# A New Approach to Rough Set Based on Remote Neighborhood Systems

Authors: Shoubin Sun; Lingqiang Li; Kai Hu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The notion of neighborhood systems is abstracted from the geometric notion of “near”, and it is primitive in the theory of topological spaces. Now, neighborhood systems have been applied in the study of rough set by many researches. The notion of remote neighborhood systems is initial in the theory of topological molecular lattice, and it is abstracted from the geometric notion of “remote”. Therefore, the notion of remote neighborhood systems can be considered as the dual notion of neighborhood systems. In this paper, we develop a theory of rough set based on remote neighborhood systems. Precisely, we construct a pair of lower and upper approximation operators and discuss their basic properties. Furthermore, we use a set of axioms to describe the lower and upper approximation operators constructed from remote neighborhood systems.


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## Body

## 1. Introduction and Background

Pawlak’s rough set theory [1, 2] was based on equivalence for dealing with vagueness and granularity in information systems. It is well known that the basic notions in rough set theory are the lower and upper approximations [3, 4]. Generally, two different basic methods have been proposed to develop rough set theories, the constructive method and the axiomatic method [5–7]. In the constructive method, binary relations, coverings, and neighborhood systems on the universe of discourse are all primitive notions [8–10]. The lower and upper approximation operators are constructed by means of these notions [10–40].Neighborhood system comes from topology. LetU be a universe of discourse and 2U denotes the power set of U. A neighborhood system on U is a mapping N: U→22U, which is defined by assigning to each x of U a nonempty collection N(x) of subset of U. Each member of N(x) is called a neighborhood of x and represents the semantic of “near”. The rough sets based on generalized neighborhood system attract the attention of many scholars. Lin and Michael [41–44] explored the approximations based on generalized neighborhood systems. Lin [45, 46] and Yao [47] discussed the approximation retrieval and information retrieval based on general neighborhood systems, respectively. Wang [48], Syau [49], and Zhang [50] investigated relationships between reflexive generalized neighborhood system-based rough sets and other rough sets. Quite recently, the second author and his coauthor [51] discussed the axiomatic characterization on approximation operators based on generalized neighborhood systems.In [52], Wang studied the theory of topological molecular lattice by the tool of remote neighborhood systems [53]. A remote neighborhood system on a universe of discourse U is generally defined as a mapping RN:U→22U with some additional condition. For any x∈U, each member of RN(x) is called a remote neighborhood of x and represents the semantic of “remote”. Therefore, the notion of remote neighborhood systems can be considered as the dual notion of neighborhood systems.Now, the notion of neighborhood systems has been extensively used in rough sets. So, as the dual notion of neighborhood systems, can the remote neighborhood systems be used in the rough sets? In this paper, we shall develop a theory of rough sets base on general remote neighborhood systems.The contents are arranged as follows. Section2 recalls some notions and results about generalized neighborhood system-based rough sets as preliminary. Section 3 defines approximation operators based on generalized remote neighborhood system operator and discusses their basic properties. Section 4 presents the axiomatic characterization on newly defined approximation operators. Section 5 gives the conclusions and future works.


## 2. Preliminary

In this section, we will recall some basic conceptions of general rough set approximations. LetU be a nonempty set of all the objects under consideration, referred to as the universe. The power set of U denoted by 2U is the collection of all subsets of U. Let R be an equivalence relation on U. We use U/R to denote the family of all equivalence classes of R and we use [x] to denote an equivalence class in R containing an element x∈U. The pair (U,R) is called an approximation space. For any X⊆U one can define the lower approximation and the upper approximation of X [1, 54] by R_X={x∣[x]⊆X}, R¯X={x∣[x]∩X≠∅}, respectively. The pair RX=(R_X,R¯X) is referred to as the rough set of X. The rough set RX denotes the description of X under the current scheme. If R_X=R¯X, then X is a definable set of R, that is, X is an exact set of R. Next, we introduce some results about rough sets based on generalized neighborhood system [48, 50].The notion of generalized neighborhood system is introduced by Sierpiński [55] and then discussed by Lin [41].Definition 1 ([41, 55]).
LetU be the universe of discourse and 2U denote the power set of U. Then a function N:U→22U is called a generalized neighborhood system operator on U if for any x∈X, N(x) is nonempty. Usually, N(x) is called generalized neighborhood system of x and each K∈N(x) is called neighborhood of x.The rough sets based on generalized neighborhood system is introduced by Lin-Yao [46] and then studied by Syau-Lin [48] and Zhang et al. [50].Definition 2 ([46, 48, 50]).
LetN:U→22U be a generalized neighborhood system operator. Then for each subset X of U, the upper and lower approximation operators N_(X) and N¯(X) are defined as follows:(1)N_X=x∈U∣∃K∈Nx,K⊆X,N¯X=x∈U∣∀K∈Nx,K∩X≠∅.The above two definitions are the text citation for [51].In [50], it is observed that rough sets based on neighborhood operator can be regarded as special case of rough sets based on generalized neighborhood system operators.


## 3. Approximation Operators Based on Generalized Remote Neighborhood System

In this section, we will define a pair of approximation operators based on generalized remote neighborhood system operator and discuss their basic properties. In the following, we assumeX′=U-X be the complement of X, for any X⊆U.For anyx∈U and K∈2U. It is easily seen that if K represents the nearness of x, then intuitively, K′ represents the remoteness of x. Thus if N:U→22U is a generalized neighborhood system operator on U, then intuitively, the function RN:U→22U defined by(2)RNx=K′∣K∈Nx,∀x∈Ucan be interpreted as a general remote system operator on U. Therefore, we give the following definition.Definition 3.
LetU be the universe of discourse. Then a function RN:U→22U is called a generalized remote neighborhood system operator on U if for any x∈X, RN(x) is nonempty. Usually, RN(x) is called generalized remote neighborhood system of x and each K∈RN(x) is called a remote neighborhood of x.Remark 4.
(1) The idea of remote neighborhood is initiated by Wang [52] in the context of topological modular lattice. The notion of generalized remote neighborhood systems is an extension of the notion of remote neighborhood systems of topological spaces in [56].
(2) Although both generalized neighborhood system and generalized remote neighborhood system are defined as functions fromU to 22U, their sematic meanings are different. The difference will be exhibited more obviously in the following study on serial condition, reflexive condition, transitive condition, etc.Definition 5.
LetRN:U→22U be a generalized remote neighborhood system operator. Then for each subset X of U, the upper and lower approximation operators RN¯(X) and RN_(X) are defined as follows:(3)RN¯X=x∈U∣∀K∈RNx,X⊈K.RN_X=x∈U∣∃K∈RNx,X′⊆K.X is called a definable set if RN_(X)=RN¯(X), and it is rough set otherwise.Example 6.
LetU={a,b,c} and RN(a)={∅,{b},{c}}, RN(b)={∅,{a,c},{c}}  RN(c)={∅,{a},{b}}.
Next, we showRN_({a,b})={a,b} and RN¯({a,b})={a,b,c} as example.
LetX={a,b}. Then X′={c}. By Definition 5 of RN_(X), we have X′={c}⊆RN(a), X′={c}⊆RN(b), and X′={c}⊈RN(c). Therefore RN_({a,b})={a,b}.
SinceX={a,b}, for any K∈RN(a), we have X⊈K, so a∈RN¯({a,b}). The same can be verified b,c∈RN¯({a,b}). Thus RN¯({a,b})={a,b,c}.
Then(4)RN_∅=∅,RN_a=∅,RN_b=b,RN_c=∅,RN_a,b=a,b,RN_a,c=a,c,RN_b,c=b,c,RN_U=U.RN¯∅=∅,RN¯a=a,RN¯b=b,RN¯c=c,RN¯a,b=a,b,c,RN¯a,c=a,c,RN¯b,c=a,b,c,RN¯U=U.Next, we give the basic properties of the defined approximation operators.Proposition 7.
LetRN be a generalized remote neighborhood system operator on U. Then
(1)RN¯(∅)=∅, RN_(U)=U,
(2) for anyX,Y⊆U and X⊆Y⇒RN_(X)⊆RN_(Y), RN¯(X)⊆RN¯(Y).Proof.
(1) Now we proveRN¯(∅)=∅ and RN_(U)=U.
By Definition5 of RN¯(∅), for any x∈U and K∈RN(x), then ∅⊈K. It is contradictory for ∅⊆K. Therefore RN¯(∅)=∅.
BecauseRN_(U)={x∈U∣∃K∈RN(x),U′⊆K} and for any x∈U, we have U′=∅⊆K, so RN_(U)=U.
(2) Firstly, we prove that for anyX,Y⊆U and X⊆Y⇒RN_(X)⊆RN_(Y).
For anyx∈RN_(X), then there exists K∈RN(x) such that X′⊆K. By X⊆Y, we have Y′⊆X′. Therefore Y′⊆K. Then x∈RN_(Y). We obtain RN_(X)⊆RN_(Y).
Secondly, we prove that for anyX,Y⊆U and X⊆Y⇒RN¯(X)⊆RN¯(Y).
For anyx∈RN¯(X) and for any K∈RN(x), we have X⊈K. By X⊆Y, we have Y⊈K. This tells us x∈RN¯(Y). Thus RN¯(X)⊆RN¯(Y).Theorem8 shows that RN¯ and RN_ are dual.Theorem 8.
LetRN be a generalized remote neighborhood system operator on U. Then
(1)RN¯X′′=RN_(X).
(2)RN_X′′=RN¯(X).Proof.
We only prove (1) as an example; the proof of (2) is similar to (1).
On the one hand, for anyx∈(RN¯(X′))′, then x∉RN¯(X′). We have ∃K∈RN(x) such that X′⊆K. Therefore x∈RN_(X). Thus (RN¯(X′))′⊆RN_(X).
On the other hand, for anyx∈RN_(X), then there exists K∈RN(x) such that X′⊆K. We have x∉RN¯(X′). Thus x∈(RN¯(X′))′. Therefore (RN¯(X′))′⊇RN_(X). We obtain that (RN¯(X′))′=RN_(X).In the recent papers [48, 50], the researchers studied serial, reflexive, symmetric, transitive, weak-transitive, and weak-unary generalized neighborhood system and related rough sets.LetN be a generalized neighborhood system operator on U. Then(1)N is said to be serial, if for any x∈U and K∈N(x), K≠∅.(2)N is said to be reflexive, if for any x∈U and K∈N(x), x∈K.(3)N is said to be symmetric, if for any x,y∈U, K∈N(x), and V∈N(y), x∈V⇒y∈K.(4)N is said to be transitive, if for any x,y,z∈U, K∈N(y), and L∈N(z), x∈K and y∈L⇒x∈L.(5)N is said to be weak-transitive, if for any x∈U and K∈N(x), there exists an V∈N(x) such that for any y∈V there exists an Vy∈N(y) with Vy⊆K.(6)N is said to be weak-unary, if for any x∈U and K,V∈N(x), there exists an M∈N(x) such that M⊆K∩V.The above six concepts are the text citation for [51]Now, for generalized remote neighborhood system operator we define the condition of serial, reflexive, symmetric, transitive, weak-transitive, and weak-unary, respectively.LetRN be a generalized remote neighborhood system operator on U. Then(1)RN is said to be serial, if for any x∈U and K∈RN(x), K≠U.(2)RN is said to be reflexive, if for any x∈U and K∈RN(x), x∉K.(3)RN is said to be symmetric, if for any x,y∈U, K∈RN(x), and V∈RN(y), x∉V⇒y∉K.(4)RN is said to be transitive, if for any x,y,z∈U, K∈RN(y), and L∈RN(z), x∉K and y∉L⇒x∉L.(5)RN is said to be weak-transitive, if for any x∈U and K∈RN(x), there exists a V∈RN(x) such that for any y∉V there exists a Vy∈RN(y) with K⊆Vy.(6)RN is said to be weak-unary, if for any x∈U and K,V∈RN(x), there exists an M∈RN(x) such that K∪V⊆M.Theorem 9 ([50]).
LetN be a generalized neighborhood system operator on U. Then
(1)N is serial ⇔N_(∅)=∅⇔N¯(U)=U.
(2)N is reflexive ⇔∀X⊆U, N_(X)⊆X ⇔ ∀X⊆U, X⊆N¯(X).
(3)N is weak-transitive ⇒ ∀X⊆U, N_(X)⊆N_(N_(X)) and ∀X⊆U, N¯(X)⊇N¯(N¯(X)).
(4)N is weak-unary ⇔ ∀X,Y⊆U, N_(X∩Y)=N_(X)∩N_(Y) ⇔ ∀X,Y⊆U, N¯(X∪Y)=N¯(X)∪N¯(Y).The above theorem is the text citation for [51]Proposition 10 ([50]).
LetN be a generalized neighborhood system operator on U. Then
(1) IfN is symmetric then ∀X⊆U, X⊆N_(N¯(X)), and X⊇N¯(N_(X)).
(2) IfN is transitive then ∀X⊆U, N_(X)⊆N_(N_(X)), and N¯(X)⊇N¯(N¯(X)).The above preposition is the text citation for [51]In [50] Zhang used the generalized neighborhood to get the conclusions above.Now we use the remote neighborhood to get the following conclusions.Proposition 11.
LetRN be a generalized remote neighborhood system operator on U. Then the following are equivalent:
(1)RN is serial,
(2)RN_(∅)=∅,
(3)RN¯(U)=U.Proof.
We only prove (1)⇔ (2) as an example. (1) ⇔ (3) is similar to (1) ⇔ (2).
(1)⇒ (2) If RN is serial, then ∀x∈U, K∈RN(x), K≠U. It is obvious that RN_(∅)=∅.
(2)⇒ (1) Since RN_(∅)=∅. Then there is not any K∈RN(x) such that U⊆K. Therefore ∀K∈RN(x), K≠U. So RN is serial.Proposition 12.
LetRN be a generalized remote neighborhood system operator on U. Then the following are equivalent:
(1)RN is reflexive,
(2)∀X⊆U, RN_(X)⊆X,
(3)∀X⊆U, X⊆RN¯(X).Proof.
We only prove (1)⇔ (2) as an example. (1) ⇔ (3) is similar to (1) ⇔ (2).
(1)⇒ (2) For any X⊆U and x∉X, by RN is reflexive, we have ∀K∈RN(x), x∉K. Then X′⊈K; this tells us x∉RN_(X), so RN_(X)⊆X.
(2)⇒ (1) For any x∈U and K∈RN_(x), since (K′)′⊆K, so x∈RN_(K′). By (2), we have RN_(K′)⊆K′. Then x∈K′. Therefore x∉K. This shows that RN is reflexive.Proposition 13.
LetRN be a generalized remote neighborhood system operator on U. Then the following are equivalent:
(1)RN is weak-unary,
(2)∀X,Y⊆U, RN_(X∩Y)=RN_(X)∩RN_(Y),
(3)∀X,Y⊆U, RN¯(X∪Y)=RN¯(X)∪RN¯(Y).Proof.
We only prove (1)⇔ (2) as an example. (1) ⇔ (3) is similar to (1) ⇔ (2).
(1)⇒ (2) ∀X,Y⊆U, we only need to prove RN_(X∩Y)⊇RN_(X)∩RN_(Y).
∀ x ∈ R N _ ( X ) ∩ R N _ ( Y ). Then x∈RN_(X) and x∈RN_(Y). There exists K,V∈RN(x) such that X′⊆K and Y′⊆V. We have X′∪Y′⊆K∪V. Because RN is weak-unary, so for K,V∈RN(x), there exists an M∈RN(x) such that K∪V⊆M. We have X′∪Y′⊆M. Thus (X∩Y)′⊆M. Hence x∈RN_(X∩Y). Then RN_(X∩Y)⊇RN_(X)∩RN_(Y).
(2)⇒ (1) For any x∈U and K,V∈RN(x), we prove that there exists an M∈RN(x) such that K∪V⊆M.
For anyx∈U and K,V∈RN(x), by (K′)′⊆K, we have x∈RN_(K′). In the same way, we obtain x∈RN_(V′). Therefore x∈RN_(K′)∩RN_(V′)=RN_(K′∩V′). Then there exists M∈RN(x) such that (K′∩V′)′⊆M. Thus K∪V⊆M. Then RN is weak-unary.Proposition 14.
LetRN be a generalized remote neighborhood system operator on U. Then the following are equivalent:
(1)RN is weak-transitive,
(2)∀X⊆U, RN_(X)⊆RN_(RN_(X)),
(3)∀X⊆U, RN¯(X)⊇RN¯(RN¯(X)).Proof.
We only prove (1)⇔ (2) as an example. (1) ⇔ (3) is similar to (1) ⇔ (2).
(1)⇒(2) For anyx∈RN_(X), there exists a K∈RN(x) such that X′⊆K. Since RN is weak-transition, for the K∈RN(x), there exists a V∈RN(x) such that for any y∉V, there exists a Vy∈RN(y) with K⊆Vy, so X′⊆Vy. Hence y∈RN_(X). We have V′⊆RN_(X), i.e., (RN_(X))′⊆V. Then x∈RN_(RN_(X)). Therefore RN_(X)⊇RN_(RN_(X)).
(2)⇒(1) For anyx∈U and ∀K∈RN(x). Then x∈RN_(K′)⊆RN_(RN_(K′)). There exists V∈RN(x) such that (RN_(K′))′⊆V, i.e., V′⊆RN_(K′). For any y∉V, we have y∈RN_(K′), then there exists Vy∈RN(y) such that K=(K′)′⊆Vy. Thus RN is weak-transitive.Proposition 15.
LetRN be a generalized remote neighborhood system operator on U. If RN is symmetric then ∀X⊆U, X⊆RN_(RN¯(X)) and X⊇RN¯(RN_(X)).Proof.
We only prove that ifRN is symmetric then ∀X⊆U, X⊇RN¯(RN_(X)) as an example.
For anyx∈RN¯(RN_(X)) and for any K∈RN(x), we have RN_(X)⊈K. Then there exists a y∈RN_(X) such that y∉K. By y∈RN_(X), there exists V∈RN(y) such that X′⊆V. Because RN is symmetric, so for V∈RN(y), we have x∉V. Thus x∉X′. So x∈X. Therefore X⊇RN¯(RN_(X)).The converse of the above statements is not true; next Example16 shows this.Example 16.
LetU={a,b}, RN(a)={{b},∅}, RN(b)={{a},∅}. Then RN¯(RN_(∅))=∅,(5)RN¯RN_a=a,RN¯RN_b=b,RN¯RN_U=U.Therefore X⊇RN¯(RN_(X)) for all X⊆U.
Sincea∉∅∈RN(b) and b∈{b}⊆RN(a), we conclude that RN is not symmetric.Proposition 17.
LetRN be a generalized remote neighborhood system operator on U. If RN is transitive then ∀X⊆U, RN_(X)⊆RN_(RN_(X)) and RN¯(X)⊇RN¯(RN¯(X)).Proof.
We only proveRN¯(X)⊇RN¯(RN¯(X)) as an example. ∀x∈RN¯(RN¯(X)) and K∈RN(x), we have RN¯(X)⊈K. Then there exists a y∈RN¯(X) such that y∉K. By y∈RN¯(X), we have for all M∈RN(y), X⊈M, then there exists z∈X but z∉M. Then z∉K. We obtain X⊈K. Thus x∈RN¯(X). Hence RN¯(X)⊇RN¯(RN¯(X)).The converse of the Proposition17 is not true.Example 18.
LetU={a,b,c} and RN(a)=∅, RN(b)=∅, RN(c)={∅,{a},{a,c},{b,c},{a,b}}. Then(6)RN_∅=∅=RN_RN_∅,RN_a=c=RN_RN_a,RN_b=c=RN_RN_b,RN_c=c=RN_RN_c,RN_a,b=c=RN_RN_a,b,RN_a,c=c=RN_RN_a,c,RN_b,c=c=RN_RN_b,c,RN_U=U=RN_RN_U.Hence RN_(X)⊆RN_(RN_(X)) for all X⊆U. Since a∉∅=RN(b), b∉{a,c}⊆RN(c), but a∈{a,c}. Hence RN(x) is not transitive.


## 4. Axiomatic Characterization on Approximation Operators Based on General Remote Neighborhood Systems

In this section, we will give an axiomatic characterization on approximation operators based on general remote neighborhood systems.


### 4.1. On Upper Approximation Operator

Theorem 19.
Letf:2U→2U be an operator. Then there exists a generalized remote neighborhood system operator RN such that f=RN¯ if and only if f satisfies
(T1):f(∅)=∅; (T2): A⊆B ⇒ f(A)⊆f(B).Proof.
( ⇒ ). It follows immediately from Proposition 7.
( ⇐ ). Let f:2U→2U be an operator satisfying (T1) and (T2). Then we define an operator RNf:U→22U as follows:
∀ x ∈ U , A ∈ 2 U , A ∈ R N f ( x ) ⇔ ∃ B ∈ 2 U , A ⊆ B and x∉f(B).
By (T1), it is clear that for anyx∈U, ∅∈RNf(x) and so RNf(x) is nonempty. Hence RNf is a generalized remote neighborhood system operator. Next, we prove that RNf¯=f. Obviously, we need only check that ∀A∈2U, x∉RNf¯(A)⇔x∉f(A).
Letx∉RNf¯(A). Then there exists a B∈RNf(x) such that A⊆B. So A∈RNf(x). By the definition of RNf, we have x∉f(B). By (T2) we have f(A)⊆f(B). Therefore x∉f(A), as desired.
Conversely, letx∉f(A). Then A∈RNf(x). By the definition of RNf¯(A), ∀B∈RNf(x), A⊈B. Expressly, for B=A. Then A⊆B. Thus x∉RNf¯(A).Theorem 20.
Letf:2U→2U be an operator. Then there exists a serial generalized remote neighborhood system operator RN such that f=RN¯ if and only if f satisfies (T1), (T2), and (T3): f(U)=U.Proof.
Necessity. Let RN be a serial generalized remote neighborhood system operator on U and f=RN¯. By Propositions 7 and 11, we have f=RN¯ satisfying (T1), (T2), and (T3).
Sufficiency. Let f be an operator satisfying (T1), (T2), and (T3) and let RNf be defined as that in Theorem 19. It is easily seen that we need only check that (T3) implies the serial condition. Indeed, for each x∈U, by f(U)=U we have x∈f(U); it follows that U∉RNf(x). Therefore RN is serial.Theorem 21.
Letf:2U→2U be an operator. Then there exists a reflexive generalized remote neighborhood system operator RN such that f=RN¯ if and only if f satisfies (T1), (T2), and (T4): X⊆f(X), for all X∈2U.Proof.
Necessity. Let RN be a reflexive generalized remote neighborhood system operator on U and f=RN¯. Then it follows by Propositions 7 and 12 that f=RN¯f satisfies (T1), (T2), and (T4).
Sufficiency. Let f be an operator satisfying (T1), (T2), and (T4). Let RNf be defined as that in Theorem 19. It is easily seen that we need only to check that (T4) implies the reflexive condition. Indeed, ∀x∈U, let A∈RNf(x), by the definition of RNf, then there exists an B∈2U such that A⊆B and x∉f(B). By (T4), we have B⊆f(B). Thus x∉B, we have x∉A. Hence RNf is reflexive.Theorem 22.
Letf:2U→2U be an operator. Then there exists a weak-transitive generalized remote neighborhood system operator RN such that f=RN¯ if and only if f satisfies (T1), (T2), and (T5): f(X)⊇f(f(X)), for all X∈U.Proof.
Necessity. Let RN be a weak-transitive generalized remote neighborhood system operator on U and f=RN¯. Then it follows by Propositions 7 and 14 that f=RN¯f satisfies (T1), (T2), and (T5).
Sufficiency. Let f be an operator satisfying (T1), (T2), and (T5) and let RNf be defined as that in Theorem 19. It is easily seen that we need only check that (T5) implies the weak-transitive condition. ∀x∈U, let A∈RNf(x). For x∉f(A)=RN¯f(A), by (T5), we obtain x∉RN¯f(RN¯f(A)); by Definition 5, there exists B∈RNf(x) such that RN¯f(A)⊆B. ∀y∉B, then y∉RN¯f(A). Therefore there exists Vy∈RNf(y) such that A⊆Vy. Thus RNf is weak-transitive.Theorem 23.
Letf:2U→2U be an operator. Then there exists a weak-unary generalized remote neighborhood system operator RN such that f=RN¯ if and only if f satisfies (T1), (T2), and (T6): f(X∪Y)=f(X)∪f(Y), for all X,Y∈2U.Proof.
Necessity. Let RN be a weak-unary generalized remote neighborhood system operator on U and f=RN¯. Then it follows by Propositions 7 and 13 that f=RN¯f satisfies (T1), (T2), and (T6).
Sufficiency. Let f be an operator satisfying (T1), (T2), and (T6) and RNf be defined as that in Theorem 19. It is easy to see that we need only check that (T6) implies the weak-unary condition. Indeed, ∀x∈U, let K,V∈RNf(x), x∉f(K), and x∉f(V). By (T6), we have x∉(f(K)∪f(V))=f(K∪V). Then there exists M∈RNf(x) such that K∪V⊆M. Therefore RNf is weak-unary.


### 4.2. On Lower Approximation Operators

Similar to the axiomatic characterizations on upper approximation operators, we can obtain the axiomatic characterizations on lower approximation operators.Theorem 24.
Letf:2U→2U be an operator. Then there exists a generalized remote neighborhood system operator RN such that f=RN_ if and only if f satisfies
(R1):f(U)=U; (R2): A⊆B⇒f(A)⊆f(B).Proof.
Necessity. It follows immediately from Proposition 7.
Sufficiency. Let f:2U→2U be an operator satisfying R1 and R2. Then we define an operator RNf:U→22U as follows:
∀ x ∈ U, A∈2U, A∈RNf(x)⇔ there exist B∈2U, A⊆B, and x∈f(B′).
Next, we prove thatRN_f=f. Indeed for any A∈2U and x∈RN_f(A). Then there exists B∈RNf(x) such that A′⊆B. So A′∈RNf(x). By the definition of RNf, we have x∈f(B′). Since A′⊆B, so B′⊆A. By (R2), we have f(B′)⊆f(A) and x∈f(A). Thus ∀A∈2U, RN_f(A)⊆f(A).
Conversely, letx∈f(A),A⊆U. By the definition of RNf, there exist K∈RNf(x) and K⊆A′. By the definition of RNf_, we have x∈RNf_(A). Therefore f(A)⊆RN_f(A).The proof of the following theorem is similar to that of the upper approximation operator; we omit the proofs.Theorem 25.
Letf:2U→2U be an operator. Then
(1) There exists a serial generalized remote neighborhood system operatorRN such that f=RN_ if and only if f satisfies (R1), (R2), and (R3): f(∅)=∅.
(2) There exists a reflexive generalized remote neighborhood system operatorRN such that f=RN_ if and only if f satisfies (R1), (R2), and (R4): f(X)⊆X, for all X∈2U.
(3) There exists a weak-transitive generalized remote neighborhood system operatorRN such that f=RN_ if and only if f satisfies (R1), (R2), and (R5): f(X)⊆f(f(X)), for all X∈2U.
(4) There exists a weak-unary generalized remote neighborhood system operatorRN such that f=RN_ if and only if f satisfies (R1), (R2), and (R6): f(X∩Y)=f(X)∩f(Y), for all X,Y∈2U.


## 4.1. On Upper Approximation Operator

Theorem 19.
Letf:2U→2U be an operator. Then there exists a generalized remote neighborhood system operator RN such that f=RN¯ if and only if f satisfies
(T1):f(∅)=∅; (T2): A⊆B ⇒ f(A)⊆f(B).Proof.
( ⇒ ). It follows immediately from Proposition 7.
( ⇐ ). Let f:2U→2U be an operator satisfying (T1) and (T2). Then we define an operator RNf:U→22U as follows:
∀ x ∈ U , A ∈ 2 U , A ∈ R N f ( x ) ⇔ ∃ B ∈ 2 U , A ⊆ B and x∉f(B).
By (T1), it is clear that for anyx∈U, ∅∈RNf(x) and so RNf(x) is nonempty. Hence RNf is a generalized remote neighborhood system operator. Next, we prove that RNf¯=f. Obviously, we need only check that ∀A∈2U, x∉RNf¯(A)⇔x∉f(A).
Letx∉RNf¯(A). Then there exists a B∈RNf(x) such that A⊆B. So A∈RNf(x). By the definition of RNf, we have x∉f(B). By (T2) we have f(A)⊆f(B). Therefore x∉f(A), as desired.
Conversely, letx∉f(A). Then A∈RNf(x). By the definition of RNf¯(A), ∀B∈RNf(x), A⊈B. Expressly, for B=A. Then A⊆B. Thus x∉RNf¯(A).Theorem 20.
Letf:2U→2U be an operator. Then there exists a serial generalized remote neighborhood system operator RN such that f=RN¯ if and only if f satisfies (T1), (T2), and (T3): f(U)=U.Proof.
Necessity. Let RN be a serial generalized remote neighborhood system operator on U and f=RN¯. By Propositions 7 and 11, we have f=RN¯ satisfying (T1), (T2), and (T3).
Sufficiency. Let f be an operator satisfying (T1), (T2), and (T3) and let RNf be defined as that in Theorem 19. It is easily seen that we need only check that (T3) implies the serial condition. Indeed, for each x∈U, by f(U)=U we have x∈f(U); it follows that U∉RNf(x). Therefore RN is serial.Theorem 21.
Letf:2U→2U be an operator. Then there exists a reflexive generalized remote neighborhood system operator RN such that f=RN¯ if and only if f satisfies (T1), (T2), and (T4): X⊆f(X), for all X∈2U.Proof.
Necessity. Let RN be a reflexive generalized remote neighborhood system operator on U and f=RN¯. Then it follows by Propositions 7 and 12 that f=RN¯f satisfies (T1), (T2), and (T4).
Sufficiency. Let f be an operator satisfying (T1), (T2), and (T4). Let RNf be defined as that in Theorem 19. It is easily seen that we need only to check that (T4) implies the reflexive condition. Indeed, ∀x∈U, let A∈RNf(x), by the definition of RNf, then there exists an B∈2U such that A⊆B and x∉f(B). By (T4), we have B⊆f(B). Thus x∉B, we have x∉A. Hence RNf is reflexive.Theorem 22.
Letf:2U→2U be an operator. Then there exists a weak-transitive generalized remote neighborhood system operator RN such that f=RN¯ if and only if f satisfies (T1), (T2), and (T5): f(X)⊇f(f(X)), for all X∈U.Proof.
Necessity. Let RN be a weak-transitive generalized remote neighborhood system operator on U and f=RN¯. Then it follows by Propositions 7 and 14 that f=RN¯f satisfies (T1), (T2), and (T5).
Sufficiency. Let f be an operator satisfying (T1), (T2), and (T5) and let RNf be defined as that in Theorem 19. It is easily seen that we need only check that (T5) implies the weak-transitive condition. ∀x∈U, let A∈RNf(x). For x∉f(A)=RN¯f(A), by (T5), we obtain x∉RN¯f(RN¯f(A)); by Definition 5, there exists B∈RNf(x) such that RN¯f(A)⊆B. ∀y∉B, then y∉RN¯f(A). Therefore there exists Vy∈RNf(y) such that A⊆Vy. Thus RNf is weak-transitive.Theorem 23.
Letf:2U→2U be an operator. Then there exists a weak-unary generalized remote neighborhood system operator RN such that f=RN¯ if and only if f satisfies (T1), (T2), and (T6): f(X∪Y)=f(X)∪f(Y), for all X,Y∈2U.Proof.
Necessity. Let RN be a weak-unary generalized remote neighborhood system operator on U and f=RN¯. Then it follows by Propositions 7 and 13 that f=RN¯f satisfies (T1), (T2), and (T6).
Sufficiency. Let f be an operator satisfying (T1), (T2), and (T6) and RNf be defined as that in Theorem 19. It is easy to see that we need only check that (T6) implies the weak-unary condition. Indeed, ∀x∈U, let K,V∈RNf(x), x∉f(K), and x∉f(V). By (T6), we have x∉(f(K)∪f(V))=f(K∪V). Then there exists M∈RNf(x) such that K∪V⊆M. Therefore RNf is weak-unary.


## 4.2. On Lower Approximation Operators

Similar to the axiomatic characterizations on upper approximation operators, we can obtain the axiomatic characterizations on lower approximation operators.Theorem 24.
Letf:2U→2U be an operator. Then there exists a generalized remote neighborhood system operator RN such that f=RN_ if and only if f satisfies
(R1):f(U)=U; (R2): A⊆B⇒f(A)⊆f(B).Proof.
Necessity. It follows immediately from Proposition 7.
Sufficiency. Let f:2U→2U be an operator satisfying R1 and R2. Then we define an operator RNf:U→22U as follows:
∀ x ∈ U, A∈2U, A∈RNf(x)⇔ there exist B∈2U, A⊆B, and x∈f(B′).
Next, we prove thatRN_f=f. Indeed for any A∈2U and x∈RN_f(A). Then there exists B∈RNf(x) such that A′⊆B. So A′∈RNf(x). By the definition of RNf, we have x∈f(B′). Since A′⊆B, so B′⊆A. By (R2), we have f(B′)⊆f(A) and x∈f(A). Thus ∀A∈2U, RN_f(A)⊆f(A).
Conversely, letx∈f(A),A⊆U. By the definition of RNf, there exist K∈RNf(x) and K⊆A′. By the definition of RNf_, we have x∈RNf_(A). Therefore f(A)⊆RN_f(A).The proof of the following theorem is similar to that of the upper approximation operator; we omit the proofs.Theorem 25.
Letf:2U→2U be an operator. Then
(1) There exists a serial generalized remote neighborhood system operatorRN such that f=RN_ if and only if f satisfies (R1), (R2), and (R3): f(∅)=∅.
(2) There exists a reflexive generalized remote neighborhood system operatorRN such that f=RN_ if and only if f satisfies (R1), (R2), and (R4): f(X)⊆X, for all X∈2U.
(3) There exists a weak-transitive generalized remote neighborhood system operatorRN such that f=RN_ if and only if f satisfies (R1), (R2), and (R5): f(X)⊆f(f(X)), for all X∈2U.
(4) There exists a weak-unary generalized remote neighborhood system operatorRN such that f=RN_ if and only if f satisfies (R1), (R2), and (R6): f(X∩Y)=f(X)∩f(Y), for all X,Y∈2U.


## 5. Conclusions

In this paper, we construct a pair of approximation operators based on general remote neighborhood systems. Then we discuss the basic properties and axiomatic characterization on this pair of approximation operators. It is well known that reduction theory is one of the most important contents in rough set. It is the basis of the application of rough set theory. In [57], the second author and his coauthor have established reduction theory of neighborhood systems-based rough sets, which can be regarded as a natural extension of reduction theory of covering-based rough set discussed in [58]. In the future work, we shall establish a reduction theory of remote neighborhood-based rough sets. Quite recently, the second author and his coauthor also fuzzify the notion of generalized neighborhood systems and then develop a theory of fuzzy rough sets based on fuzzy general neighborhood systems [59]. In [60], the author and his coauthor discussed the dual matroids and spanning; we shall also consider a theory of fuzzy rough sets and fuzzy matroids based on fuzzy general remote systems, which played an important role in the theory of fuzzy topological spaces.



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*Source: 8712010-2019-05-13.xml*