Some twin approximation operators on covering approximation spaces

International Journal of Approximate Reasoning 56 (2015) 59–70

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International Journal of Approximate Reasoning www.elsevier.com/locate/ijar

Some twin approximation operators on covering approximation spaces Liwen Ma School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, PR China

a r t i c l e

i n f o

Article history: Received 6 November 2013 Received in revised form 10 June 2014 Accepted 8 August 2014 Available online 15 August 2014 Keywords: Twin approximation operators Complementary neighborhood Neighborhood Topology

a b s t r a c t Complementary neighborhood is a conception analogous to the neighborhood that we first introduced in a former paper. In this paper, we show that two different approximation operators may have a close relationship, namely, they can be defined almost in the same way except that one uses the notion of neighborhood and another uses the complementary neighborhood. We call such two approximation operators the twin approximation operators. We give some concrete examples of the twin approximation operators. Through detailed investigation on the relationship between the neighborhood and the complementary neighborhood, we further study the properties of given twin approximation operators and investigate the relationships between different twins. We also reveal the topological properties of those twin approximation operators. © 2014 Elsevier Inc. All rights reserved.

1. Introduction Since the rough set theory was developed by Pawlak in 1982 [1], it has been successfully applied to many areas. Originally, the equivalence relation is used in the rough set theory to describe the indiscernibility of elements so as to deal with the vagueness and uncertainty in information systems. However, in real-world applications, the relations between objects are often much more complicated than the equivalence relations, and vast quantities of important information such as degrees of inclusion relations between sets and the extent of overlap of sets, etc., were not taken into account in the equivalence-relation-based rough set theory. In order to solve more and more complicated problems, researchers generalized the equivalence-relation-based rough set theory to the non-equivalent-relation-based rough set theory [2–5] covering rough set theory [6–8] and fuzzy rough set theory [9,10], etc. As the concept of neighborhood has many practical applications in feature selection, granular computing and attribute reduction, etc. [11–15], the neighborhood-related rough sets were studied. In the non-equivalent-relation-based rough set theory, the successor neighborhood and predecessor neighborhood are two important concepts [4,16], and the k-step neighborhood system was also applied to this theory [17]. In the covering rough set theory, the concept of neighborhood induced by covering plays an important role [7,18–24]. The neighborhood-based covering rough set theory has proven to be useful in the discovery of decision rules from the incomplete information systems [14] and the attribute reduction from nominal data [12]. In Ref. [21], we first introduced the concept of complementary neighborhood in the investigation of covering rough sets. During the investigation of covering rough set theory, various neighborhood-based lower and upper approximation operators have been defined and studied [4,23–26]. It is often difficult to find the relationship between different approximation

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ijar.2014.08.003 0888-613X/© 2014 Elsevier Inc. All rights reserved.

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L. Ma / International Journal of Approximate Reasoning 56 (2015) 59–70

operators. In this paper, based on the notions of neighborhood and complementary neighborhood, we simplified the definitions of some known approximation operators, and find that two different approximation operators may have a close relationship, namely, they can be defined almost in the same way except that one uses the notion of neighborhood and another uses the complementary neighborhood. We call such two approximation operators the twin approximation operators. We also find corresponding twin operators for some known approximation operators. Moreover, through detailed investigation of the relationship between the neighborhood and complementary neighborhood, we further reveal the properties of given twin operators. Topology provides many valid mathematical methods and skills for the study of rough set theory, and many researchers have investigated the rough set theory from topological viewpoint [7,21,27–31]. In this paper, we also inquire into topological properties of all involved approximation operators. It is found that the twin approximation operators have similar properties. The rest of this paper is organized as follows. In Section 2, we discuss the relationships between the neighborhood and complementary neighborhood. In Section 3, we show several pairs of twin approximations, and study the properties of them in the next section. In Section 5, we thoroughly investigate the topological properties of all those approximation operators. We conclude in the last section. 2. Neighborhood and complementary neighborhood In this section, we recall some fundamental concepts in rough set theory and discuss the relationship between neighborhood and complementary neighborhood. Pawlak’s rough sets are based on equivalence relations, or equivalently, partitions. Let U be a finite set called universe, and R be an equivalence relation on U . U / R denotes the family of all equivalence classes induced by R. Obviously U / R is a partition of U . For any X ⊆ U , the lower and upper approximations of X are defined as follows:

R∗( X ) =



{Y i ∈ U / R : Y i ⊆ X },

R∗( X ) =



{Y i ∈ U / R : Y i ∩ X = ∅}.

According to Pawlak’s definition, X is called a rough set if and only if R ∗ ( X ) = R ∗ ( X ). Covering is an extension to partition. Pawlak’s rough set model was also extended to covering based rough sets. Definition 1. (See [7].) Let U be a universe and C be a family of subsets of U . If no element in C is empty and U = then C is called a covering of U , and the ordered pair (U , C ) is called a covering approximation space.



C ∈C

C,

Following the sense of Pawlak, a set X ⊂ U is called a covering rough set if its covering-induced lower approximation and upper approximation are not equal. The concept of neighborhood plays an important role in defining approximations of sets in covering approximation spaces. In our recent paper [10], we gave a new notion of complementary neighborhood, which is analogous to neighborhood in practical and theoretical investigations of covering rough sets. It is extremely important to introduce the concept of complementary neighborhood. Based on the concepts of neighborhood and complementary neighborhood, not only can we simplify the forms of some known types of lower and upper approximation operators, but can make the relationships between some approximation operators clear and, furthermore, can define some new types of lower and upper approximation operators so as to select suitable approximations in practice. In the following, we review these two concepts, and discuss their properties and relationships. Definition 2. (See [7,8].) Let (U , C ) be a covering approximation space. We define the neighborhood of an element x ∈ U as

N (x) =



{C ∈ C : x ∈ C }.

In this paper, we use − X to denote the subset U − X , where U is the universe and X ⊂ U . We next define the concept of complementary neighborhood in a new way which is equivalent to Definition 9 in paper [21]. Definition 3. Let (U , C ) be a covering approximation space and x ∈ U . We call



M (x) = −



{C ∈ C : x ∈ / C}

the complementary neighborhood of x. We can easily find that this definition is equivalent to

M (x) =

  −C : (C ∈ C ) ∧ (x ∈ / C) ,

where M (x) = U if x ∈ C for each C ∈ C .

L. Ma / International Journal of Approximate Reasoning 56 (2015) 59–70

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The following propositions describe the relationships between neighborhood and complementary neighborhood. Proposition 1. (See [21].) Let (U , C ) be a covering approximation space and x, y ∈ U , then y ∈ N (x) if and only if x ∈ M ( y ). Proposition 2. (See [21].) Let (U , C ) be a covering approximation space. For each x ∈ U , we have M (x) = { y : x ∈ N ( y )} and N (x) = { y : x ∈ M ( y )}. In paper [7], the following property of neighborhood was proved. Proposition 3. Let (U , C ) be a covering approximation space, then for each C ∈ C , we have

C=



N (x).

x∈C

Proposition 4 which is similar to Proposition 3 deals with the property of complementary neighborhood. Proposition 4. (See [21].) Let (U , C ) be a covering approximation space. For each C ∈ C , we have

−C =



M (x).

x∈−C

According to Definition 3, Proposition 3 and Proposition 4, we have Proposition 5. Let (U , C ) be a covering approximation space, then for each x ∈ U ,

N (x) = −



M ( y ),

y ∈− N (x)

Proof.



M ( y) =

y ∈− N (x)

=



M (x) = −

N ( y ).

y ∈− M (x)





M ( y ) : y ∈ − N (x) =





M ( y) : y ∈ −



C

x∈C

M ( y) : y ∈



  (−C ) = {−C : x ∈ C } = − {C : x ∈ C } = − N (x).

x∈C

So, we have

N (x) = −



M ( y ).

y ∈− N (x)

Similarly, we can prove the equality

M (x) = −



N ( y ).

2

y ∈− M (x)

Proposition 6. (See [21,32].) Let (U , C ) be a covering approximation space and x, y , z ∈ U . Then both the neighborhood operator N and the complementary neighborhood operator M are transitive operators, i.e., if x ∈ N ( y ) and y ∈ N ( z), then x ∈ N ( z), if x ∈ M ( y ) and y ∈ M ( z), then x ∈ M ( z). The following two properties are important to describe the connections between neighborhood and complementary neighborhood. Proposition 7. Let (U , C ) be a covering approximation space and x, y ∈ U . For each x, y ∈ U , M (x) = M ( y ) if and only if N (x) = N ( y ). Proof.

M (x) = M ( y )

⇔

⇔

{C ∈ C : x ∈ / C } = {C ∈ C : y ∈ / C}

{C ∈ C : x ∈ C } = {C ∈ C : y ∈ C }

⇔

N (x) = N ( y ).

2

Proposition 8. Let (U , C ) be a covering approximation space. The following statements are equivalent: i ). N (x) ⊂ M (x) for each x ∈ U , ii ). M (x) ⊂ N (x) for each x ∈ U ,

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L. Ma / International Journal of Approximate Reasoning 56 (2015) 59–70

iii ). N (x) = M (x) for each x ∈ U , i v ). { N (x) : x ∈ U } is a partition of U , v ). { M (x) : x ∈ U } is a partition of U . Proof. iii ) → i ) and iii ) → ii ) are obvious. We shall next show i v ) → iii ), v ) → iii ), ii ) → i v ) and i ) → v ). i v ) → iii ). If { N (x) : x ∈ U } is a partition of U , then for every x, y ∈ U , we have N (x) = N ( y ) or N (x) ∩ N ( y ) = ∅. Thus, for each y ∈ N (x) we have N (x) = N ( y ). Then x ∈ N ( y ), and it is followed from Proposition 1 that y ∈ M (x). So N (x) ⊂ M (x) holds for each x ∈ U . Suppose that there is y ∈ M (x) but y ∈ / N (x) for some x ∈ U , then we have x ∈ N ( y ) and x ∈ / M ( y ). This contradicts to N ( y ) ⊂ M ( y ). Thus we have proved N (x) = M (x) for each x ∈ U . Similarly, we can prove v ) → iii ). ii ) → i v ). By contradiction. If { N (x) : x ∈ U } is not a partition of U , then there are y , z ∈ U such that N ( y ) = N ( z) and N ( y ) ∩ N ( z) = ∅. Without loss of generality, we assume that N ( y ) − N ( z) = ∅. Then y ∈ N ( y ) − N ( z). Otherwise, if y ∈ N ( z), we have N ( y ) ⊂ N ( z), and this contradicts to N ( y ) − N ( z) = ∅. We then can take x ∈ N ( y ) ∩ N ( z). This implies y ∈ M (x). We can prove that y ∈ / N (x). If y ∈ N (x), considering that x ∈ N (z), we have y ∈ N (z), and this contradicts to N ( y ) − N (z) = ∅. Thus y ∈ / N (x) and we have a contradiction to M (x) ⊂ N (x). So ii ) → i v ) is proved. We can prove i ) → v ) similarly. Thus we finished the proof of the theorem. 2 In this paper, we use superscripts to distinguish operators generated by different coverings. For example, M C , M D denote the complementary neighborhood operators in covering approximation space (U , C ) and (U , D ) respectively. Proposition 2 shows that the following Proposition 9 is correct. Proposition 9. Let U be a set, and C and D be two coverings of U . Then for each x ∈ U , M C (x) = M D (x) if and only if N C (x) = N D (x). In order to discuss the properties and relationships of approximation operators, we also introduce the definitions of neighborhoods and complementary neighborhoods of sets. Definition 4. Let (U , C ) be a covering approximation space and A ⊂ U . We call

N ( A) =





N (x) : x ∈ A ,

M ( A) =



M (x) : x ∈ A



the neighborhood and complementary neighborhood of set A respectively. Now we can construct several descriptions of each element in a universe by neighborhood and complementary neighborhood. For example, M ( N (x)), N ( M (x)), M (x) ∩ N (x), M (x) ∪ N (x), M ( N (x)) ∩ N ( M (x)), M ( N (x)) ∪ N ( M (x)), etc. Proposition 10. Let (U , C ) be a covering approximation space. Then for each x ∈ U , we have N (x) ∪ M (x) ⊂ N ( M (x)) ∩ M ( N (x)). Proof. It follows from Definition 4 that we have M (x) ⊂ N ( M (x)) and N (x) ⊂ M ( N (x)). Because of {x} ⊂ N (x) and {x} ⊂ M (x), we also have M (x) = M ({x}) ⊂ M ( N (x)) and N (x) = N ({x}) ⊂ N ( M (x)). So M (x) ⊂ N ( M (x)) ∩ M ( N (x)) and N (x) ⊂ N ( M (x)) ∩ M ( N (x)) and hence N (x) ∪ M (x) ⊂ N ( M (x)) ∩ M ( N (x)). 2 For every x ∈ U , M ( N (x)) and N ( M (x)) are not always equal. We show this by the following example. Example 1. Let U = {x1 , x2 , x3 , x4 , x5 } and C = {C 1 , C 2 , C 3 , C 4 }, where C 1 = {x1 , x2 , x3 , x4 }, C 2 = {x3 , x4 }, C 3 = {x4 , x5 } and C 4 = {x2 }. Considering the covering approximation space (U , C ) and x3 , we can easily calculate that





M N (x3 ) = {x1 , x3 , x4 , x5 },





N M (x3 ) = {x1 , x2 , x3 , x4 }.

Next, we shall investigate some sufficient conditions under which M ( N (x)) = N ( M (x)) holds for each x ∈ U . Proposition 11. Let (U , C ) be a covering approximation space and x, y ∈ U , then y ∈ N ( M (x)) if and only if x ∈ N ( M ( y )); y ∈ M ( N (x)) if and only if x ∈ M ( N ( y )). Proof. It follows from Proposition 1 that





y ∈ N M (x)

⇔

 (∃ z ∈ U ) y ∈ N ( z) ∧ z ∈ M (x)

The proof of the case y ∈ M ( N (x)) is similar to this one.

⇔

 (∃ z ∈ U ) z ∈ M ( y ) ∧ x ∈ N ( z)

⇔





x ∈ N M ( y) .

2

Definition 5. The operator N is called semi-Euclidean if y ∈ N (x) and z ∈ N (x) imply that y ∈ N ( z) or z ∈ N ( y ).

L. Ma / International Journal of Approximate Reasoning 56 (2015) 59–70

63

Fig. 1. For the readers’ convenience, we draw a figure as follows, where the lower element is a subset of the upper element when they are linked by a segment.

Proposition 12. If the operator M is semi-Euclidean, then for each x ∈ U we have M ( N (x)) ⊂ N ( M (x)); if the operator N is semiEuclidean, then for each x ∈ U we have N ( M (x)) ⊂ M ( N (x)). Proof. Suppose that the operator M is semi-Euclidean. For each z ∈ M ( N (x)), there exists y ∈ N (x) such that z ∈ M ( y ). Thus we have x ∈ M ( y ) and z ∈ M ( y ). Since M is semi-Euclidean, then z ∈ M (x) or x ∈ M ( z), namely, z ∈ M (x) or z ∈ N (x). Therefore, z ∈ N ( M (x)) holds and we have M ( N (x)) ⊂ N ( M (x)). If the operator N is semi-Euclidean, we can prove N ( M (x)) ⊂ M ( N (x)) similarly. Thus the desired result is proved. 2 Proposition 13. If the operators M and N are both semi-Euclidean, then for each x ∈ U we have









M N (x) = N M (x) . The converse of this proposition is generally not correct. We show this by the following example. Example 2. Let U = {x1 , x2 , x3 , x4 } and C = {C 1 , C 2 , C 3 }, where C 1 = {x1 , x2 , x3 , x4 }, C 2 = {x2 , x4 } and C 3 = {x3 , x4 }. Consider the covering approximation space (U , C ) and each xi (i = 1, 2, 3, 4). A straightforward calculation shows that N ( M (xi )) = M ( N (xi )) = {x1 , x2 , x3 , x4 } (i = 1, 2, 3, 4). / N (x3 ) and x3 ∈ / N (x2 ) that N is We next show that M and N are not semi-Euclidean. It follows from x2 , x3 ∈ N (x1 ), x2 ∈ not semi-Euclidean. Similarly, it follows from x2 , x3 ∈ M (x4 ), x2 ∈ / M (x3 ) and x3 ∈ / M (x2 ) that M is not semi-Euclidean. The following proposition is obvious, which shows the relationships of all above descriptions of each element. Proposition 14. For each x in the covering approximation space (U , C ), the relationships of N (x), M (x), M (x) ∩ N (x), M (x) ∪ N (x), M ( N (x)), N ( M (x)), M ( N (x)) ∩ N ( M (x)) and M ( N (x)) ∪ N ( M (x)) can be represented by the following lattice, where the lower element is a subset of the upper element if they are linked by one broken line. The following example shows that every two elements adjoined in Fig. 1 are generally not equal. Example 3. Let U = {x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 } and C = {C 1 , C 2 , C 3 , C 4 , C 5 }, where C 1 = {x1 , x2 , x3 , x4 , x5 , x7 }, C 2 = {x1 , x3 , x4 }, C 3 = {x4 , x6 , x7 }, C 4 = {x5 } and C 5 = {x8 }. Considering the covering approximation space (U , C ) and x1 , we can easily calculate that

M (x1 ) = {x1 , x2 , x3 },

N (x1 ) = {x1 , x3 , x4 },

N (x1 ) ∩ M (x1 ) = {x1 , x3 },

N (x1 ) ∪ M (x1 ) = {x1 , x2 , x3 , x4 },  M N (x1 ) = {x1 , x2 , x3 , x4 , x6 , x7 }, N M (x1 ) = {x1 , x2 , x3 , x4 , x5 , x7 }     N M (x1 ) ∩ M N (x1 ) = {x1 , x2 , x3 , x4 , x7 }, N M (x1 ) ∪ M N (x1 ) = {x1 , x2 , x3 , x4 , x5 , x6 , x7 }.



3. Twin approximation operators In this section, we use P ( X ) to denote the lower approximation of X and P ( X ) to denote upper approximation of X . In Ref. [26], there are several pairs of neighborhood-based lower and upper approximations. Here we select four of them:

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L. Ma / International Journal of Approximate Reasoning 56 (2015) 59–70







P 1 ( X ) = x : N (x) ⊂ X ,











P 1 ( X ) = x : N (x) ∩ X = ∅ .

P 2 ( X ) = x : ∀u x ∈ N (u ) → u ∈ X

 ,

P 2( X ) =





N (x) : x ∈ X .

   , P 4( X ) = N (x) : N (x) ∩ X = ∅ .   P 5( X ) = P xC : P xC ∩ X = ∅ ,

P 4 ( X ) = x : ∀u x ∈ N (u ) → N (u ) ⊂ X P 5( X ) = in which



 P xC : P xC ⊂ X ,





P xC = y : ∀C ∈ C (x ∈ C ↔ y ∈ C ) . The definition of approximation operator pair ( P 1 , P 1 ) and detailed discussion about them can be found in Refs. [4,25,26]. The operator pair ( P 2 , P 2 ) can also be found in Ref. [23]. In the following, using the relationships between neighborhood and complementary neighborhood, we give an equivalent definition of each pair of ( P 2 , P 2 ), ( P 4 , P 4 ) and ( P 5 , P 5 ). We must note that these new definitions are simpler than their original ones and all have the same forms as the definition of ( P 1 , P 1 ) which are convenient for our later discussion of their properties. Proposition 15. For each subset X of the covering approximation space (U , C ), we have the following equivalent definitions of ( P 2 , P 2 ), ( P 4 , P 4 ) and ( P 5 , P 5 ):





P 2 ( X ) = x : M (x) ⊂ X ,











P 4 ( X ) = x : N M (x) ⊂ X ,





P 2 ( X ) = x : M (x) ∩ X = ∅ ,











P 4 ( X ) = x : N M (x) ∩ X = ∅ ,



P 5 ( X ) = x : N (x) ∩ M (x) ⊂ X ,







P 5 ( X ) = x : N (x) ∩ M (x) ∩ X = ∅ .

Proof. Using Proposition 1 repeatedly, we have



     = x : ∀u u ∈ M (x) → u ∈ X = x : M (x) ⊂ X ,           P 2( X ) = N (x) : x ∈ X = N ( X ) = x : ∃ y ∈ X x ∈ N ( y ) = x : ∃ y ∈ X y ∈ M (x) = x : M (x) ∩ X = ∅ ,          P 4 ( X ) = x : ∀u x ∈ N (u ) → N (u ) ⊂ X = x : ∀u u ∈ M (x) → N (u ) ⊂ X = x : N M (x) ⊂ X ,           P 4( X ) = N (x) : N (x) ∩ X = ∅ = y : y ∈ N (x) ∧ N (x) ∩ X = ∅ = y : x ∈ M ( y ) ∧ N (x) ∩ X = ∅       = y : N M ( y ) ∩ X = ∅ = x : N M (x) ∩ X = ∅ ,         P xC = y : ∀C ∈ C (x ∈ C ↔ y ∈ C ) = y : N (x) = N ( y ) = y : N (x) ⊂ N ( y ) ∧ N ( y ) ⊂ N (x)           = y : x ∈ N ( y ) ∧ y ∈ N (x) = y : y ∈ M (x) ∧ y ∈ N (x) = y : y ∈ M (x) ∩ N (x) = N (x) ∩ M (x). 

P 2 ( X ) = x : ∀u x ∈ N (u ) → u ∈ X



It follows from N and M are both transitive operators that y ∈ N (x) ∩ M (x) implies N ( y ) ∩ M ( y ) ⊂ N (x) ∩ M (x), i.e., y ∈ P xC implies P Cy ⊂ P xC . Then

P 5( X ) =



       P xC : P xC ⊂ X = y : y ∈ P xC ∧ P xC ⊂ X ⊂ y : P Cy ⊂ X

    = y : N ( y ) ∩ M ( y ) ⊂ X = x : N (x) ∩ M (x) ⊂ X .

Since that {x : N (x) ∩ M (x) ⊂ X } = {x : P xC ⊂ X } ⊂







{ P xC : P xC ⊂ X } is obviously hold, so we have

P 5 ( X ) = x : N (x) ∩ M (x) ⊂ X . P 5( X ) =



          P xC : P xC ∩ X = ∅ = y : y ∈ P xC ∧ P xC ∩ X = ∅ = y : y ∈ P xC ∧ ∃ z ∈ X z ∈ P xC

             = y : ∃ z ∈ X y ∈ P zC = y : ∃ z ∈ X z ∈ P Cy = y : P Cy ∩ X = ∅ = y : y ∈ P xC ∧ ∃z ∈ X x ∈ P zC      2 = x : P xC ∩ X = ∅ = x : N (x) ∩ M (x) ∩ X = ∅ .

The following pairs ( P 6 ( X ), P 6 ( X )) and ( P 7 ( X ), P 7 ( X )) were also defined in paper [21]. Definition 6. For each X ⊂ (U , C ), we define the lower approximations and upper approximations of X as follows:









P 6 ( X ) = x : N (x) ⊂ X ∨ M (x) ⊂ X





P 7 ( X ) = x : N (x) ∪ M (x) ⊂ X ,

 ,











P 6 ( X ) = x : N (x) ∩ X = ∅ ∧ M (x) ∩ X = ∅ ,









P 7 ( X ) = x : N (x) ∪ M (x) ∩ X = ∅ .

We can define new lower and upper approximations by replacing neighborhood in the definition of neighborhood-based lower and upper approximations with complementary neighborhood. For example, in Ref. [26], there is a pair of lower and upper approximations:

L. Ma / International Journal of Approximate Reasoning 56 (2015) 59–70









C n ( X ) = x ∈ U : ∃u u ∈ N (x) ∧ N (u ) ⊂ X

 ,

65



C n ( X ) = x ∈ U : ∀u u ∈ N (x) → N (u ) ∩ X = ∅ . Then we can also define the following pair of lower and upper approximations:









C m ( X ) = x ∈ U : ∃u u ∈ M (x) ∧ M (u ) ⊂ X

 ,



C m ( X ) = x ∈ U : ∀u u ∈ M (x) → M (u ) ∩ X = ∅ . If the definition of a pair of lower and upper approximations is based on both neighborhood and complementary neighborhood, then we can define another pair of lower and upper approximations by replacing N , M with M , N respectively. For example, according to the following pair of lower and upper approximations









D n ( X ) = x ∈ U : ∃u u ∈ M (x) ∧ N (u ) ⊂ X

 ,



D n ( X ) = x ∈ U : ∀u u ∈ M (x) → N (u ) ∩ X = ∅ , we can define a new pair:









D m ( X ) = x ∈ U : ∃u u ∈ N (x) ∧ M (u ) ⊂ X

 ,



D m ( X ) = x ∈ U : ∀u u ∈ N (x) → M (u ) ∩ X = ∅ . These two approximation operator pairs (C n , C n ) and (C m , C m ) as well as ( D n , D n ) and ( D m , D m ) are called twin pairs. Definition 7. If the definition of a lower (or upper) approximation operator P is based on neighborhood or complementary neighborhood, we can define another lower (or upper) approximation operator Q by replacing N with M and replacing M with N in the definition of P . The approximation operator Q is called twin approximation operator of P , or the two lower approximations P and Q are called twin approximation operators. If two pairs of lower and upper approximation operators ( P , P ) and ( Q , Q ) are twin approximation operators respectively, we call them twin pairs. We further propose the following types of lower and upper approximations based on neighborhood operators and complementary neighborhood operators. Definition 8. For each X ⊂ (U , C ), we define the lower approximations and upper approximations of X as follows:



















 





 







P 3 ( X ) = x : M N (x) ⊂ X ,





P 8 ( X ) = x : N M (x) ∩ M N (x) ⊂ X ,



 





 



 ,

 







P 10 ( X ) = x : N M (x) ∪ M N (x) ⊂ X

 ,

 







 ∩ X = ∅ ,



P 9 ( X ) = x : N M (x) ∩ X = ∅ ∧ M N (x) ∩ X = ∅ ,





P 8 ( X ) = x : N M (x) ∩ M N (x)

P 9 ( X ) = x : N M (x) ⊂ X ∨ M N (x) ⊂ X





P 3 ( X ) = x : M N (x) ∩ X = ∅ ,



 







P 10 ( X ) = x : N M (x) ∪ M N (x)

 ∩ X = ∅ .

Obviously, pairs ( P 1 , P 1 ) and ( P 2 , P 2 ) as well as ( P 3 , P 3 ) and ( P 4 , P 4 ) are twin pairs. The twin pair of ( P i , P i ) (5 ≤ i ≤ 10) is itself. We can find out from these definitions that the introduction of the concept of complementary neighborhood can not only simplify the forms of some known types of neighborhood-based lower and upper approximations, but define some new types as well. In the following, we will see that it can help us to make the relationships of some approximations clear so as to select suitable approximations in practice. It surely facilitates our study of the rough set theory. 4. Relationships between different approximation operators Followed from Proposition 14, we can easily find the relationships of P i ( X ) and P i ( X ) (1 ≤ i ≤ 10) as follows. (See Fig. 2.) Proposition 16. For each subset X in a covering approximation space (U , C ), the relationships of all lower and upper approximations P i ( X ), P i ( X ) (1 ≤ i ≤ 10) of X can be represented by the following lattice, where the lower element is a subset of the upper element if they are linked by one broken line. According to Proposition 13, the next proposition can be easily obtained.

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L. Ma / International Journal of Approximate Reasoning 56 (2015) 59–70

Fig. 2. For the readers’ convenience, we draw a figure as follows, where the lower element is a subset of the upper element when they are linked by a segment.

Proposition 17. If M, N are both semi-Euclidean operators, then

P 3 ( X ) = P 4 ( X ),

P 3 ( X ) = P 4 ( X ).

We can obtain the following properties of lower and upper approximations. Proposition 18. For every X in the covering approximation space (U , C ), we have

























































P 2 P 1 ( X ) = P 4 ( X ),

P 1 P 2 ( X ) = P 3 ( X ),

P 1 P 3 ( X ) = P 3 ( X ),

P 2 P 4 ( X ) = P 4 ( X ),

P 4 P 1 ( X ) = P 4 ( X ),

P 3 P 2 ( X ) = P 3 ( X ),

P 1 P 5 ( X ) = P 1 ( X ),

P 5 P 1 ( X ) = P 1 ( X ),

P 2 P 5 ( X ) = P 2 ( X ),

P 5 P 2 ( X ) = P 2 ( X ),

P 1 P 6 ( X ) = P 1 ( X ),

P 2 P 6 ( X ) = P 2 ( X ),

P 1 P 7 ( X ) = P 3 ( X ),

P 2 P 7 ( X ) = P 4 ( X ).

Proof. We only need to show P 2 ( P 1 ( X )) = P 4 ( X ). The proofs of other equalities are similar to this one. For each x ∈ P 2 ( P 1 ( X )), we have M (x) ⊂ P 1 ( X ). Then for every y ∈ M (x), y ∈ P 1 ( X ) holds and N ( y ) ⊂ X follows. Therefore, N ( M (x)) ⊂ X holds and hence x ∈ P 4 ( X ). Thus P 2 ( P 1 ( X )) ⊂ P 4 ( X ) is correct. Conversely, for each x ∈ P 4 ( X ), we have N ( M (x)) ⊂ X . This means that for each y ∈ M (x), N ( y ) ⊂ X holds and y ∈ P 1 ( X ) follows. Then M (x) ⊂ P 1 ( X ), thus x ∈ P 2 ( P 1 ( X )) is correct and P 4 ( X ) ⊂ P 2 ( P 1 ( X )) holds. Hence P 2 ( P 1 ( X )) = P 4 ( X ) is verified. 2 Proposition 19. For every X in the covering approximation space (U , C ), we have

























































P 2 P 1 ( X ) = P 4 ( X ),

P 1 P 2 ( X ) = P 3 ( X ),

P 1 P 3 ( X ) = P 3 ( X ),

P 2 P 4 ( X ) = P 4 ( X ),

P 4 P 1 ( X ) = P 4 ( X ),

P 3 P 2 ( X ) = P 3 ( X ),

P 1 P 5 ( X ) = P 1 ( X ),

P 5 P 1 ( X ) = P 1 ( X ),

P 2 P 5 ( X ) = P 2 ( X ),

P 5 P 2 ( X ) = P 2 ( X ),

P 1 P 6 ( X ) = P 1 ( X ),

P 2 P 6 ( X ) = P 2 ( X ),

P 1 P 7 ( X ) = P 3 ( X ),

P 2 P 7 ( X ) = P 4 ( X ).

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Proof. We only need to show that P 2 ( P 1 ( X )) = P 4 ( X ). If x ∈ P 2 ( P 1 ( X )), then M (x) ∩ P 1 ( X ) = ∅. This means that there exists a y ∈ M (x) such that N ( y ) ∩ X = ∅. Therefore, N ( M (x)) ∩ X = ∅ and then x ∈ P 4 ( X ) holds. So P 2 ( P 1 ( X )) ⊂ P 4 ( X ) is correct. For each x ∈ P 4 ( X ), we have N ( M (x)) ∩ X = ∅. There is a y ∈ M (x) such that N ( y ) ∩ X = ∅. Therefore, y ∈ P 1 ( X ) and M (x) ∩ P 1 ( X ) = ∅ follows. This means x ∈ P 2 ( P 1 ( X )), thus P 4 ( X ) ⊂ P 2 ( P 1 ( X )) is correct. So P 2 ( P 1 ( X )) = P 4 ( X ) is proved. 2 We can find that the twin approximation operators have the similar properties. We next discuss the relationship between twin pairs under dual coverings. Definition 9. Suppose that U is a set and C is a covering of U . If {−C : C ∈ C } is also a covering of U , then we call that C and D = {−C : C ∈ C } are dual coverings of U ; if {−C : C ∈ C } is not a covering of U , then we call that C and D = {−C : C ∈ C } ∪ {U } are dual coverings of U . In these two situations, we also call that C and D are dual to each other. The following interesting results are based on the relationship of neighborhood and complementary neighborhood. The proof of the next property is easy and we omit it. Proposition 20. If C and D are dual coverings of U , then for each x ∈ U and A ⊂ U , we have

M C (x) = N D (x),

N C (x) = M D (x),

M C ( A ) = N D ( A ),

N C ( A ) = M D ( A ).

From this proposition we can deduce the relationships between twin pairs under dual coverings. Generally, we have D C nC ( X ) = C m ( X ),

D C nC ( X ) = C m ( X );

D D nC ( X ) = D m ( X ),

D D nC ( X ) = D m ( X ).

This proposition also leads to the following properties. Proposition 21. If C and D are dual coverings of U , then for each X ⊂ U , we have D PC 1 ( X ) = P 2 ( X ),

D PC 1 ( X ) = P 2 ( X ),

D PC 3 ( X ) = P 4 ( X ),

D PC 3 ( X ) = P 4 ( X ),

D PC i ( X ) = P i ( X ),

D PC i (X) = P i (X)

(5 ≤ i ≤ 10).

From Propositions 20 and 21, we can conclude that studying a pair of lower or upper approximations under a covering is equal to studying the pair of their twin approximations under the dual covering. 5. Topological properties of approximation operators According to the discussion in Section 4, the twin approximation operators have similar properties. In this section, we can see that the topological properties of the twin approximation operators are also similar. So we can select one of the twins as a representative in theoretical study and real applications. We then recall some topological concepts which can be found in Ref. [33]. Definition 10. Let cl : P (U ) → P (U ) and int : P (U ) → P (U ) be two operators. X , Y ⊆ U . If the rules

(C 1 ) (C 2 ) (C 3 ) (C 4 )

cl(∅) = ∅, X ⊆ cl( X ), cl( X ∪ Y ) = cl( X ) ∪ cl(Y ), cl(cl( X )) = cl( X )

hold, we call cl a closure operator on U . If the rules

(I1) (I2) (I3) (I4)

int(U ) = U , int( X ) ⊆ X , int( X ∩ Y ) = int( X ) ∩ int(Y ), int(int( X )) = int( X )

hold, we call int an interior operator on U .

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For any subset X of a topological space, the closure operator cl and the interior operator int satisfy the formulas

cl( X ) = −int(− X ),

int( X ) = −cl(− X ).

It follows from the results in paper [21] that the operators P i (i = 1, 2, 5) are interior operators, the operators P i (i = 1, 2, 5) are closure operators, P i (i = 6, 7) and P i (i = 6, 7) are generally not interior operators and closure operators respectively. We recall the following two theorems in paper [21]. Theorem 1. (See [21].) Let (U , C ) be a covering approximation space. The following statements are equivalent: i ). ii ). iii ). i v ). v ).

P 6 is a closure operator on U . P 6 is an interior operator on U . P 6 = P 5. P 6 = P 5. For any x ∈ U , either M (x) ⊆ N (x) or N (x) ⊆ M (x) holds.

Combining this theorem and Proposition 8, we can easily obtain the following corollary. Corollary 1. For a covering approximation space (U , C ), the following statements are equivalent: i ). P 6 is a closure operator on U . ii ). P 6 is an interior operator on U . iii ). M (x) = N (x) for each x ∈ U , i.e., { N (x) : x ∈ U } is a partition of U . Theorem 2. (See [21].) Let (U , C ) be a covering approximation space. The following statements are equivalent: i ). P 7 is a closure operator on U . ii ). P 7 is an interior operator on U . iii ). The operators M , N are both semi-Euclidean. We next discuss topological properties of the approximation operators P i and P i (i = 3, 4, 8, 9, 10). Theorem 3. Let U be a set and C be a covering of U . Then the following statements are equivalent: i ). P 3 is a closure operator, ii ). P 3 is an interior operator, iii ). N ( M (x)) ⊂ M ( N (x)) for each x ∈ U . Proof. i ) ⇔ ii ) is obvious since P 3 and P 3 are dual to each other. We shall next show i ) ⇔ iii ). i ) → iii ). By contradiction, we assume that iii ) does not hold. Then there exist x, y such that y ∈ N ( M (x)) and y ∈ / M ( N (x)). Put X = { y }, then x ∈ P 3 ( P 3 ( X )), but x ∈ / P 3 ( X ). Therefore P 3 ( P 3 ( X )) = P 3 ( X ). Thus P 3 is not a closure operator and this is a contradiction. iii ) → i ). Since P 3 (∅) = ∅, X ⊂ P 3 ( X ) and P 3 ( X ∪ Y ) = P 3 ( X ) ∪ P 3 (Y ) are obvious for each X , Y ⊂ U , we only show P 3 ( P 3 ( X )) = P 3 ( X ) for every X ⊂ U . P 3 ( X ) ⊂ P 3 ( P 3 ( X )) is also obvious, we next show P 3 ( P 3 ( X )) ⊂ P 3 ( X ). Take x ∈ P 3 ( P 3 ( X )), then M ( N (x)) ∩ P 3 ( X ) = ∅. Thus there is y ∈ P 3 ( X ) such that y ∈ M ( N (x)). Then there is z ∈ X satisfying z ∈ M ( N ( y )). It follows from y ∈ M ( N (x)) and z ∈ M ( N ( y )) that there are y 1 ∈ N (x) 1, and z1 ∈ N ( y ) 2, satisfying y ∈ M ( y 1 ) and z ∈ M ( z1 ). This means that y 1 ∈ N ( y ) 3, and z1 ∈ N ( z) 4. Combining 2 and 3, we have z1 ∈ N ( M ( y 1 )). It follows from iii ) that z1 ∈ M ( N ( y 1 )). Then there is z0 ∈ N ( z1 ) 5, such that z0 ∈ N ( y 1 ) 6. Noticed that N is also a transitive operator. Combining 4 and 5, 6 and 1, we can deduce that z0 ∈ N ( z) and z0 ∈ N (x). Thus z ∈ M ( N (x)) holds. Hence x ∈ P 3 ( X ) follows and P 3 ( P 3 ( X )) ⊂ P 3 ( X ) is correct. 2 The following corollary is due to this theorem and Proposition 12. Corollary 2. If the operator N is semi-Euclidean in the covering approximation space (U , C ), then P 3 and P 3 are interior operator and closure operator respectively. Similar to Theorem 3, we can prove the following theorem. Theorem 4. Let (U , C ) be a covering approximation space, then the following statements are equivalent:

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i ). P 4 is a closure operator, ii ). P 4 is an interior operator, iii ). M ( N (x)) ⊂ N ( M (x)) for each x ∈ U . The next corollary is directly followed from this theorem and Proposition 12. Corollary 3. If M is a semi-Euclidean operator in the covering approximation space (U , C ), then P 4 and P 4 are interior operator and closure operator respectively. Theorem 5. Let (U , C ) be a covering approximation space, then the following statements are equivalent: i ). P 8 is a closure operator, ii ). P 8 is an interior operator, iii ). P 9 is a closure operator, i v ). P 9 is an interior operator, v ). P 3 and P 4 are both closure operators, vi ). P 3 and P 4 are both interior operators, vii ). M ( N (x)) = N ( M (x)) for each x ∈ U . Proof. i ) ⇔ ii ), iii ) ⇔ i v ) and v ) ⇔ vi ) are obvious. We only show i ) ⇔ vii ), iii ) ⇔ vii ) and v ) ⇔ vii ). To prove i ) → vii ) and iii ) → vii ), we first assume that P 8 and P 9 are closure operators. Suppose that the formula M ( N (x)) = N ( M (x)) does not hold for some x ∈ U . Without loss of generality, we may assume that M ( N (x)) − N ( M (x)) = ∅. Then there is y ∈ M ( N (x)) such that y ∈ / N ( M (x)). Put X = { y }, then x ∈ / P 8 ( X ) and x ∈ / P 9 ( X ). Since y ∈ M ( N (x)), there is z ∈ N (x) such that y ∈ M ( z). Then z ∈ P 8 ( X ), z ∈ P 9 ( X ) and thus x ∈ P 8 ( P 8 ( X )), x ∈ P 9 ( P 9 ( X )). So P 8 ( P 8 ( X )) = P 8 ( X ), P 9 ( P 9 ( X )) = P 9 ( X ) and hence neither P 8 nor P 9 is closed operator. Therefore, for each x ∈ U , we have M ( N (x)) = N ( M (x)). If vii ) holds, then P 3 = P 4 = P 8 = P 9 . Then vii ) → i ) and vii ) → iii ) follow obviously. v ) ⇔ vii ) is a direct consequence of Theorem 3 and Theorem 4. Therefore the proof of the theorem is finished. 2 According to this theorem and Proposition 13, we have the next corollary. Corollary 4. If M and N are both semi-Euclidean operators in the covering approximation space (U , C ), then the approximation operators P i (i = 3, 4, 8, 9) are all interior operators, and P i (i = 3, 4, 8, 9) are all closure operators. Theorem 6. Let (U , C ) be a covering approximation space, then the following statements are equivalent: i ). P 10 is a closure operator, ii ). P 10 is an interior operator, iii ). for each x ∈ U , there is M ( N ( M (x))) ⊂ N ( M (x)) or M ( N ( M (x))) ⊂ M ( N (x)). Proof. i ) ⇔ ii ) is obvious. We next show i ) ⇔ iii ). i ) → iii ). By contradiction, suppose that iii ) is not correct. Then there is x ∈ U such that M ( N ( M (x))) ⊂ N ( M (x)) and M ( N ( M (x))) ⊂ M ( N (x)). So there is y ∈ M ( N ( M (x))) satisfying y ∈ / M ( N (x)) and y ∈ / N ( M (x)). Put X = { y }, then x ∈ P 10 ( P 10 ( X )), but x ∈ / P 10 ( X ). Therefore, P 10 ( P 10 ( X )) = P 10 ( X ) and hence P 10 is not a closure operator. iii ) → i ). We need to show that the following a), b), c ) and d) hold. a). b ). c ). d ).

P 10 (∅) = ∅, X ⊂ P 10 ( X ) for each X ⊂ U , P 10 ( X ∪ Y ) = P 10 ( X ) ∪ P 10 (Y ) for any X , Y ⊂ U , P 10 ( P 10 ( X )) = P 10 ( X ) for each X ⊂ U .

Since a), b) and c ) are obvious, so we next show that d) holds. As P 10 ( P 10 ( X )) ⊃ P 10 ( X ) is obvious, we only show P 10 ( P 10 ( X )) ⊂ P 10 ( X ). Take x ∈ P 10 ( P 10 ( X )), then ( M ( N (x)) ∪ N ( M (x))) ∩ P 10 ( X ) = ∅. There exists y ∈ M ( N (x)) ∪ N ( M (x)) such that y ∈ P 10 ( X ). This means ( M ( N ( y )) ∪ N ( M ( y ))) ∩ X = ∅ and thus there exists z ∈ X such that z ∈ M ( N ( y )) ∪ N ( M ( y )). Therefore we have y ∈ M ( N (x)) 1, or y ∈ N ( M (x)) 2, satisfying z ∈ M ( N ( y )) 3, or z ∈ N ( M ( y )) 4. Next we show that 1 combined with 3, 1 combined with 4, 2 combined with 3 and 2 combined with 4 can all lead to x ∈ P 10 ( X ). As these four cases are similar, we only prove the case of 1 combined with 3. 1 and 3 mean that there are x1 , z1 ∈ U such that x1 ∈ N (x), y ∈ M (x1 ), z1 ∈ N ( y ) and z ∈ M (z1 ). Thus x ∈ M ( N ( M ( z1 ))). It follows from iii ) that there exists y 1 ∈ U such that y 1 ∈ N ( z1 ), x ∈ M ( y 1 ) or y 1 ∈ M ( z1 ), x ∈ N ( y 1 ). Combining z ∈ M ( z1 ), both of the two subcases lead to z ∈ M ( N ( M (x))) and hence z ∈ N ( M (x)) or z ∈ M ( N (x)). Therefore x ∈ P 10 ( X ) and P 10 ( P 10 ( X )) ⊂ P 10 ( X ) are correct. Thus the proof of the theorem is completed. 2

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Corollary 5. If either operator M or N is semi-Euclidean, then P 10 is interior operator and P 10 is closure operator. Proof. According to Proposition 12, if M is semi-Euclidean, then for each x ∈ U , M ( N ( M (x))) ⊂ N ( M ( M (x))) = N ( M (x)); if M is semi-Euclidean, then for each x ∈ U , M ( N ( M (x))) ⊂ M ( M ( N (x))) = M ( N (x)). Thus P 10 is interior operator and P 10 is a closure operator. 2 6. Conclusion Based on the relationship between neighborhood and complementary neighborhood, we defined the concept of twin approximation operators, and investigated the properties of some given twin approximation operators. Through our discussion we can see that the concepts of neighborhood and complementary neighborhood relate closely to each other, and they have great importance in the study of covering rough set theory. Topological methods are valid mathematical tools to study the rough set theory, we also discussed the topological properties of all related approximation operators. In the future, we may study the properties of finite products of covering approximation spaces as well as the functions between two such spaces. Acknowledgements The author is grateful to the Editor and anonymous referees for their insightful comments and suggestions. This research is supported by the Fundamental Research Funds for the Central Universities (BUPT2013RC0902). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

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