \frac{0.7}{x_{3}}, \frac{0.9}{x_{4}}}$ .

Therefore,

$\bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \bar{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$X \subseteq \bar{M 5_{\sum_{i = 1}^{m} C_{i}}} (X) \bar{M 6_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \bar{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋃ \underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \underline{M 5_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋂ \underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) \supseteq \underline{M 6_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋃ \underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋂ \underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) \supseteq \underline{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋂ \bar{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) \supseteq \bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) ⋂ \bar{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) \subseteq \bar{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ .

Remark 5. $X \subseteq \bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (X)$ and $X \subseteq \bar{M 3_{\sum_{i = 1}^{m} C_{i}}} (X)$ may not satisfied.

Example 7. (Continued from Example 1) From Example 1, we have that $X = {\frac{0.7}{x_{1}}, \frac{0.5}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.2}{x_{4}}}$ , $\bar{M 1_{C_{1} + C_{2}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{2}}, \frac{0.3}{x_{3}}, \frac{0.3}{x_{4}}}$ , $\bar{M 3_{C_{1} + C_{2}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{2}}, \frac{0.5}{x_{3}}, \frac{0.5}{x_{4}}}$ .

Therefore, $X \subseteq \bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (X)$ and $X \subseteq \bar{M 3_{\sum_{i = 1}^{m} C_{i}}} (X)$ are all not satisfied.

Definition 16. [28] Let C be a covering of a set U. C is said to be unary if for any x ∈ U, |md (x) | = 1, where | · | denotes the cardinality of a set.

C is also said to be unary if for any x ∈ U,|MD (x) | = 1.

Theorem 9. Let 〈U, C〉 be a covering approximation space, and C₁, C₂, ⋯ , C_m ∈ C. For any X ⊆ F (U), if C₁, C₂, ⋯ , C_m are all unary, then we obtain

$\underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) = \underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) = \bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X)$ ;

$\underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) = \underline{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\bar{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) = \bar{M 4_{\sum_{i = 1}^{m} C_{i}}} (X)$ ;

$\underline{M 5_{\sum_{i = 1}^{m} C_{i}}} (X) = \underline{M 6_{\sum_{i = 1}^{m} C_{i}}} (X)$ ,

$\bar{M 5_{\sum_{i = 1}^{m} C_{i}}} (X) = \bar{M 6_{\sum_{i = 1}^{m} C_{i}}} (X)$ .

Proof: Here only the proofs of (1) and (2) are presented, the rest could be proved in a similar way.

If C₁, C₂, ⋯ , C_m are all unary, then for any x ∈ U, we have that ∩{K|K ∈ md_{C
_i} (x)} = ∪{K|K ∈ md_{C
_i} (x)}. Therefore, according to Definition 7, 8, 9 and 10, one has $\begin{matrix} \underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) = \underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X), \\ \bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) = \bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) . \end{matrix}$

Theorem 9 proposes a sufficient condition for which type-1 and type-2 CMFRS produce the identical approximations. Moreover, Theorem 9 also gives the sufficient condition for which type-3 and type-4 CMFRS and type-5 and type-6 produce identical approximations.

Remark 6. The reverse of Theorem 9 may not satisfied, i.e., the lower(upper) approximation of type-1 MGCRS and the lower(upper) approximation of type-2 MGCRS are identical, and the lower(upper) approximation of type-3 MGCRS and the lower(upper) approximation of type-4 MGCRS are identical, and the lower(upper) approximation of type-5 MGCRS and the lower(upper) approximation of type-6 MGCRS are identical, but none of C₁, C₂, ⋯ , C_m is unary.

Remark 6 could be illustrated by means ofExample 8.

Example 8. Let 〈U, C〉 be a covering approximation space, and C₁, C₂ ∈ C, where U = {x₁, x₂, x₃}, C₁ = {{x₁, x₂} , {x₁, x₃}}, C₂ = {{x₁, x₃} , {x₂, x₃}}.

Suppose that $X = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{3}}}$ , one has $\begin{matrix} m d_{C_{1}} (x_{1}) = {{x_{1}, x_{2}}, {x_{1}, x_{3}}}, \\ m d_{C_{1}} (x_{2}) = {{x_{1}, x_{2}}}, \\ m d_{C_{1}} (x_{3}) = {{x_{1}, x_{3}}}; \\ m d_{C_{2}} (x_{1}) = {{x_{1}, x_{3}}}, \\ m d_{C_{2}} (x_{2}) = {{x_{2}, x_{3}}}, \\ m d_{C_{2}} (x_{3}) = {{x_{1}, x_{3}}, {x_{2}, x_{3}}}; \\ M D_{C_{1}} (x_{1}) = {{x_{1}, x_{2}}, {x_{1}, x_{3}}}, \\ M D_{C_{1}} (x_{2}) = {{x_{1}, x_{2}}}, \\ M D_{C_{1}} (x_{3}) = {{x_{1}, x_{3}}}; \\ M D_{C_{2}} (x_{1}) = {{x_{1}, x_{3}}}, \\ M D_{C_{2}} (x_{2}) = {{x_{2}, x_{3}}}, \\ M D_{C_{2}} (x_{3}) = {{x_{1}, x_{3}}, {x_{2}, x_{3}}} . \end{matrix}$

Obviously, both C₁ and C₂ are not unary. But, owing to the definitions of CMFRS, we have $\begin{matrix} \underline{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) = \underline{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{3}}}, \\ \bar{M 1_{\sum_{i = 1}^{m} C_{i}}} (X) = \bar{M 2_{\sum_{i = 1}^{m} C_{i}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{3}}}; \\ \underline{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) = \underline{M 4_{\sum_{i = 1}^{m} C_{i}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{3}}}, \\ \bar{M 3_{\sum_{i = 1}^{m} C_{i}}} (X) = \bar{M 4_{\sum_{i = 1}^{m} C_{i}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{3}}}; \\ \underline{M 5_{\sum_{i = 1}^{m} C_{i}}} (X) = \underline{M 6_{\sum_{i = 1}^{m} C_{i}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{3}}}, \\ \bar{M 5_{\sum_{i = 1}^{m} C_{i}}} (X) = \bar{M 6_{\sum_{i = 1}^{m} C_{i}}} (X) = {\frac{0.5}{x_{1}}, \frac{0.5}{x_{3}}} . \end{matrix}$

Theorem 10. Let 〈U, C〉 be a covering approximationspace, and C₁, C₂, ⋯ , C_m ∈ C. If inf{X (y) |y ∈ ∩{K|K ∈ md_{C
_i} (x)}= inf{X (y) |y ∈ ∪{K|K ∈ md_{C
_i} (x)} and sup{X (y) |y ∈ ∩{K|K ∈ md_{C
_i} (x)}= sup{X (y) |y ∈ ∪{K|K ∈ md_{C
_i} (x)}, then for any X ⊆ U, $M 1_{\sum_{i = 1}^{m} C_{i}}$ , $M 2_{\sum_{i = 1}^{m} C_{i}}$ and $M 5_{\sum_{i = 1}^{m} C_{i}}$ produce the identical covering multi-granulation fuzzy rough lower and upper approximations.

Proof. It can be proved easily with the aid of Definitions 7, 8 and 11.

Theorem 11. Let 〈U, C〉 be a covering approximationspace, and C₁, C₂, ⋯ , C_m ∈ C. If inf{X (y) |y ∈ ∩{K|K ∈ MD_{C
_i} (x)}= inf{X (y) |y ∈ ∪{K|K ∈ MD_{C
_i} (x)} and sup{X (y) |y ∈ ∩{K|K ∈ MD_{C
_i} (x)}= sup{X (y) |y ∈ ∪{K|K ∈ MD_{C
_i} (x)}, then for any X ⊆ U, $M 3_{\sum_{i = 1}^{m} C_{i}}$ , $M 4_{\sum_{i = 1}^{m} C_{i}}$ and $M 6_{\sum_{i = 1}^{m} C_{i}}$ produce the identical covering multi-granulation fuzzy rough lower and upper approximations.

5 Conclusions

We have introduced six types of covering multi-granulation fuzzy rough set models and studied some important properties of these models. We have shown that our CMFRS models are natural extensions of MGRS models in covering approximation space. Moreover, the sufficient conditions for two distinct CMFRS models which produce identical lower and upper approximations of a given concept in a covering approximation space are disclosed. Finally, the relationships among various CMFRS models were discussed in details. The outputs show that for any X ⊆ F (U), the set of all those lower and upper approximations equipped with the binary relation ⊆ of inclusion, can not construct a lattice.

Acknowledgments

This work is supported by China National Natural Science Foundation of Youth Science Foundation under Grant No.: 61305052, 61403329 and the State Scholarship Fund of China (File No. 201409865003).

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