Several rough set models in quotient space

In order to deal with coarse ‐ grained and multi ‐ grained calculation problems, as well as granularity transformation problems in information system, quotient space theory is introduced in rough set theory. The main idea of this research is to try to maintain the important properties of the original space into the quotient space. Aimed to preserve the micro properties and the macro properties, two pairs of approximation operators on the quotient space are defined. When it comes to the composite of quotient spaces, the idea of these operators shows greater advantages. Examples are cited to illustrate possible applications of these operators, and their matrix representations are also given to make the calculations easy. Finally, all approximation operators on the quotient space involved so far are compared and their relationships are shown through a diagram.


| INTRODUCTION
Since rough set theory [1] was introduced by Z. Pawlak to deal with inconsistent and incomplete information in 1982, its effectiveness has been widely proved in many fields, such as computer science, engineering calculation, and information technology. [2][3][4][5][6]. The classical rough set theory is based on an equivalence relation that constitutes a partition of the universe, and the partition is regarded as a kind of knowledge being composed of definable sets, then the uncertain sets are approximated by the definable sets in this theory. However, in many practical problems, forming an equivalence relation is too strict to apply the theory. In this situation, researchers have generalized classical rough set theory from multiple angles [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Covering rough set theory which replaces partition with covering is one of generalized rough set theories [7,8,12,13,20,21,23] and based on a covering, the notion of neighbourhood of each element in the universe was given. Then many types of approximation operators [7,8,[24][25][26][27] based on the neighbourhoods were defined and their properties were investigated. To simplify the calculation, matrix methods have also been applied as a useful tool to study rough set theory [9,28,29].
Quotient space theory is used to provide a suitable granularity space [30,31] to describe and solve problems, as well as to establish the connection between different granularity spaces to simplify the problems [30,[32][33][34]. In reality, humans are always confronted with large amounts of complex information with limited cognitive abilities. In this case, complex information can be thought as coarse-grained objects which are composed of fine-grained objects, then the knowledge of fine-grained objects is taken to describe coarsegrained objects. Under an equivalence relation, the quotient space of a covering approximation space can describe the information granulation well, and it is wished that the excellent rough properties in original space can be well preserved in the quotient space. At present, quotient space is seldom introduced into rough set theory. In previous studies, Wang et al. [35] classified elements with the same neighbourhood into the same equivalent class by defining a consistent function, and then defined the images of the covering blocks of the original space as the covering of the image space. This article analysed the rough sets in a very particular image space which is essentially a special quotient space. It is a good innovation, however, the general quotient space under this type of covering cannot preserve the properties of the original space, either at the micro level or at the macro level.
Here, the study of rough sets in quotient approximation spaces of a covering approximation space is carried out, and it is known that it must be closely related to the original space to make sense. Therefore, the main idea of this research is to try to maintain the important properties of the original space. On the basis of the preliminary study in article [36], a new covering of a given quotient space is defined so that descriptors' characteristics in original space are preserved completely in quotient space, and the quotient space with this type of covering is called as quotient covering approximation space. Then, two pairs of approximation operators on the quotient space are defined and it is found that the approximation effect of the second pair of operators is better than that of the first pair. Also, by combining these two pairs of operators, a reasonable solution is proposed to a coarse-grained problem. Next, the composite of quotient spaces to solve a coarser-grained problem is discussed. In addition, depending on the results of Ma's article [28], the matrix formulas of those two pairs of approximation operators on the quotient approximation space are defined to make the calculation easier. All approximation operators involved on the quotient space were also compared and a diagram to show their relationships is given.
The rest of this article is organized as follows. In Section 2, some basic concepts and properties are recalled. In Section 3, a quotient covering approximation space is defined and then the neighbourhood in this space is given. In Section 4, two pairs of approximation operators on the quotient spaceare given, as well as research on their basic properties and applications are carried out. The matrix representations of those two pairs of approximation operators are given in Section 5 and the composite of quotient spaces is introduced in Section 6. In Section 7, all the approximation operators involved on quotient space are compared and the conclusion of this article is made in Section 8.

| PRELIMINARIES
In this section, some fundamental knowledge that will be used in this article are recollected. The notation 'A ⊂ B' in this article represents that 'the set A is a strict subset of the set B' or 'the set A is equal to the set B'. Definition 1 [16,17]. Let U be a finite set and C be a family of subsets of U. If none of the elements in C is empty and U ¼ [ C2C C, then C is called a covering of U, and the ordered pair ðU; CÞ is called a covering approximation space.
Definition 2 [17]. Let ðU; CÞ be a covering approximation space. For each element x 2 U, a neighbourhood of x is defined as NðxÞ ¼ ∩fC 2 C : x 2 Cg.
Neighbourhood N(x) is the minimum descriptor while describing characteristics of x. Depending on N(x), a pair of approximation operators is defined and has been widely used in rough set theory. Definition 3 [11]. Let ðU; CÞ be a covering approximation space and X be a subset of U. The lower approximation P − X ð Þ and the upper approximation P þ X ð Þ of X are defined as From definitions of the lower approximation operator and upper approximation operator, we can easily get the following properties: Next, some basic concepts related to the quotient set are recalled.
Definition 4 [37]. Suppose R is an equivalence relation on U, for each element x 2 U, the subset which can be described as y 2 U : xRy f g is called the equivalent class of x, and the family of equivalent classes is defined as a quotient set U/R determined by R.
Proposition 1 [37]. Given an equivalence relation R on U, Proposition 2 [37]. Given an equivalence relation R on U, and set V = U/R, then the inverse of R satisfies the formula:

| NEIGHBOURHOOD IN QUOTIENT COVERING APPROXIMATION SPACE
In this section, a quotient covering approximation space is constructed with the type of covering of a given quotient space in [36] for the sake of preserving characteristic of neighbourhoods in original space. Then the neighbourhood of y in quotient covering approximation space is given.
According to the covering of quotient space, the neighbourhood of y in quotient space can be further defined.

Definition 6
Let ðV ; C R Þ be a quotient covering approximation space of ðU; CÞ. For each element y 2 V, the neighbourhood of y is defined as nðyÞ ¼ ∩ Here a practical example is given to introduce the meaning of each formula in the original space and in the corresponding quotient space. Example 2 Leading industries of a city can be used to judge whether a city is polluted or not. If Ministry of Environmental Protection wants to govern polluted provinces under limited conditions, such as, it only knows the leading industries of each city and has limited material conditions, then the first step is to get the details of the leading industries of the provinces. Suppose U is a collection of some cities in the country and C is the collection of leading industries of all cities in U. Let V be the collection of provinces to which all cities in U belong. Then it is assumed that ðU; CÞ and ðV ; C R Þ are the same as that in Example 1.
If x 2 C i ∩ C j , then city x has both leading industries C i and C j . So NðxÞ ¼ ∩fC 2 C : x 2 Cg reflects the leading industries of x, on the other hand, it can also represent the cities which own the leading industries of x. In definition, C R is a collection of the ranges of leading industries of each province in V. As given in Example 1, the leading industries of y 1 are in the range of leading industries C 1 ; then y i 's leading industries and y j 's leading industries are within the range D m but not necessarily identical. By this means, n(y) reflects the smallest range of leading industries of y in V.
Certainly, it also represents the smallest set of provinces whose leading industries are also within the range of leading industries of y.
Now the connection between neighbourhoods n(y) and N x ð Þ under the equivalence relation R is taken into consideration.
Proposition 3 [36] Let ðU; CÞ be a covering approximation space and ðV ; C R Þ be the quotient covering approximation space of ðU; CÞ. For each element y 2 V, we have n y ð Þ ¼ [ Proof. According to the proposition in [36], which Act R on both sides of the formula, then n y ð Þ ¼ [ This proposition obviously indicates that if This means that properties of the minimum descriptor N x ð Þ in the original space are completely preserved by n y ð Þ in the quotient space, and this is why such type of covering is defined on the quotient ZHAO AND MA -3 space. Next, the data from Example 1 is used for testing this conclusion and give an actual explanation about the advantages of this proposition according to Example 2. Here, only consider y 1 2 V as a representative to show this point.
In Example 2, these results mean that when city x 4 owns the leading industries of the city x 1 , there will be the leading industries of province y 2 containing the city x 4 are also within the smallest range of leading industries of the province y 1 containing the city x 1 , so does x 5 . Also, when city x 3 owns the leading industries of city x 2 , then the leading industries of province y 2 containing the city x 3 are also within the smallest range of leading industries of the province y 1 containing the city x 2 and so on. Hence, from the perspective of the range of the leading industries of provinces in V, Ministry of Environmental Protection can adopt the same governance as y 1 to govern y 2 and y 3 .

Corollary 1
Let ðU; CÞ be a covering approximation space and ðV ; C R Þ be the quotient covering approximation space of ðU; CÞ. For any y i , y j 2 V, if we take Proof. It directly follows from Proposition 3 that

| TWO PAIRS OF APPROXIMATION OPERATORS ON QUOTIENT SPACE
Based on n(y), now the first pair of approximation operators is given on ðV ; C R Þ. This pair of operators can be understood as preserving the properties of the original space from the micro perspective.

Definition 7
Let ðU; CÞ be a covering approximation space and ðV ; C R Þ be the quotient covering approximation space of ðU; CÞ. For each Y ⊂ V, we define the lower approximation p − (Y) and the upper approximation p + (Y) of Y as In order to make the approximation operators on the quotient space retain the properties of approximation operators on the original space from the macro perspective, the approximation operators on the original space are used directly for defining the approximation operators on the quotient space. So the second pair of lower approximation p − *(Y) and upper approximation p +* (Y) of Y are given as follows.

Definition 8
Let V be a quotient space of a covering approximation space ðU; CÞ. For each Y ⊂ V, the lower approximation of Y is defined as: It is worth mentioning that this pair of operators requires only a covering of U, not a covering of V.
It can be seen that the first pair of approximation operators on quotient space has the same form as the pair of approximation operators in Definition 3. Also, relying on the excellent properties of n(y), it can be proved that p − has properties P 1 ð Þ − P 6 ð Þ, and p + has properties Q 1 ð Þ − Q 6 ð Þ. Now the properties of the second pair of approximation operators p -* ; p þ* À � are studied À p − * ; p þ * � � .

Proposition 4
Let V be a quotient space of a covering approximation space ðU; CÞ. For each Y ⊂ V, we have Proof. The first equation is to be proved. For each element. y 0 2 p − * ðY Þ Then the second equation is to be proved. For each element y 0 2 p þ * ðY Þ Notice that the equivalence of Equations (2) and (3) is determined by Proposition 2, and it must be noted that R is a special relation-the equivalence relation.
Exactly as discussed above, the second pair of approximation operators can be transformed into the expressions determined by the image of neighbourhoods of elements in original space. Therefore, it will be easier to study the following propositions.

Proposition 5
Let V be a quotient space of a covering approximation space ðU; CÞ. For each Y ⊂ V, the lower approximation operator p − * satisfies properties P 1 ð Þ − P 4 ð Þ.
Proof. Conditions P 1 ð Þ − P 3 ð Þ are obvious, so the proof of P 4 ð Þ can be given. First, prove ð Þ, then act R on both sides of the formula, and the proof will be finished. According to the fact that Let us act R on both sides of the formula, it follows that Here an example to illustrate that p − * is given and it does not satisfy the property P 5 ð Þ. To prove the property Q 4 ð Þ is equivalent to prove the Then, it needs to be proved that □ An example is given to show that p − * and p þ * do not satisfy the duality, namely, P 6 ð Þ and Q 6 ð Þ.
Proposition 3 presents a definite formula between n y ð Þ and Þ. It is very important, and it can be taken to build the relationship between the two pairs of approximation operators.
Proof. For each element y 0 2 p − Y ð Þ, it follows that n y 0 ð Þ ⊂ Y . By Proposition 3, we can get . Now the inclusion relationships on the right is proved. For each element y 0 2 p þ* Y ð Þ, there exists x 0 2 R −1 y 0 ð Þ such that R N x 0 ð Þ ð Þ ∩ Y ≠ ∅. By Proposition 3, we have n y 0 ð Þ ∩ Y ≠ ∅.
Example 6 In Example 1, consider the two pair of lower and upper approximations of ð Þ ¼ y 1 ; y 2 ; y 7 ; y 8 ; y 9 � � and p þ Y 1 ð Þ ¼ y 1 ; y 2 ; � y 3 ; y 7 ; y 8 ; y 9 g. Therefore This example also shows that every containment relationship in the formula of Proposition 7 is generally not an equality.
Proposition 7 tells us the fact that the second pair of op- has better approximation effect than that of the first pair p − ; p þ ð Þ. By taking advantage of that, these two pairs of operators can be combined to solve the practical problem in Example 1.

Example 7
In Example 2, it is assumed that C 1 , C 5 , C 7 are the leading industries that cause cities to get polluted, which indicates that if city x 2 C i , i = 1, 5, 7, then it is a polluted city. However, the province to which the city x belongs is not necessarily polluted, for whether a province is polluted is determined by its leading industries. The information Tables 1 and 2 are provided to show the pollution of cities in U and the pollution of provinces in V according to the data in Example 1, and the information Table 2 is inconsistent.
Under limited material conditions, if Ministry of Environmental Protection wants to give priority to govern polluted provinces, then the models of the two pairs of approximation operators can be taken into account.
As for the polluted provinces y 1 , y 7 in Table 2 and the results of Example 6, the province y 7 can be governed firstly, which is selected by the second pair of operators from the polluted provinces y 1 , y 7 . Province y 1 can be governed secondly, provinces y 2 , y 8 , y 9 can be governed thirdly, province y 3 can be governed fourthly, and provinces y 4 , y 5 , y 6 need not be governed.

| MATRIX REPRESENTATIONS OF THE TWO PAIRS OF APPROXIMATION OPERATORS
In this section, matrix representations of the two new pairs of approximation operators introduced in Section 4 are given. Through matrix calculation, the abstract calculations of definitions simple and straightforward can be made, especially when the data is very big.
Definition 9 [13,15]. Suppose that C ¼ C 1 ; C 2 ; f …; C l g is a family of subsets of a finite universe The matrix representation depends on the orders of the elements in U. In fact, the main results of the calculation T A B L E 1 A table of the pollution of cities in U

City
The leading industries Polluted

Province
The smallest range of leading industries Polluted formulas in this section are not affected by these orders. So any given order will be determined.
Definition 10 [28]. Given Boolean matrixes n�n is defined as: where ∨ and ∧ mean the max and the min operations, apparently, the Boolean product C n�n is still a Boolean matrix.
Definition 11 [29]. Suppose that X is a subset of U. We represent X as the column vector χ X ¼ a 1 ; a 2 ; ð …; a n Þ T , where a i = 1 when x i 2 X and a i = 0 when x i ∉ X, and this column vector is called the characteristic function vector of the subset X.
In Ma's article [28], it has been found that the way to calculate the upper and lower approximations of a rough set by defining the product of Boolean matrices.
Proposition 9 [28]. Let ðU; CÞ be a covering approximation space, if M C is a matrix representation for C, then for each X ⊂ U, it follows Then it can just take the relational matrix N R of R into account to study the matrix representations of p − * ; p þ * � � .

Proposition 10
Let V be a quotient space of a covering approximation space ðU; CÞ, where U ¼ x 1 ; x 2 ; …; x n f g and V ¼ y 1 ; y 2 ; …; y m � � . Suppose that M C is the matrix representation of the covering C ¼ fC 1 ; C 2 ; …; C l g of U and N R is the relational matrix of R, then for each Y ⊂ V, we have Proof. It follows from Proposition 8 that So, according to the Proposition 9 in article [28], the result is obtained. □ Next, an algorithm for calculating the neighbourhood n y ð Þ of each element y in the quotient space V of ðU; CÞ is given.
Then the matrix representations of p − ; p þ ð Þ can be surely obtained on the basis of the Boolean matrix n …; mg is the family of neighbourhoods of elements in V ¼ y 1 ; y 2 ; …; y m � � .

Proposition 11
Let ðV ; C R Þ be a quotient covering approximation space of a covering approximation space ðU; CÞ, then for each Y ⊂ V, we have And χ p − Y ð Þ ¼ M B T ⋅ χ Y follows directly from the duality of p − Y ð Þ and p þ Y ð Þ. □ Next, a concrete example to show how to use Boolean matrix for computing the upper and lower approximations of subsets in quotient covering approximation space under the two pairs of operators is given.