Extended results on the relationship between information systems

Information Sciences 290 (2015) 156–173

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Information Sciences journal homepage: www.elsevier.com/locate/ins

Extended results on the relationship between information systems Anhui Tan a,b, Jinjin Li b,⇑, Guoping Lin b,c a

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, Fujian, China c School of computer and Information Technology, Shanxi University, Taiyuan 030006, Shanxi, China b

a r t i c l e

i n f o

Article history: Received 17 October 2013 Received in revised form 15 June 2014 Accepted 15 August 2014 Available online 26 August 2014 Keywords: Information system Neighborhood-continuous function Relation mapping Neighborhood-homomorphism Attribute reduct

a b s t r a c t The relationship between information systems is an important topic in rough set field and it is studied through mappings. Many kinds of consistent functions have been proposed and they can be used to describe the invariance of attribute reducts between two information systems efficiently. However they cannot reflect relationships between neighborhoods generated from two information systems. In this paper, to achieve this aim, firstly, we introduce a neighborhood-continuous function that is inspired by the concept of continuous function in topology. Then, we address its properties and discuss relationships between neighborhood-continuous functions and several kinds of existing consistent functions. Furthermore, we investigate some important properties of neighborhood-continuous function with respect to relation mappings. Finally, based on neighborhood-continuous functions, the notion of a neighborhood-homomorphism between different information systems is proposed. It is noted that the reduct feature of a single information system can be described by neighborhood-continuous functions, while the reduct invariance of two different information systems can be described by neighborhood-homomorphisms. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Rough set theory, proposed by Pawlak [15,16], is a useful mathematical tool for uncertain, inexact and fuzzy knowledge. It has been widely used in data analysis, data mining and artificial intelligence [9,11,13,20,29,31,37,38]. The original idea of rough set theory is based on an indiscernibility relation among elements of the universe. By using the indiscernibility relation, two definable sets are induced to approximate any given subset of the universe, called lower and upper approximations. The lower approximation is the greatest definable set contained in the given set, while the upper approximation is the smallest definable set containing the given set. Generally speaking, the upper approximation is larger than lower approximation. If the lower and upper approximations of a subset are just the same, then the subset is called a definable set; otherwise, called a rough set. Rough set-based data analysis starts from a data table, which is also called an information system. Most studies of an information system focus on internal structures and properties of approximation spaces [1,3,9,14,18,21,30,33–35,40,45]. It is attractive that as a new research direction, the investigation on homomorphisms or mappings between information systems is receiving more attentions in recent years [6–8,12,19,23–27,36,39,42,43]. From mathematical viewpoint, ⇑ Corresponding author. E-mail addresses: [email protected] (A. Tan), [email protected] (J. Li), [email protected] (G. Lin). http://dx.doi.org/10.1016/j.ins.2014.08.038 0020-0255/Ó 2014 Elsevier Inc. All rights reserved.

A. Tan et al. / Information Sciences 290 (2015) 156–173

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homomorphisms or mappings can be considered as to compare structures and properties between different information systems, which have many important applications such as data compression, data fusion and information transmission [4,17,18,28]. The notion of homomorphism or mapping between information systems based on rough sets was first proposed by Grzymala-Busse [6]. In 1988, Grzymala-Busse depicted the conditions which make an information system to be selective in terms of endomorphism of the system [7]. Li et al. talked about the redundancy and reduct of information systems under homomorphisms or mappings [12]. Wang et al. investigated invariant properties of relation information systems under homomorphisms [22,24]. In order to disclose symmetric relationships between two kinds of important mappings proposed by Wang in [22], Zhu et al. introduced the notion of a neighborhood consistent function and unified Wang’s mappings into neighborhood-consistent functions [41]. Homomorphisms or mappings cannot only reflect relationships between classical information systems, but also between more generalized information systems. For example, by using homomorphisms, Wang et al. presented some conditions under which attribute reducts of two covering-based information systems are equivalent to each other. Wang and Zhu discussed the properties of fuzzy information systems under homomorphisms [25,43]. Moreover, several invariant properties of ordered information systems and including degree-based information systems are systematically studied in [5,26,44]. A main contribution of these existing mappings was a description of invariance of attribute reducts in an original system and an image system. However, this kind of description is based on relation mappings with which a corresponding binary relation on one information system is induced by a binary relation on another information system. Therefore, if relations on both an original system and its image system have been given, these existing mappings cannot reflect relationships between the two binary relations. In this paper, in order to achieve this aim, we propose a neighborhood-continuous function that is inspired by the notion of continuous function in topology. We point out that neighborhood-continuous functions can unify some existing mappings between information systems and can compare two classes of granules between information systems. Moreover, based on neighborhood-continuous functions, the notion of a neighborhood-homomorphism between different information systems is proposed. We show that the notions of neighborhood-continuous functions and neighborhood-homomorphism are in accordance with structures of relation mappings. It is noted that the reduct feature of a single information system can be described by neighborhood-continuous functions, while the reduct invariance of different information systems can be described by neighborhood-homomorphisms. The remainder of this paper is organized as follows. In Section 2, we review some relevant concepts about information systems and several kinds of mappings between information systems. Besides, we present definitions of continuous functions in topology. In Section 3, we introduce a neighborhood-continuous function and study its properties. In Section 4, we investigate the relationship between neighborhood-continuous functions and several existing consistent functions. Section 5 addresses some extended properties about relation mappings and studies neighborhood-continuous functions with respect to relation mappings. The results indicate that the notion of neighborhood-continuous function is in accordance with the structure of relation mapping. In Section 6, we describe the reduct feature of a single information system by using neighborhood-continuous functions and present the reduct invariance of different information systems by using neighborhoodhomomorphisms.

2. Preliminaries This section reviews some basic concepts related to this paper.

2.1. Basic concepts in information systems The notion of a binary relation plays a basic role in information systems. Generally, for an arbitrary binary relation RU on an object set U; RU is called: ð1Þ ð2Þ ð3Þ ð4Þ

reflexive if ðx; xÞ 2 RU for all x 2 U; symmetric if ðx; yÞ 2 RU implies ðy; xÞ 2 RU for all x; y 2 U; anti-symmetric if ðx; yÞ 2 RU and ðy; xÞ 2 RU imply x ¼ y for all x; y 2 U; transitive if ðx; yÞ 2 RU and ðy; zÞ 2 RU imply ðx; zÞ 2 RU for all x; y; z 2 U.

It is easy to see that relation RU is an equivalence relation if and only if it is reflexive, symmetric and transitive simultaneously. Based on relation RU on U, the predecessor and successor neighborhoods of each x 2 U can be defined as [33]:

RUp ðxÞ ¼ fy 2 Ujðy; xÞ 2 RU g; RUs ðxÞ ¼ fy 2 Ujðx; yÞ 2 RU g: An information system is a triple S ¼ ðU; A; V ¼ fV a ja 2 AgÞ, where U is a nonempty and finite set of objects called the universe, A ¼ fa1 ; a2 ; . . . ; an g is a nonempty finite set of attributes such that a : U ! V a for any a 2 A, i.e., aðxÞ 2 V a ; x 2 U, where V a is called the domain for attribute a. If A is the union of two kinds of attributes, A ¼ C [ D, where C is the so-called condition attribute set, D is the so-called decision attribute set and C \ D ¼ ;, then S is called a decision information system. For simplicity, we write an information system S ¼ ðU; A; V ¼ fV a ja 2 AgÞ as S ¼ ðU; AÞ [32,36].

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Each nonempty subset B # A in an information system determines an indiscernibility relation: RB ¼ fðx; yÞ 2 U  U : aðxÞ ¼ aðyÞ; 8a 2 Bg. Since RB is an equivalence relation on U, it forms a partition U=RB ¼ f½xB : x 2 Ug, where ½xB is the equivalence class of x with respect to B, i.e., ½xB ¼ fy 2 U : ðx; yÞ 2 RB g. U=RB reflects basic granules of knowledge w.r.t. B in an information system. Based on the indiscernibility relation, one can consider the concepts of reducts and approximations in an information system. In an information system S ¼ ðU; AÞ, if an attribute set B satisfies ½xB ¼ ½xA for all x 2 U, then B is called a consistent set. If B is consistent and any C  B is not consistent, then B is called a reduct. A decision information system S ¼ ðU; A ¼ C [ DÞ is called consistent if ½xC # ½xD for all x 2 U [32,36]. In an information system ðU; AÞ, the lower and upper approximations of the subset X # U are defined as follows [15,16]:

RðXÞ ¼ fx 2 Uj½xA # Xg; RðXÞ ¼ fx 2 Uj½xA \ X – ;g: If RðXÞ ¼ RðXÞ, then X is a definable set; otherwise, X is a rough set. Classical rough set based on an equivalence relation still restricts its extensive applications. To address this issue, many generalized rough sets have been proposed, such as covering-based rough sets and relation-based rough sets. In the following, we recall some basic notions related to covering-based rough sets and dominance relation-based rough sets which can be found in [26,28]. For an universe U; C is called a covering of U if C is a family of nonempty subsets of U and [C ¼ U. In a covering C, for each x 2 U, denote C x ¼ \fK 2 Cjx 2 Kg as the neighborhood of x. It is clear that the concept of a covering is an extension of the concept of a partition. Let RU be a relation on U. If RU is simultaneously reflexive, anti-symmetric and transitive, then RU is actually a dominance relation on U, and RUs ðxÞ denotes the dominance class of x 2 U. One can see that the neighborhood of an element in a covering and the dominance class of an element under a dominance relation are actually two kinds of special neighborhoods. 2.2. Mappings between information systems The aim of establishing mappings between two information systems is to characterize relationships between information systems. In this subsection, we review some basic concepts and properties about mappings between information systems. We first present some notations.  Let U and V be two universes and f : U ! V be a mapping. Sets f ðSÞ and f ðTÞ denote ff ðxÞ 2 Vjx 2 Sg and fx 2 Ujf ðxÞ 2 Tg,  respectively, where S # U and T # V. We call f ðSÞ the image of S and f ðTÞ the original image of T. If mapping f : U ! V is    bijective, denote a mapping f : V ! U such that f ðyÞ ¼ x iff f ðxÞ ¼ y for x 2 U; y 2 V and call f the inverse of f. In an universe U, for each element x 2 U, we associate x with a subset nU ðxÞ of U and call it a neighborhood of x. The mapping nU : U ! 2U is said to be a neighborhood operator, where 2U is the power set of U. Thus predecessor and successor neighborhoods are two kinds of particular neighborhoods [22,37]. In terms of the notions of neighborhoods, the consistent functions can be defined. Definition 1 [22]. Let U and V be two universes, RU be a binary relation on U and f : U ! V be a mapping. Let

½xf ¼ fy 2 U : f ðyÞ ¼ f ðxÞg; ½xUs ¼ fy 2 U : RUs ðyÞ ¼ RUs ðxÞg:

(1) If ½xf # RUs ðyÞ or ½xf \ RUs ðyÞ ¼ ; for any x; y 2 U, then f is called a type-1 consistent function with respect to RU on U. (2) If ½xf # ½xUs for any x 2 U, then f is called a type-2 consistent function with respect to RU on U. Besides, Zhu et al. disclosed a symmetric relationship between type-1 and type-2 consistent functions by introducing the notion of a neighborhood-consistent function [3]. Definition 2 [41]. Let U and V be two universes and nU : U ! 2U be a neighborhood operator. A mapping f : U ! V is called a neighborhood-consistent function with respect to nU if nU ðxÞ ¼ nU ðyÞ whenever f ðxÞ ¼ f ðyÞ for all x; y 2 U. In other words, a mapping f is neighborhood-consistent if and only if any two elements of U with the same image have the same neighborhood in U. Based on predecessor and successor neighborhood operators, predecessor and successor neighborhood-consistent functions can be defined, respectively. Definition 3 [41]. Let U and V be two universes, RU be a binary relation on U and f : U ! V be a mapping.

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(1) f is called a predecessor-consistent function with respect to RU iff for any x; y 2 U; RUp ðxÞ ¼ RUp ðyÞ whenever f ðxÞ ¼ f ðyÞ. (2) f is called a successor-consistent function with respect to RU iff for any x; y 2 U; RUs ðxÞ ¼ RUs ðyÞ whenever f ðxÞ ¼ f ðyÞ. So predecessor and successor-consistent functions are two kinds of particular neighborhood-consistent functions. Definition 3 means that a mapping f : U ! V is predecessor-consistent (respectively, successor-consistent) if and only if any two elements of U with the same image owns the same predecessor (respectively, successor) neighborhood in U. It is pointed out that the concept of type-1 (respectively, type-2) consistent function is equivalent to that of predecessorconsistent (respectively, successor-consistent) function [3]. Theorem 1 [41]. Let U and V be two universes, RU be a binary relation on U and f : U ! V be a mapping. (1) f is a predecessor-consistent function with respect to RU if and only if it is a type-1 consistent function with respect to RU ; (2) f is a successor-consistent function with respect to RU if and only if it is a type-2 consistent function with respect to RU . Therefore, the notions of type-1 and type-2 consistent functions are unified to that of neighborhood-consistent functions. We unitedly call type-1 and type-2 consistent functions [4] and neighborhood-consistent functions [3] as consistent functions. As a generalization of the consistent functions established on classical information systems, the consistent functions can also be employed to study relationships between coverings. Definition 4 [28]. Let U and V be universes, f : U ! V be a mapping and C be a covering of U. Mapping f is called a consistent function with respect to C if C x ¼ C y whenever f ðxÞ ¼ f ðyÞ for any x; y 2 U.

2.3. Continuous function in topology It is known that an important purpose of topology is to describe the relationship between different topological spaces. Thus the consistent functions referred to above have a close relationship with topology. In order to disclose more relationships between information systems, we introduce some related background in topology. Definition 5 [10]. A topological space is a pair ðU; CÞ consisting a set U and a family C of subsets of U satisfying: (1) ; 2 C and U 2 C. (2) If X; Y 2 C, then X \ Y 2 C. (3) If C1 # C, then [C1 2 C.

The elements of C are called open sets of C. One basic function that could connect two topological spaces is continuous function. There are several kinds of equivalent definitions for a continuous function. For our purpose, we introduce some of them. Definition 6 [10]. Let ðU; CU Þ and ðV; CV Þ be two topological spaces. A function f : U ! V is continuous if the original image of any open set of Y is also open in U. In a topological space ðU; CU Þ, a subset nU ðxÞ of U is called a neighborhood of x if there is an open set X # U such that x 2 X # nU ðxÞ. Obviously, all open sets that contain x are certainly neighborhoods of x. With use of the notions of neighborhoods, a continuous function has another equivalence description. Definition 7 [10]. Let ðU; CU Þ and ðV; CV Þ be two topological spaces. A function f : U ! V is continuous if for all x 2 U, the original image of any neighborhood of f ðxÞ is also a neighborhood of x. In Definition 7, an intuitive interpretation of the concept of continuous function f : U ! V can be described as follows. For

8x 2 U and any neighborhood nV ðf ðxÞÞ of f ðxÞ in the image space, there exists a neighborhood of x in the original image space such that it is wholly mapped into nV ðf ðxÞÞ. Proposition 1 [10]. Let ðU; CU Þ; ðV; CV Þ and ðW; CW Þ be three topological spaces. If two functions f : U ! V and g : V ! W are continuous, then f  g is continuous, where f  g : U ! W is a so-called composite function such that ðf  gðxÞÞ ¼ f ðgðxÞÞ for x 2 U. Continuous functions can reflect the relationship between two classes of neighborhoods and characterize the consistency of mapping between different neighborhoods. In the following, we will introduce a kind of mapping between information systems that is inspired by the concept of continuous function in topology.

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3. Neighborhood-continuous function and its properties We can find a common limitation of the existing consistent functions, that is, these consistent functions are based on unilateral neighborhoods of an original system and do not consider the neighborhoods of its image system. To overcome this limitation and further study relationship between information systems, we introduce the notion of a neighborhoodcontinuous function which can reflect relationships between neighborhoods of two information systems. Definition 8. Let U and V be two universes, nU ; nV be neighborhood operators on U and V, respectively, and f be a mapping from U to V. f : ðU; nU Þ ! ðV; nV Þ is called a neighborhood-continuous function iff y 2 nU ðxÞ whenever f ðyÞ 2 nV ðf ðxÞÞ for all x; y 2 U. Definition 8 means that a neighborhood-continuous function f only maps the elements that belong to the neighborhood of x 2 X into the neighborhood of f ðxÞ. So the notion of neighborhood-continuous function derives from that of continuous function in topology and it can reflect some relationships between neighborhoods of two information systems. Theorem 2. Let U and V be two universes, nU and nV be neighborhood operators on U and V, respectively, and f be a mapping from  U to V. f : ðU; nU Þ ! ðV; nV Þ is a neighborhood-continuous function iff f ðnV ðf ðxÞÞÞ # nU ðxÞ for all x 2 U. Proof. Necessity. Suppose that f is a neighborhood-continuous function. By Definition 8, it follows f ðyÞ 2 nV ðf ðxÞÞ )   y 2 nU ðxÞ for x; y 2 U. That is y 2 f ðnV ðf ðxÞÞÞ ) y 2 nU ðxÞ. Hence, f ðnV ðf ðxÞÞÞ # nU ðxÞ.  Sufficiency. Assume f ðnV ðf ðxÞÞÞ # nU ðxÞ for x 2 U. For x; y 2 U and f ðyÞ 2 nV ðf ðxÞÞ, we need to prove y 2 nU ðxÞ. If  f ðyÞ 2 nV ðf ðxÞÞ, then y 2 f ðnV ðf ðxÞÞ, it follows y 2 nU ðxÞ. By Definition 8, f is a neighborhood-continuous function. h.  In Theorem 2, the conclusion f ðnV ðf ðxÞÞÞ # nU ðxÞ means that the original image of the neighborhood of f ðxÞ is contained in the neighborhood of x. Thus for the neighborhood of f ðxÞ, the neighborhood-continuous function f only maps the elements in the neighborhood of x into it. Hence, the notion of neighborhood-continuous function intuitively reflects consistency of mapping between two classes of neighborhoods. Proposition 2. Let U and V be two universes, nU and nV be neighborhood operators on U and V, respectively, and f be a mapping from U to V. If f : ðU; nU Þ ! ðV; nV Þ is a neighborhood-continuous function, then (1) (2) (3) (4) (5)



f ðf ðxÞÞ # nU ðxÞ; nV ðf ðxÞÞ # f ðnU ðxÞÞ;  f ðf ðnU ðxÞÞÞ # [y2nU ðxÞ nU ðyÞ;  f ð[x2X nV ðf ðxÞÞÞ # [x2X nU ðxÞ for any X # U;  f ðnV ðf ðxÞÞÞ # \y2f  ðf ðxÞÞ nU ðyÞ.

Proof. 

(1) Assume that f is a neighborhood-continuous function. By Theorem 2, it holds f ðnV ðf ðxÞÞÞ # nU ðxÞ, with the fact of  f ðxÞ 2 nv ðf ðxÞÞ, it follows f ðf ðxÞÞ # nU ðxÞ.  (2) Assume that f is a neighborhood-continuous function. By Theorem 2, it holds f ðnV ðf ðxÞÞÞ # nU ðxÞ. Then   f ðf ðnV ðf ðxÞÞÞÞ # f ðnU ðxÞÞ. Thus f ðf ðnV ðf ðxÞÞÞÞ ¼ nV ðf ðxÞÞ # f ðnU ðxÞÞ.    (3) By (1), f ðnU ðxÞÞ ¼ [y2nU ðxÞ f ðyÞ, it follows f ðf ðnU ðxÞÞÞ ¼ f ð[y2nU ðxÞ f ðyÞÞ ¼ [y2nU ðxÞ f ðf ðyÞÞ # [y2nU ðxÞ nU ðyÞ.   (4) By (2), we can conclude f ð[x2X nV ðf ðxÞÞÞ ¼ [x2X f ðnV ðf ðxÞÞÞ # [x2X nU ðxÞ.     (5) If y 2 f ðf ðxÞÞ, that is f ðxÞ ¼ f ðyÞ, then f ðnV ðf ðxÞÞÞ ¼ f ðnV ðf ðyÞÞÞ # nU ðyÞ. Thus f ðnV ðf ðxÞÞÞ # \y2f  ðf ðxÞÞ nU ðyÞ. h Neighborhood-continuous functions are based on neighborhoods and neighborhood operators. In fact, a relation on an universe uniquely determines a class of predecessor (or, successor) neighborhoods and a predecessor (or, successor) neighborhood operator. Thus according to Definition 8, we can write a mapping by a form f : ðU; RU Þ ! ðV; RV Þ when talking about properties of predecessor (respectively, successor) neighborhoods, where RU and RV are two relations on universes U and V, respectively. In an universe U, if RU is an equivalence relation on U, we always regard the equivalence class ½xRU of each x 2 U as the neighborhood of x, i.e., nU ðxÞ ¼ ½xRU . Theorem 3. Let U and V be two universes, RU and RV be equivalence relations on U and V, respectively, and f be a mapping from U to V. If f : ðU; RU Þ ! ðV; RV Þ is a neighborhood-continuous function, then for any x; y 2 U, 

ð1Þ f ðf ð½xRU ÞÞ ¼ ½xRU ; ð2Þ f ð½xRU Þ \ f ð½yRU Þ ¼ ; if ½xRU – ½yRU ;   ð3Þ f ðf ð½xRU ÞÞ \ f ðf ð½yRU ÞÞ ¼ ; if ½xRU – ½yRU .

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Proof. 

ð1Þ For x 2 U, from Proposition 2(1), it follows f ðf ðxÞÞ # nU ðxÞ ¼ ½xRU , with the fact that f ð½xRU Þ ¼ [y2½xR f ðxÞ, we have U     f ðf ð½xRU ÞÞ ¼ f ð[y2½xR f ðyÞÞ ¼ [y2½xR f ðf ðyÞÞ # [y2½xR ½yRU ¼ ½xRU , that is f ðf ð½xRU ÞÞ # ½xRU . Moreover, it is easy to 

U

U

U



see ½xRU # f ðf ð½xRU ÞÞ. As a result, f ðf ð½xRU ÞÞ ¼ ½xRU . ð2Þ Suppose, by contradiction, that ½xRU – ½yRU , but f ð½xRU Þ \ f ð½ yRU Þ – ;. Thus there exists v 2 V satisfying v 2 f ð½xRU Þ and v 2 f ð½yRU Þ, then f  ðv Þ # f  ðf ð½xRU ÞÞ; f  ðv Þ # f  ðf ð½ yRU ÞÞ and f  ðv Þ – ;, it gives rise to f  ðf ð½xRU ÞÞ \ f  ðf ð½yRU ÞÞ 







 f ðv Þ. By (1), f ðf ð½xRU ÞÞ ¼ ½xRU and f ðf ð½yRU ÞÞ ¼ ½yRU , thus ½xRU \ ½yRU  f ðv Þ – ;. It contradicts with ½xRU \ ½yRU ¼ ;. Hence f ð½xRU Þ \ f ð½yRU Þ ¼ ; if ½xRU – ½yRU . ð3Þ For two equivalence classes ½xRU and ½yRU , if ½xRU – ½ yRU , it follows ½xRU \ ½yRU ¼ ;. From ð1Þ, we have     f ðf ð½xRU ÞÞ # ½xRU and f ðf ð½yRU ÞÞ # ½yRU . So f ðf ð½xRU ÞÞ \ f ðf ð½ yRU ÞÞ ¼ ;. h A neighborhood of an information system is an information granule, hence all neighborhoods of an information system form a class of information granules. In essence, the relationship between two information systems is attributed to the relationship between two classes of information granules. Let U and V be two information granules in universes U and V, respectively, and f be a mapping from U to V. We say that f is a granule-homomorphism if ff ðXÞjX 2 Ug ¼ V and f ðXÞ – f ðYÞ for any X – Y and X; Y 2 U. Intuitively speaking, a granulehomomorphism reflects the equivalence of granularity between two classes of information granules. Corollary 1. Let U and V be two universes, RU and RV be equivalence relations on U and V, respectively, and f be a mapping from U to V. Denote the partitions on U and V by U=RU ¼ fU 1 ; U 2 ; . . . ; U n g and V=RV ¼ fV 1 ; V 2 ; . . . ; V m g, respectively. If f : ðU; RU Þ ! ðV; RV Þ is a surjective neighborhood-continuous function, then ð1Þ ð2Þ ð3Þ ð4Þ

for any U i 2 U=RU , there exists V 0 # V=RV such that f ðU i Þ ¼ [V 0 ; f ðU i Þ \ f ðU j Þ ¼ ; for any U i – U j ; U i ; U j 2 U=RU ; we have n 6 m; if n ¼ m, then f is a granule-homomorphism.

Proof. Suppose that f is a surjective neighborhood-continuous function. Since V=RV is a partition, then for any x 2 U i ; 1 6 i 6 n, there exists only one 1 6 ix 6 m such that f ðxÞ 2 V ix , that is nV ðf ðxÞÞ ¼ V ix . 



ð1Þ By Theorem 2, we have f ðnV ðf ðxÞÞÞ # nU ðxÞ ¼ U i , that is f ðV ix Þ # U i .

We get V ix # f ðU i Þ:

ðaÞ

On the other hand; f ðU i Þ ¼ [x2Ui f ðxÞ # [x2Ui V ix :

ðbÞ

Combining (a) with (b), we can obtain that f ðU i Þ ¼ [x2Ui V ix . Thus ð1Þ holds. ð2Þ Assume U i ; U j 2 U=RU and U i – U j . By the proof of ð1Þ, we have f ðU i Þ ¼ [x2U i V ix and f ðU j Þ ¼ [y2Uj V iy . So if we want to verify f ðU i Þ \ f ðU j Þ ¼ ;, we need only to prove [x2U i V ix \ [y2Uj V iy ¼ ;. Moreover, V=RV is a partition, thus we need only to prove that x 2 U i ; y 2 U j ) V ix – V iy . By x 2 U i and y 2 U j , obviously, it holds x – y. Suppose, by contradiction, that V ix \ V iy – ;, with the fact that V=RV is a   partition, we have V ix ¼ V iy . By Theorem 2, x 2 f ðV ix Þ # U i and y 2 f ðV iy Þ # U j . Combining with V ix ¼ V iy , we can see   that ; – f ðV ix Þ ¼ f ðV iy Þ # U i \ U j . Hence U i \ U j – ;. It is a contradiction with the fact that U=RU is a partition on U. Therefore, ð2Þ is concluded. ð3Þ Denote V i ¼ fV i 2 V=RV jV i # f ðU i Þg for all U i 2 U=RU . By (1), we can prove that jV i j P 1 and f ðU i Þ ¼ [V i . By (2), for U i ; U j 2 U=RU and U i – U j , it follows f ðU i Þ \ f ðU j Þ ¼ ;. Thus V i – V j for any V i 2 V i ; V j 2 V j . Besides, combinP ing with jV i j P 1, we can conclude that n ¼ jU=RU j 6 16i6m V i 6 jV=RV j ¼ m. ð4Þ Denote V i ¼ fV i 2 V=RV jV i # f ðU i Þg, then jV i j P 1. Now we prove jV i j ¼ 1 for all 1 6 i 6 n if n ¼ m. 0

If not; assume there is one i such that jV i0 j > 1:

ð1aÞ

Byð3Þ; we can see that f ðU i Þ ¼ [V i :

ð1bÞ

Byð2Þ; V i \ V j ¼ ; for i – j:

ð1cÞ P

By 1a, 1b and 1c, it holds n ¼ jU=RU j < i jV i j 6 V=RV ¼ m, that is n < m. It contradicts to the condition n ¼ m. Thus jV i j ¼ 1 for all i, Combining with n ¼ m, it follows that f is a granule-homomorphism. h

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Remark 1. From Corollary 1, we can see that in terms of neighborhood-continuous functions, we can compare information granules between two information systems. Based on neighborhood-continuous functions, we can also define predecessor and successor neighborhood-continuous functions, respectively. Definition 9. Let U and V be two universes, RU and RV be relations on U and V, respectively, and f be a mapping from U to V. (1) f : ðU; RU Þ ! ðV; RV Þ is called a predecessor neighborhood-continuous function if y 2 RUp ðxÞ whenever f ðyÞ 2 RVp ðf ðxÞÞ for any x; y 2 U. (2) f : ðU; RU Þ ! ðV; RV Þ is called a successor neighborhood-continuous function if y 2 RUs ðxÞ whenever f ðyÞ 2 RUs ðf ðxÞÞ for any x; y 2 U.

Example 1. Let U ¼ fx1 ; x2 ; x3 ; x4 ; x5 g; V ¼ fy1 ; y2 ; y3 g. RU and RV are relations on U and V, respectively, and RU ¼ fðx1 ; x2 Þ; ðx2 ; x5 Þ; ðx2 ; x3 Þ; ðx3 ; x2 Þ; ðx4 ; x2 Þ; ðx5 ; x3 Þ; ðx5 ; x5 Þg, RV ¼ fðy1 ; y2 Þ; ðy2 ; y3 Þ; ðy3 ; y3 Þg. Define f i : U ! V, i = 1, 2, 3, as follows:

f 1 ðxj Þ ¼ y1 for j ¼ 1; 3; 4; f 1 ðx2 Þ ¼ y3 ; f 1 ðx5 Þ ¼ y2 : f 2 ðx1 Þ ¼ y1 ; f 2 ðxj Þ ¼ y3 for j ¼ 2; 3; 4; f 2 ðx5 Þ ¼ y2 : f 3 ðxj Þ ¼ y1 for j ¼ 1; 3; 4; f 3 ðx2 Þ ¼ y2 ; f 3 ðx5 Þ ¼ y3 : Then by definition, it is easy to check that mapping f 1 : ðU; RU Þ ! ðV; RV Þ is predecessor neighborhood-continuous (not predecessor-consistent), mapping f 2 : ðU; RU Þ ! ðV; RV Þ is successor neighborhood-continuous (not successor-consistent), and f 3 : ðU; RU Þ ! ðV; RV Þ is both predecessor neighborhood-continuous and successor neighborhood-continuous. Theorem 4. Let U and V be two universes, RU and RV be relations on U and V, respectively, and f be a mapping from U to V. 

ð1Þ f : ðU; RU Þ ! ðV; RV Þ is a predecessor neighborhood-continuous function iff f ðRVp ðf ðxÞÞÞ # RUp ðxÞ for all x 2 U;  ð2Þ f : ðU; RU Þ ! ðV; RV Þ is a successor neighborhood-continuous function iff f ðRVs ðf ðxÞÞÞ # RUs ðxÞ for all x 2 U. ð3Þ If f : ðU; RU Þ ! ðV; RV Þ is a predecessor neighborhood-continuous function, RU is transitive and RV is reflexive, then for any  x 2 U, there is a set V x # V such that f ð[y2V x RVp ðyÞÞ ¼ RUp ðxÞ; ð4Þ If f : ðU; RU Þ ! ðV; RV Þ is a successor neighborhood-continuous function, RU is transitive and RV is reflexive, then for any  x 2 U, there is a set V x # V such that f ð[y2V x RVs ðyÞÞ ¼ RUs ðxÞ. Proof. ð1Þ and ð2Þ can be concluded by Theorem 2.   ð3Þ Since x 2 f ðf ðxÞÞ, thus RUp ðxÞ # f ð[x0 2RUp ðxÞ f ðx0 ÞÞ. 

We have RUp ðxÞ # f ð[x0 2RUp ðxÞ RVp f ðx0 ÞÞ:

ð4aÞ 

On the other hand, by (1), for any x0 2 RUp ðxÞ, f ðRVp ðf ðx0 ÞÞÞ # RUp ðx0 Þ. Assume that RU is transitive, if x0 2 RUp ðxÞ, then RUp ðx0 Þ # RUp ðxÞ. 

Hence; f ð[x0 2RUp ðxÞ RVp ðf ðx0 ÞÞÞ # RUp ðxÞ:

ð4bÞ 

0



Combining (4a) and (4b), we obtain that RUp ðxÞ # f ð[x0 2RUp ðxÞ ðRVp f ðx ÞÞÞ. Let V x ¼ f ðRUp ðxÞÞ, then RUp ðxÞ # f ð[y2V 0 RVp ðyÞ finishing the proof of the theorem. ð4Þ The proof is similar to (3). h. 4. Relationships among neighborhood-continuous functions and some existing consistent functions In Section 3, we have explored some basic properties of neighborhood-continuous functions. From the analysis above, we can see that the structure of neighborhood-continuous function is based on bilateral neighborhoods of both an original system and its image system and it can reflect relationships between granular knowledges of two information systems. In this section, we will show that a neighborhood-continuous function has close relationships with some existing consistent functions and it can represent and unify these consistent functions through reducing suitable neighborhood operators. As the notion of discreteness in topology [24], the class of all neighborhoods in an universe U is called discrete if the neighborhood of x is nU ðxÞ ¼ fxg for all x 2 U. Let U be an universe and nU be a neighborhood operator on U. Define another neighborhood operator n0U on U induced by nU as n0U ðxÞ ¼ fy 2 UjnU ðxÞ ¼ nU ðyÞg for 8x 2 U. Since the existing consistent functions are only based on the neighborhoods of an original system and do not consider neighborhoods of its image system, while neighborhood-continuous functions are based on two classes of an original and

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image systems, thus in this section, in order to facilitate the study of relationships between neighborhood-continuous functions and existing consistent functions, we always assume that the neighborhoods of an image system is discrete. Lemma 1. Let U and V be two universes, RU ; RV be relations on U and V, respectively. Then for any x; y 2 U, ð1Þ y 2 n0U ðxÞ () nU ðxÞ ¼ nU ðyÞ; ð2Þ f ðyÞ 2 nV ðf ðxÞÞ () f ðyÞ ¼ f ðxÞ.

Proof. It is straightforward. h 4.1. The relationship between neighborhood-continuous functions and some existing neighborhood-consistent functions In this subsection, we explore the relationship between neighborhood-continuous functions and some existing neighborhood-consistent functions. Theorem 5. Let U and V be two universes and nU be a neighborhood operator on U. A mapping f : U ! V is a neighborhoodconsistent function with respect to nU iff it is a neighborhood-continuous function with respect to n0U . Proof. Necessity. By Theorem 2, we need to prove f ðyÞ 2 nV ðf ðxÞÞ ) y 2 n0U ðxÞ, thus we need only to prove f ðyÞ ¼ f ðxÞ ) nU ðxÞ ¼ nU ðyÞ. Assume that f is a neighborhood-consistent function with respect to nU . For any x; y 2 U, if f ðxÞ ¼ f ðyÞ, it follows nU ðxÞ ¼ nU ðyÞ. By Lemma 1(1), we have y 2 n0U ðxÞ. We obtain that f is a neighborhood continuous function with respect to n0U . Sufficiency. If we want to examine that f is a neighborhood-consistent function, we need only to prove f ðxÞ ¼ f ðyÞ ) nU ðxÞ ¼ nU ðyÞ. Suppose that f is a neighborhood-continuous function with respect to n0U , it follows f ðyÞ 2 nV ðf ðxÞÞ ) y 2 n0U ðxÞ, besides by Lemma 1(1) and (2), we can conclude f ðxÞ ¼ f ðyÞ ) nU ðxÞ ¼ nU ðyÞ. Hence f is a neighborhood-consistent function. h. By Theorem 5, we can particularly represent the consistent functions based on coverings and dominance relations by neighborhood-consistent functions. Corollary 2. Let U and V be universes, f : U ! V be a mapping and C be a covering of U. Then f is a consistent function with respect to C iff f is a neighborhood-continuous function with respect to n0U , where n0U ðxÞ ¼ fy 2 Uj8K 2 Cðx 2 K () y 2 KÞg for any x 2 U. Proof. We can easily conclude that the proposition of 8K 2 Cðx 2 K () y 2 KÞ is equal to C x ¼ C y . By Definition 4 and Theorem 5, we can assert it. h Corollary 3. Let U and V be universes, f : U ! V be a mapping and RU be a dominance relation on U. Then f is a consistent function with respect to RU iff f is a neighborhood-continuous function with respect to n0U , where n0U ðxÞ ¼ fy 2 Uj8z 2 Uððx; zÞ 2 RU () ðy; zÞ 2 RU Þg for any x 2 U. Proof. It is similar to Corollary 2. h By Corollaries 2 and 3, the consistent functions with respect to coverings and dominance relations can also be represented by neighborhood-continuous functions, so neighborhood-continuous functions are generalizations of consistent functions. Corollary 4. Let U and V be two universes, RU be a relation on U and f : U ! V be a mapping. Then ð1Þ f is a predecessor neighborhood-consistent function with respect to RUp iff it is a predecessor neighborhood-continuous function with respect to R0Up ; ð2Þ f is a successor neighborhood-consistent function with respect to RUs iff it is a successor neighborhood continuous function with respect to R0Us .

Proof. Since predecessor and successor neighborhood operators are two kinds of neighborhood operators, by Theorem 5, they hold. h The above conclusions are based on an arbitrary binary relation. In what follows, we give some properties of neighborhood-continuous functions with respect to some kinds of special relations. Proposition 3. Let U and V be two universes, RU be a reflexive and transitive relation on U and f : U ! V be a mapping. Then

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(1) f is a predecessor neighborhood-continuous function () f is a predecessor neighborhood-consistent function; (2) f is a successor neighborhood-continuous function () f is a successor neighborhood-consistent function.

Proof. We only prove (2) for simplicity, the proof of (1) is paralleled. (2) Necessity. If we want to prove that f is a successor neighborhood-consistent function, we need only to prove f ðxÞ ¼ f ðyÞ ) RUs ðxÞ ¼ RUs ðyÞ. Assume that f is a neighborhood-continuous function. Since V is discrete, then for any x; y 2 U, f ðxÞ ¼ f ðyÞ () RVs ðf ðyÞÞ ¼ RVs ðf ðxÞ, that is

f ðxÞ ¼ f ðyÞ ) f ðyÞ 2 RVs ðf ðxÞÞ and f ðxÞ 2 RVs ðf ðyÞÞ:

ð3aÞ

Since f is a neighborhood-continuous function, we have

f ðyÞ 2 RVs ðf ðxÞÞ ) y 2 RUs ðxÞ; and f ðxÞ 2 RVs ðf ðyÞÞ ) x 2 RUs ðyÞ:

ð3bÞ

By (3a) and (3b), it holds

f ðxÞ ¼ f ðyÞ ) y 2 RUs ðxÞ and x 2 RUs ðyÞ:

ð3cÞ

With the fact that RU is transitive, then for any z 2 RUs ðyÞ, it holds y 2 RUs ðxÞ ) z 2 RUs ðxÞ. So y 2 RUs ðxÞ ) RUs ðyÞ # RUs ðxÞ. By the symmetry of x and y, we can also obtain x 2 RUs ðyÞ ) RUs ðxÞ # RUs ðyÞ. Thus,

y 2 RUs ðxÞ and x 2 RUs ðyÞ ) RUs ðxÞ ¼ RUs ðyÞ:

ð3fÞ

By (3c) and (3f), we have f ðxÞ ¼ f ðyÞ ) RUs ðxÞ ¼ RUs ðyÞ. By Definition 3(2), f is a successor neighborhood-consistent function. Sufficiency. If f is a successor neighborhood-consistent function, then

f ðxÞ ¼ f ðyÞ ) RUs ðxÞ ¼ RUs ðyÞ:

0

ð3a Þ

RU is reflexive, it follows y 2 RUs ðyÞ. 0

Thus; RUs ðxÞ ¼ RUs ðyÞ ) y 2 RUs ðxÞ:

ð3b Þ

Since V is discrete; we can conclude that f ðyÞ 2 RVs ðf ðxÞÞ ) f ðyÞ ¼ f ðxÞ:

ð4c Þ

0

0

0

By (3a ) and (3c ), we have

f ðyÞ 2 RVs ðf ðxÞÞ ) RUs ðxÞ ¼ RUs ðyÞ: 0

0

ð3d Þ

0

By (3b ) and (3d ), we have f ðyÞ 2 RVs ðf ðxÞÞ ) y 2 RUs ðxÞ. Therefore, f is a successor neighborhood-continuous function. h. Since an equivalence relation is reflexive and transitive, the results in Proposition 3 also hold with respect to an equivalence relation. Remark 2. From Proposition 3, with respect to a reflexive and transitive binary relation, a neighborhood-continuous function degenerates into a neighborhood-consistent function. However, the degeneration is based on a hypothesis that the class of all neighborhoods in an image system is discrete and this hypothesis is very limited. Hence, neighborhoodcontinuous functions can reflect the relationship between two classes of neighborhoods of information systems more extensively. Moreover, It should be pointed out that in Proposition 3, if predecessor (or, successor) neighborhoods generated by relation RV on V is not limited to discrete, but under the condition that RV is reflexive, the necessity also holds. Therefore, a neighborhood-continuous function can be seen as a generalization of a neighborhood-consistent function in some degree.

4.2. The relationship between predecessor and successor neighborhood-continuous functions

Lemma 2 [41]. Let RU be a reflexive and transitive relation on universe U. Then for any x; y 2 U; RUp ðxÞ ¼ RUp ðyÞ iff RUs ðxÞ ¼ RUs ðyÞ. In [22], Wang et al. pointed out that type-1 consistent function and type-2 consistent function are equal if the relation on an original system is reflexive and transitive. A beautiful conclusion is described as follows.

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Theorem 6 [22]. Let U and V be two universes. If RU is a reflexive and transitive relation on U, then f : U ! V is a predecessorconsistent function iff it is a successor-consistent function. In the following, we present a kind of relationship between predecessor neighborhood-continuous and successor neighborhood-continuous functions based on a reflexive and transitive relation on an original system. Corollary 5. Let U and V be two universes. If RU is a reflexive and transitive relation on U, then f : U ! V is a predecessor neighborhood-continuous function iff it is a successor neighborhood-continuous function. Proof. If RU is reflexive and transitive, then by Proposition 3(2), f is a successor neighborhood-continuous function () f is a successor neighborhood-consistent function; by Theorem 6, f is a successor neighborhood-consistent function () f is a predecessor neighborhood-consistent function; by Proposition 3(1), f is a predecessor neighborhood-consistent function () f is a predecessor neighborhood-continuous function. Therefore, f is successor neighborhood-continuous () f is predecessor neighborhood-continuous. h. 5. Some extended properties of relation mappings and neighborhood-continuous functions with respect to relation mappings In an information system, an attribute uniquely corresponds to a binary relation. Hence, a mapping is essentially to examine the relationship between two classes of binary relations. Wang et al. introduced the notions of relation mappings by which some relations on codomains can be constructed [22,41]. In this section, we will address more properties of relation mappings and talk about some properties of neighborhood-continuous functions with respect to relation mappings. Let U and V be two universes, the classes of all binary relations on U and V are denoted by RðUÞ and RðVÞ, respectively. Definition 10 [22]. Let U; V be two universes and f : U ! V a mapping. Two mappings from RðUÞ to RðVÞ and from RðVÞ to RðUÞ can be induce by f: (1) ^f : RðUÞ ! RðVÞ, RU j ! ^f ðRU Þ 2 RðVÞ; 8RU 2 RðUÞ;

^f ðR Þ ¼ [ f ðxÞ  f ðR ðxÞÞ: U x2U Us (2) ^f  : RðVÞ ! RðUÞ, RV j ! ^f  ðRV Þ 2 RðUÞ; 8RV 2 RðVÞ;

^f  ðRV Þ ¼ [y2V f  ðyÞ  f  ðRVs ðyÞÞ: We call ^f and ^f  the induced relation mappings by f. We give the following intuitive descriptions of relation mappings. Proposition 4. Let U and V be two universes, RU ; RV be relations on U and V, respectively and f : U ! V be a mapping. Assume that ^f and ^f  are the induced relation mapping by f. Then ð1Þ ð2Þ ð3Þ ð4Þ

 ð^f  ðRV ÞÞs ðxÞ ¼ f ðRVs ðf ðxÞÞÞ for x 2 U;   ^ ðf ðRV ÞÞp ðxÞ ¼ f ðRVp ðf ðxÞÞÞ for x 2 U; ð^f ðRU ÞÞs ðyÞ ¼ [x2f  ðyÞ f ðRUs ðxÞÞ for y 2 V; ð^f ðRU ÞÞp ðyÞ ¼ [x2f  ðyÞ f ðRUp ðxÞ for y 2 V.

Proof. 



ð1Þ For any x 2 U, there is only one y 2 V satisfying f ðxÞ ¼ y. Then x 2 f ðyÞ and x R f ðy0 Þ for all y0 – y; y0 2 V.  ^f  ðRV Þ ¼ [y2V f  ðyÞ  f  ðRVs ðyÞÞ. By Definition 10(2), Thus, ðx; f ðRVs ðyÞÞÞ 2 ^f  ðRV Þ holds. We have   ^ ðf ðRV ÞÞs ðxÞ  f ðRVs ðyÞÞ.   On the other hand, if z 2 U and z 2 ð^f  ðRV ÞÞs ðxÞ, that is ðx; zÞ 2 ^f  ðRV Þ. By Definition 10(2), ðf ðyÞ; f ðRVs ðyÞÞÞ 2 ^f  ðRV Þ,     besides, x 2 f ðyÞ and x R f ðy0 Þ for y0 – y; y0 2 V, it follows z 2 f ðRVs ðyÞÞ. Thus ð^f  ðRV ÞÞs ðxÞ # f ðRVs ðyÞÞ.  That is ð^f  ðRV ÞÞs ðxÞ ¼ f ðRVs ðf ðxÞÞÞ. ð2Þ For any x 2 U, we need to obtain ^f  ðRV Þp ðxÞ.

It follows that y 2 ð^f  ðRV ÞÞp ðxÞ () x 2 ð^f  ðRV ÞÞs ðyÞ:

ð4aÞ

 By ð1Þ; ð^f  ðRV ÞÞs ðyÞ ¼ f ðRs ðf ðyÞÞÞ:

ð4bÞ

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Combining (4a) with (4b), we have  y 2 ð^f  ðRV ÞÞp ðxÞ () x 2 f ðRs ðf ðyÞÞÞ:

ð4cÞ



ð4dÞ

In addition; x 2 f ðRs ðf ðyÞÞÞ () f ðxÞ 2 Rs ðf ðyÞÞ:

From (4c) and (4d), We can obtain that y 2 ð^f  ðRV ÞÞp ðxÞ () f ðxÞ 2 Rs ðf ðyÞÞ. Moreover f ðxÞ 2 Rs ðf ðyÞÞ () f ðyÞ 2 Rp ðf ðxÞÞ. Thus y 2 ð^f  ðRV ÞÞp ðxÞ () f ðyÞ 2 Rp ðf ðxÞÞ. So it holds.   ð3Þ For y 2 V, if z 2 ð^f ðRU ÞÞs ðyÞ, then ðy; zÞ 2 ^f ðRU Þ. By Definition 10(1), there is x1 2 f ðyÞ; x2 2 f ðzÞ and x1 ; x2 2 U such that ðx1 ; x2 Þ 2 RU , that is x2 2 RUs ðx1 Þ, then z ¼ f ðx2 Þ 2 f ðRUs ðx1 ÞÞ. Thus z ¼ f ðx2 Þ 2 [  f ðRUs ðxÞÞ for any z 2 ð^f ðRU ÞÞ ðyÞ. That x2f ðyÞ

s

means ð^f ðRU ÞÞs ðyÞ # [x2f  ðyÞ f ðRUs ðxÞÞ for y 2 V.  On the other hand, for any y 2 V, any x 2 f ðyÞ, and any z 2 V with z 2 f ðRUs ðxÞÞ, then there is x0 2 U; x0 2 RUs ðxÞ such 0 0 that f ðx Þ ¼ z. by Definition 10(1), if x 2 RUs ðxÞ, then f ðx0 Þ 2 ð^f ðRU ÞÞs ðf ðxÞÞ, thus z 2 ð^f ðRU ÞÞs ðyÞ for any z 2 f ðRUs ðxÞÞ.  Therefore, f ðRUs ðxÞÞ # ð^f ðRU ÞÞ ðyÞ for any x 2 f ðyÞ. That means [  f ðRUs ðxÞÞ # ð^f ðRU ÞÞ ðyÞ for any y 2 V. x2f ðyÞ

s

s

We have ð^f ðRU ÞÞs ðyÞ ¼ [x2f  ðyÞ f ðRUs ðxÞÞ for y 2 V. ð4Þ Since z 2 ð^f ðRU ÞÞp ðyÞ () y 2 ð^f ðRU ÞÞs ðzÞ, by (3) and the proof of (2), it holds. h Based on relation mappings, we give the following conclusions for neighborhood-continuous functions. Theorem 7. Let U; V be two universes, RU ; RV be equivalence relations on U and V, respectively and f be a mapping from U to V. If f : ðU; RU Þ ! ðV; RV Þ is a surjective neighborhood-continuous function, then ð1Þ ð2Þ ð3Þ ð4Þ

f ðRU ðXÞÞ ¼ RV ðf ðXÞÞ; RV ðf ðXÞÞ # f ðRU ðXÞÞ;  f ðf ðRU ðXÞÞÞ ¼ RU ðXÞ;  f ðf ðRU ðXÞÞÞ ¼ RU ðXÞ.

Proof. Denote the partitions on U and V by U=RU ¼ fU 1 ; U 2 ; . . . ; U n g and V=RV ¼ fV 1 ; V 2 ; . . . ; V m g, respectively. Then RX is the union of some U i , assume RX ¼ [i2C U i , where C # f1; 2; . . . ng. By Corollary 1, for any f ðU i Þ, there exists V i # V=RV such that f ðU i Þ ¼ [V i for 1 6 i 6 n. (1) f ðRU XÞ ¼ f ð[i2C U i Þ ¼ [i2C f ðU i Þ ¼ [i2C V i . In fact for all i 2 C, it holds U i # X, then f ðU i Þ # f ðXÞ, thus [V i # f ðXÞ. It is easy to conclude [V i # RV ðf ðXÞÞ for all 1 6 i 6 n. Hence, we have f ðRU ðXÞÞ # RV ðf ðXÞÞ. (2) We can conclude that RU ðXÞ ¼ [x2X ½xRU and RV ðf ðXÞÞ ¼ [x2X ½f ðxÞRV . f ðRU ðXÞÞ ¼ f ð[x2X ½xRU Þ ¼ [x2X f ð½xRU Þ. By Theorem 3, we have f ð½xRU Þ  ½f ðxÞRV . Hence f ðRU ðXÞÞ ¼ [x2X f ð½xRU Þ  [x2X ½f ðxÞRV ¼ RV ðf ðXÞÞ. Thus it holds.    (3) By Corollary 1, f ðf ðRU ðXÞÞÞ ¼ f ðf ð[i2C U i ÞÞ ¼ [i2C f ðf ðU i ÞÞ ¼ [i2C U i ¼ RU ðXÞ. (4) Also we have RU ðXÞ is the union of some U i , assume RU ðXÞ ¼ [i2C 0 U i , C 0 # f1; 2; . . . ng. By Corollary 1, it follows that    f ðf ðRU ðXÞÞÞ ¼ f ðf ð[i2C 0 U i ÞÞ ¼ [i2C 0 f ðf ðU i ÞÞ ¼ [i2C 0 U i ¼ RU ðXÞ. h In the following, we will investigate some properties of neighborhood-continuous functions with respect to relation mappings. Theorem 8. Let U and V be two universes, RU ; RV be relations on U and V, respectively and f : U ! V be a mapping. Assume that ^f and ^f  are the induced relation mapping by f. Then (1) f : ðU; ^f  ðRV ÞÞ ! ðV; RV Þ is a predecessor neighborhood-continuous function; (2) f : ðU; ^f  ðRV ÞÞ ! ðV; RV Þ is a successor neighborhood continuous function.  If there is the inverse of f, denoted by f , then  ^ (3) f : ðV; f ðRU Þ ! ðU; RU Þ is a predecessor neighborhood-continuous function;  (4) f : ðV; ^f ðRU Þ ! ðU; RU Þ is a successor neighborhood-continuous function.

Proof. ð1Þ By Theorem 4(1), we need only to proof f ðRVp ðf ðxÞÞÞ # ð^f  ðRV ÞÞp ðxÞ for any x 2 U.  By Proposition 4(2), ð^f  ðRV ÞÞp ðxÞ ¼ f ðRVp ðf ðxÞÞÞ, Thus we can conclude it. 

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ð2Þ It is similar to (1).  By the symmetry between f and f ; U and V, and by the proof of ð1Þ and ð2Þ, we can assert that ð3Þ and ð4Þ hold.

h.

Example 2. Recall that in Example 1, U ¼ fx1 ; x2 ; x3 ; x4 ; x5 g, V ¼ fy1 ; y2 ; y3 g; RU and RV are relations on U and V, respectively, and RU ¼ fðx1 ; x2 Þ; ðx2 ; x5 Þ; ðx2 ; x3 Þ; ðx3 ; x2 Þ; ðx4 ; x2 Þ; ðx5 ; x3 Þ; ðx5 ; x5 Þg, RV ¼ fðy1 ; y2 Þ; ðy2 ; y3 Þ; ðy3 ; y3 Þg. Consider f 1 : ðU; RU Þ ! ðV; RV Þ defined by

f 1 ðxj Þ ¼ y1 for j ¼ 1; 3; 4; f 1 ðx2 Þ ¼ y3 ; f 1 ðx5 Þ ¼ y2 : We can obtain that ^f  1 ðRV Þ ¼ fðx1 ; x5 Þ; ðx2 ; x2 Þ; ðx3 ; x5 Þ; ðx4 ; x5 Þ; ðx5 ; x2 Þg. We can verify that f 1 : ðU; ^f  1 ðRV ÞÞ ! ðV; RV Þ is both predecessor neighborhood-continuous and successor neighborhoodcontinuous. Remark 3. Theorem 8 implies that the concept of neighborhood-continuous function is in line with structures of relation mappings.

6. Attribute reducts based on neighborhood-continuous functions and neighborhood-homomorphisms In this section, we investigate the attribute reducts of information systems by neighborhood-continuous functions and neighborhood-homomorphisms. 6.1. Attribute reduct of a single information system based on neighborhood-continuous functions We decompose a decision system S ¼ ðU; A ¼ C [ DÞ into two subsystems SC ¼ ðU; CÞ and SD ¼ ðU; DÞ. Hence, one subsystem is constructed by condition attribute set C, denoted by a condition subsystem and the other is constructed by decision attribute set D, denoted by a decision subsystem. In the following, we will investigate attribute reducts of information systems through neighborhood-continuous functions. In an information system ðU; AÞ; B; C # A, according to Definition 8, denote a mapping f from U to U by f : ðU; BÞ ! ðU; CÞ, where attribute sets B and C determine neighborhoods (which are virtually equivalence classes) of the original system and its image system, respectively. The followings describe the feature of attribute reducts in terms of neighborhood-continuous functions. Theorem 9. In an information system ðU; AÞ; B # A, suppose that f : ðU; AÞ ! ðU; BÞ is an identity function. Then (1) B is a consistent set iff f : ðU; AÞ ! ðU; BÞ is a neighborhood-continuous function; (2) B is a reduct iff f : ðU; AÞ ! ðU; BÞ is a neighborhood-continuous function and for any C  B; f : ðU; AÞ ! ðU; CÞ is no longer a neighborhood-continuous function.

Proof. 

(1) Sufficiency. If f : ðU; AÞ ! ðU; BÞ is a neighborhood-continuous function, by Theorem 2, f ð½f ðxÞB Þ # ½xA . f is the iden  tity function, then f ð½f ðxÞB Þ ¼ f ð½xB Þ ¼ ½xB . So ½xB # ½xA , that means B is a consistent set. Necessity. If B is a consistent set, then RA ¼ RB and ½xB # ½xA . With the fact that f is an identity function, it follows    f ð½f ðxÞB Þ ¼ f ð½xB Þ ¼ ½xB . We can obtain that f ð½f ðxÞB Þ # ½xA , by Theorem 2, f is a neighborhood-continuous function. (2) It is straightforward from (1). h

Theorem 10. In a consistent decision system ðU; C [ DÞ; B # C, suppose f : ðU; DÞ ! ðU; BÞ is an identity function. Then (1) B is a consistent set iff f : ðU; DÞ ! ðU; BÞ is a neighborhood-continuous function; (2) B is a reduct iff f : ðU; DÞ ! ðU; BÞ is a neighborhood-continuous function and for any E  B; f : ðU; DÞ ! ðU; EÞ is no longer a neighborhood-continuous function.

Proof. It follows from Theorem 9. h In Theorems 9 and 10, we assume that f is an identity mapping. In what follows, we will make a further effort to explore whether these theorems hold if f is not limited to an identify function.

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Firstly, we give the formula of sum of squares. For three positive numbers a; b; c, if a ¼ b þ c, then a2 P b þ c2 , besides 2 a2 ¼ b þ c2 if and only if b ¼ 0 or c ¼ 0 [2]. In an information system ðU; AÞ, for attribute sets B; C # A, we say that the partition U=B generated by B is finer than the partition U=C generated by C, denoted by U=B  U=C, if for any X 2 U=B, there exists Y 2 U=C satisfying X # Y. For the following discussion, we first give a lemma. Lemma 3. Let ðU; AÞ be an information system, B; C # A and U=B and U=C be the partitions generated by B and C, respectively. P P (1) If U=B  U=C, then C i 2U=C jC i j2 P Bj 2U=B jBj j2 . P P (2) If B # C, then U=B ¼ U=C () C i 2U=C jC i j2 ¼ Bj 2U=B jBj j2 .

Proof. (1) If U=B ¼ U=C, it is straightforward. Suppose U=B – U=C. Let the partition generated by B be U=B ¼ fB1 ; B2 ; B3 ; . . . ; Bn g, where Bi – ; for all Bi 2 U=B. Besides U=B  U=C, then there exists at least one C i 2 U=C such that C i is the union of some elements in U=B. Without loss of generality, we assume U=C ¼ fC 1 ; B3 ; B4 ; . . . ; Bn g, where C 1 ¼ B1 [ B2 . Thus jC 1 j ¼ jB1 j þ jB2 j. In fact B1 ; B2 – ; and by the formula of sum of squares, we have jC 1 j2 > jB1 j2 þ jB2 j2 . P P 2 2 2 2 Since Bj 2U=B jBj j2 ¼ jB1 j2 þ jB2 j2 þ jB3 j2 þ    þ jBn j2 ; C i 2U=C jC i j ¼ jC 1 j þ jB3 j þ    þ jBn j , P P 2 2 we can conclude C i 2U=C jC i j P Bj 2U=C jBj j . (2) By the proof of (1), it holds. h

Theorem 11. let ðU; AÞ be an information system and B # A. Then (1) B is a consistent set iff there exists a surjective neighborhood-continuous function f : ðU; AÞ ! ðU; BÞ; (2) B is a reduct iff there exists a neighborhood-continuous function f : ðU; AÞ ! ðU; BÞ and for any C  B, any function f : ðU; AÞ ! ðU; CÞ is not a surjective neighborhood-continuous function.

Proof. P P (1) Sufficiency. If we want to prove that B is consistent, by Lemma 3(2), we need only to verify Bi 2U=B jBi j2 ¼ Aj 2U=A jAj j2 . P P 2 2 From B # A, then U=A  U=B, by Lemma 3(1), we have Bi 2U=B jBi j P Aj 2U=A jAj j . P P In the following, if we can prove Bi 2U=B jBi j2 6 Aj 2U=A jAj j2 , then we can assert the proposition holds. By Corollary 1(1), for each Aj 2 U=A, there exists some Bj # U=B such that f ðAj Þ ¼ [Bi 2Bj Bi . P With the fact that one original image has only one image under a function, thus jAj j ¼ Bi 2Bj jBi j. Moreover, by the formula of sum of squares, we have jAj j2 P [Bi 2Bj jBi j2 . P P Besides f is surjective, it follows that Ai 2U=A jAi j2 P Bj 2U=B jBj j2 . Therefore, it holds. Necessity. If B is consistent, by Theorem 10(1), identity function f : ðU; AÞ ! ðU; BÞ is neighborhood-continuous and obviously surjective. Thus it holds. (2) We can assert it by the proof of (1). h

Theorem 12. let ðU; C [ DÞ be a consistent decision system and B # C. Then (1) B is a consistent set iff there exists a neighborhood-continuous function f : ðU; DÞ ! ðU; BÞ; (2) B is a reduct iff there exists a neighborhood-continuous function f : ðU; DÞ ! ðU; BÞ and for any E  B, any function f : ðU; DÞ ! ðU; EÞ is not a surjective neighborhood-continuous function.

Proof. It holds by Theorem 11. h Remark 4. Theorem 11 means that for an attribute set B # A, if there exists at least one surjective neighborhood-continuous function f : ðU; AÞ ! ðU; BÞ, then we can assert that B is consistent. By Theorems 11 and 12, we can see that the notions of neighborhood-continuous functions can describe reduct features of a single information system precisely.

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6.2. Reduct invariance between different information systems based on neighborhood-homomorphisms In this section, based on the notions of neighborhood-continuous functions, we define a neighborhood-homomorphism between different information systems and investigate some invariant properties of reducts between two different information systems under the condition of neighborhood-homomorphisms. As an extension of classical information systems, a relation information system can be presented by a form S ¼ ðU; RU Þ, where U is an universe and RU is a relation set on U [22]. Thus, a classical information system is a particular relation information system. Definition 11 [22]. Let ðU; RU Þ be a relation information system. The subset P U # RU is referred to as a consistent set of RU if \P U ¼ \RU .

Definition 12 [22]. Let ðU; RU Þ be a relation information system. The subset P U # RU is referred to as a reduct of RU if P U satisfies the following conditions: (1) \P U ¼ \RU ; (2) 8Ri 2 P U ; \P U  \ðP U  Ri Þ.

Let ðU; RU Þ be a relation information system and f be a mapping from U to V. Denote f ðRU Þ ¼ f^f ðRi ÞjRi 2 RU g, where ^f is the relation mapping induced by f. Mapping f is called a homomorphism from ðU; RU Þ to ðV; f ðRU Þ if f is both type-1 and type2 consistent with respect to 8Ri 2 RU [22]. Thus the concept of homomorphism is based on two classes of relations. The following result points out that attribute reducts are equivalent between an original system and its induced image system under homomorphisms. Theorem 13 [22]. Let ðU; RU Þ be a relation information system and ðV; f ðRU ÞÞ be the induced relation information system by ðU; RU Þ. If f is a homomorphism from ðU; RU Þ to ðV; f ðRU ÞÞ, then P U # RU is a reduct of RU iff f ðRU Þ is a reduct of f ðRÞ. Based on the notions of neighborhood-continuous functions, we introduce the following concept. Definition 13. Let ðU; RU Þ; ðV; RV Þ be two relation information systems and f be a mapping from U to V. If there is some orders of RU and RV arranging as RU ¼ fRU1 ; RU2 ; . . . ; RUn g and RV ¼ fRV1 ; RV2 ; . . . ; RVn g such that each f : ðU; RUi Þ ! ðV; RVi Þ is both predecessor and successor neighborhood-continuous functions, then f is referred to as a neighborhood-homomorphism from ðU; RU Þ to ðV; RV Þ.

For a neighborhood-homomorphism f : ðU; RU Þ ! ðV; RV Þ, with respect to each f : ðU; RUi Þ ! ðV; RVi Þ, we call RVi 2 RV as the corresponding relation with respect to RUi 2 RU , denoted by RVi ¼ f  RUi . For a neighborhood-homomorphism f : ðU; RU Þ ! ðV; RV Þ and any P U # RU , denote P V ¼ ff  RU 2 RV jRU 2 P U g as the collection of all corresponding relations in RV with respect to P U . We can easily conclude that if f : ðU; RU Þ ! ðV; RV Þ is a neighborhood-homomorphism, then f : ðU; P U Þ ! ðV; P V Þ is still a neighborhood-homomorphism. The properties of neighborhood-homomorphisms with respect to relation mappings are as follows. 

Theorem 14. let ðU; RU Þ; ðV; RV Þ be two relation information systems and f be a mapping from U to V. f ðRV Þ ¼  ff^ ðRV ÞjRV 2 RV g is the relation set on U induced by RV , and f ðRU Þ ¼ f^f ðRU ÞjRU 2 RU g is the relation set on V induced by RU ,  where ^f and f^ are relation mappings induced by f. Then 

ð1Þ f : ðU; f ðRV ÞÞ ! ðV; RV Þ is a neighborhood-homomorphism;   ð2Þ If f exists the inverse f , then f : ðV; f ðRU ÞÞ ! ðU; RU Þ is a neighborhood-homomorphism.

Proof.   (1) Suppose f ðRV Þ ¼ ff^ ðRV ÞjRV 2 RV g.

By Theorem 8(1) and (2), for each RV 2 RV ; f : ðU; ^f  ðRV ÞÞ ! ðV; RV Þ is both predecessor and successor neighborhoodcontinuous. Thus f : ðU; RU Þ ! ðV; f ðRU ÞÞ is a neighborhood-homomorphism. (2) The proof is similar to (1). h The following theorem describes reduct features between two information systems under the condition of neighborhoodhomomorphisms.

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Theorem 15. let ðU; RU Þ; ðV; RV Þ be two relation information systems and f be a mapping from U to V. Assume that each relation in RU is transitive, each relation in RV is reflexive, and f : ðU; RU Þ ! ðV; RV Þ is a neighborhood-homomorphism. Then,

P U # RU is a consistent set of RU if P V is a consistent set of RV ; where P V ¼ ff  RU 2 RV jRU 2 P U g is the collection of all corresponding relations in RV with respect to P U : Proof. We have: P U is a consistent set of RU () ð\P U Þp ðxÞ ¼ ð\RU Þp ðxÞ and ð\P U Þs ðxÞ ¼ ð\RU Þs ðxÞ for all x 2 U; P V is a consistent set of RV () ð\P V Þp ðyÞ ¼ ð\RV Þp ðyÞ and ð\P V Þs ðyÞ ¼ ð\RV Þs ðyÞ, for all y 2 V. Therefore, we need only to prove:

ð\P V Þp ðyÞ ¼ ð\RV Þp ðyÞ and ð\P V Þs ðyÞ ¼ ð\RV Þs ðyÞ; for all y 2 V ) ð\P U Þp ðxÞ ¼ ð\RU Þp ðxÞ and ð\P U Þs ðxÞ ¼ ð\RU Þs ðxÞ for all x 2 U: Suppose that f : ðU; RU Þ ! ðV; RV Þ is a neighborhood-homomorphism, then for each RU 2 P U ; f : ðU; RU Þ ! ðV; f  RU Þ is both predecessor and successor neighborhood-continuous.  Assume that f : ðU; RU Þ ! ðV; RV Þ is successor neighborhood-continuous and x 2 U. By Theorem 4, f ðRVs ðf ðx0 ÞÞÞ # RUs ðx0 Þ  for any x0 2 RUs ðxÞ, since RU is transitive, then RUs ðx0 Þ # RUs ðxÞ, which implies that f ðRVs ðf ðx0 ÞÞÞ # RUs ðxÞ for any x0 2 RUs ðxÞ.  We have \RV 2RV f ðRVs ðf ðx0 ÞÞÞ # \RU 2RU RUs ðxÞ for any x0 2 RUs ðxÞ. It follows that 

[x0 2RUs ðxÞ ð\RV 2RV f ðRVs ðf ðx0 ÞÞÞÞ # \RU 2RU RUs ðxÞ:

ð15aÞ

In the following, we will verify the inverse inclusion. For any x0 2 \RU 2RU RUs ðxÞ, that means x0 2 RUs ðxÞ for any RU 2 RU . with that fact that each relation in RV is reflexive, then    f ðx0 Þ 2 RVs ðf ðx0 ÞÞ, We have that x0 2 f ððf ðx0 ÞÞÞ and x0 2 f ðRVs ðf ðx0 ÞÞÞ. That means x0 2 \RV 2RV f ðRVs ðf ðx0 ÞÞÞ, which implies that 

\RU 2RU RUs ðxÞ # [x0 2RUs ðxÞ ð\RV 2RV f ðRVs ðf ðx0 ÞÞÞÞ:

ð15bÞ 

0

By (15a) and (15b), we have that \RU 2RU RUs ðxÞ ¼ [x0 2RUs ðxÞ ð\RV 2RV f ðRVs ðf ðx ÞÞÞÞ, that is  \RU 2RU RUs ðxÞ ¼ [x0 2RUs ðxÞ ðf ð\RV 2RV RVs ðf ðx0 ÞÞÞÞ for any x 2 U. It is equal to 

ð\RU Þs ðxÞ ¼ [x0 2RUs ðxÞ ðf ðð\RV Þs ðf ðx0 ÞÞÞÞ for any x 2 U: If P V is a consistent set of RV , then ð\P V Þs ðf ðxÞÞ ¼ ð\RV Þs ðf ðxÞÞ for any x 2 U.   By (15c), ð\RU Þs ðxÞ ¼ [x0 2RUs ðxÞ ðf ðð\RV Þs ðf ðx0 ÞÞÞÞ ¼ [x0 2RUs ðxÞ ðf ðð\P V Þs ðf ðx0 ÞÞÞÞ ¼ ð\P U Þs ðxÞ, that ð\P U Þs ðxÞ. We can similarly examine that ð\RU Þp ðxÞ ¼ ð\P U Þp ðxÞ. Then we can assert this conclusion. h.

ð15cÞ means

ð\RU Þs ðxÞ ¼

Example 3. Let U ¼ fx1 ; x2 ; x3 ; x4 ; x5 g; V ¼ fy1 ; y2 ; y3 g. RU ¼ fRU1 ; RU2 ; RU3 g and RV ¼ fRV1 ; RV2 ; RV 3 g are two relation sets on U and V, respectively.

RU1 ¼ fðx1 ; x2 Þ; ðx1 ; x3 Þ; ðx1 ; x5 Þ; ðx2 ; x2 Þ; ðx2 ; x3 Þ; ðx2 ; x5 Þ; ðx3 ; x2 Þ; ðx3 ; x3 Þ; ðx3 ; x5 Þ; ðx4 ; x2 Þ; ðx4 ; x3 Þ; ðx4 ; x5 Þ; ðx5 ; x3 Þ; ðx5 ; x2 Þ; ðx5 ; x5 Þg; RU2 ¼ fðx1 ; x1 Þ; ðx1 ; x2 Þ; ðx1 ; x3 Þ; ðx1 ; x5 Þ; ðx2 ; x2 Þ; ðx2 ; x3 Þ; ðx2 ; x5 Þ; ðx3 ; x2 Þ; ðx3 ; x3 Þ; ðx3 ; x5 Þ; ðx4 ; x2 Þ; ðx4 ; x3 Þ; ðx4 ; x5 Þ; ðx5 ; x3 Þ; ðx5 ; x2 Þ; ðx5 ; x5 Þg; RU3 ¼ fðx1 ; x1 Þ; ðx1 ; x2 Þ; ðx1 ; x3 Þ; ðx1 ; x5 Þ; ðx2 ; x2 Þ; ðx2 ; x3 Þ; ðx2 ; x5 Þ; ðx3 ; x2 Þ; ðx3 ; x3 Þ; ðx3 ; x5 Þ; ðx4 ; x2 Þ; ðx4 ; x3 Þ; ðx4 ; x4 Þ; ðx4 ; x5 Þ; ðx5 ; x3 Þ; ðx5 ; x2 Þ; ðx5 ; x5 Þg: RV1 ¼ fðy1 ; y1 Þ; ðy1 ; y2 Þ; ðy2 ; y2 Þ; ðy2 ; y3 Þ; ðy3 ; y3 Þg; RV2 ¼ fðy1 ; y1 Þ; ðy1 ; y2 Þ; ðy1 ; y3 Þ; ðy2 ; y2 Þ; ðy2 ; y3 Þ; ðy3 ; y3 Þg; RV3 ¼ fðy1 ; y1 Þ; ðy1 ; y2 Þ; ðy1 ; y3 Þ; ðy2 ; y2 Þ; ðy2 ; y3 Þ; ðy3 ; y2 Þ; ðy3 ; y3 Þg: Define f i : U ! V, i = 1, 2, 3, as follows:

f 1 ðxj Þ ¼ y1 for j ¼ 1; 3; 4; f 1 ðx2 Þ ¼ y2 ; f 1 ðx5 Þ ¼ y3 ; f 2 ðxj Þ ¼ y1 for j ¼ 1; 3; 4; f 2 ðx2 Þ ¼ y3 ; f 2 ðx5 Þ ¼ y2 ; f 3 ðxj Þ ¼ y1 for j ¼ 1; 3; 4; f 2 ðx2 Þ ¼ y2 ; f 2 ðx5 Þ ¼ y2 :

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Then by definition, we can check that f i : ðU; RUi Þ ! ðV; RVi Þ, i = 1,2,3, are all neighborhood-homomorphisms. We can see that fRV1 g; fRV1 ; RV 2 g; fRV 1 ; RV 2 ; RV3 g are three consistent sets of RV . Correspondingly, fRU1 g; fRU1 ; RU2 g; fRU1 ; RU2 ; RU3 g are three consistent sets of RU . Corollary 6. Let ðU; RU Þ; ðV; RV Þ be two relation information systems. Assume that a mapping f : U ! V exists its inverse  f : V ! U, each relation in RU is transitive, and each relation in RV is reflexive.  If f : ðV; RV Þ ! ðU; RU Þ is a neighborhood-homomorphism, then P V # RV is a consistent set of RV if P U is a consistent set of RU , where P U ¼ ff  RU 2 RU jRV 2 P V g is the collection of all the corresponding relations in RU with respect to P V . Proof. By the symmetry of U and V, and by Theorem 15, we can conclude it. h The following tells us that attribute reducts between an original system and its image system are equivalent to each other under the condition of neighborhood-homomorphisms. Theorem 16. Let ðU; RU Þ be a relation information system, f be a mapping from U to V, and f ðRU Þ ¼ f^f ðRU ÞjRU 2 RU g be a relation set on V, where ^f is the induce relation mapping by f. If f : ðU; RU Þ ! ðV; f ðRU ÞÞ is a neighborhood-homomorphism, then any P U # RU is a reduct of RU iff f ðP U Þ is a reduct of RV . Proof. We now prove that: f : ðU; RU Þ ! ðV; ^f ðRU Þ is successor neighborhood-continuous) f : ðU; RU Þ ! ðV; ^f ðRU Þ is successor neighborhood-consistent. Assume that f : ðU; RU Þ ! ðV; ^f ðRU Þ is successor neighborhood-continuous. For any x1 ; x2 2 U, we need only to prove that f ðx1 Þ ¼ f ðx2 Þ ) RUs ðx1 Þ ¼ RUs ðx2 Þ. Suppose that f ðx1 Þ ¼ f ðx2 Þ ¼ y for some y 2 V. By Proposition 4(3), for any y 2 V; ð^f ðRU ÞÞs ðyÞ ¼ [x2f  ðyÞ f ðRUs ðxÞÞ, that means   ð^f ðRU ÞÞs ðyÞ  f ðRUs ðx1 ÞÞ and ð^f ðRU ÞÞs ðyÞ  f ðRUs ðx2 ÞÞ. Thus f ðð^f ðRU ÞÞs ðyÞÞ  RUs ðx1 Þ and f ðð^f ðRU ÞÞs ðyÞÞ  RUs ðx2 Þ. On the other hand, sine f : ðU; RU Þ ! ðV; ^f ðRU Þ is successor neighborhood-continuous, by Theorem 2,   f ðð^f ðRU ÞÞs ðyÞÞ # RUs ðx1 Þ and f ðð^f ðRU ÞÞs ðyÞÞ # RUs ðx2 Þ:   We have f ðð^f ðRU ÞÞs ðyÞÞ ¼ RUs ðx1 Þ and f ðð^f ðRU ÞÞs ðyÞÞ ¼ RUs ðx2 Þ, which implies RUs ðx1 Þ ¼ RUs ðx2 Þ. We can also verify that f : ðU; RU Þ ! ðV; ^f ðRU Þ is predecessor neighborhood-continuous) f : ðU; RU Þ ! ðV; ^f ðRU Þ is predecessor neighborhood-consistent. Therefore we have that f : ðU; RU Þ ! ðV; ^f ðRU Þ is a neighborhood-homomorphism) f : ðU; RU Þ ! ðV; ^f ðRU Þ is a homomorphism. By Theorem 13, we can conclude it. h.

Example 4. Let U ¼ fx1 ; x2 ; x3 ; x4 ; x5 g; V ¼ fy1 ; y2 ; y3 g, and RU ¼ fRU1 ; RU2 ; RU3 g be a relation set on U.

RU1 ¼ fðx1 ; x2 Þ; ðx1 ; x3 Þ; ðx2 ; x3 Þ; ðx2 ; x5 Þ; ðx3 ; x2 Þ; ðx3 ; x3 Þ; ðx4 ; x2 Þ; ðx4 ; x3 Þg; RU2 ¼ fðx1 ; x2 Þ; ðx1 ; x3 Þ; ðx2 ; x3 Þ; ðx2 ; x5 Þ; ðx3 ; x2 Þ; ðx3 ; x3 Þ; ðx4 ; x2 Þ; ðx4 ; x3 Þ; ðx5 ; x2 Þg; RU3 ¼ fðx1 ; x2 Þ; ðx1 ; x3 Þ; ðx2 ; x3 Þ; ðx2 ; x5 Þ; ðx3 ; x2 Þ; ðx3 ; x3 Þ; ðx4 ; x2 Þ; ðx4 ; x3 Þ; ðx5 ; x5 Þ; g: Define f : U ! V as follows:

f ðxj Þ ¼ y1 for j ¼ 1; 3; 4; f ðx2 Þ ¼ y2 ; f ðx5 Þ ¼ y3 : We can obtain that:

^f ðRU1 Þ ¼ fðy ; y Þ; ðy ; y Þ; ðy ; y Þ; ðy ; y Þg; 1 1 1 2 2 1 2 3 ^f ðR Þ ¼ fðy ; y Þ; ðy ; y Þ; ðy ; y Þ; ðy ; y Þ; ðy ; y Þg; U2 1 1 1 2 2 1 2 3 3 2 ^f ðRU3 Þ ¼ fðy ; y Þ; ðy ; y Þ; ðy ; y Þ; ðy ; y Þ; ðy ; y Þg: 1

1

1

2

2

1

2

3

3

3

We can examine that fRU1 g and fRU2 ; RU3 g are two reducts of RU . Correspondingly, f^f ðRU1 Þg and f^f ðRU2 Þ; ^f ðRU3 Þg are two reducts of RV . 

Corollary 7. Let ðV; RV Þ be a relation information system and f be a mapping from U to V. If f exists its inverse f , denote  f ðRV Þ ¼ f^f  ðRV ÞjRV 2 RV g as a relation set on U, where ^f  is the relation mapping induced by f.    If f : ðV; RV Þ ! ðU; f ðRV Þ is a neighborhood-homomorphism, then any P V # RV is a reduct of RV iff f ðP V Þ is a reduct of RU .

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Proof. By Theorem 16, it holds. h Remark 5. By Theorem 15, neighborhood-homomorphisms can describe reduct features between two different information systems. By Theorem 16, neighborhood-homomorphisms can describe reduct invariance of an original system and an image system.

7. Conclusion In this paper, in order to compare granular knowledges between two information systems, we proposed a neighborhoodcontinuous function between information systems that was inspired by the concept of continuous function in topology. Moreover, based on neighborhood-continuous functions, we defined a neighborhood-homomorphism between different information systems. Our main results are as follows. Firstly, we gave some properties of neighborhood-continuous functions which could compare two classes of information granules and investigated relationships between neighborhood-continuous functions and several kinds of existing consistent functions. Secondly, some properties of neighborhood-continuous functions and neighborhood-homomorphisms under the condition of relation mappings were given in Theorems 8 and 14, respectively. In addition, based on neighborhood-continuous functions, the reduct feature of a single information system was described in Theorems 11 and 12, while based on neighborhood-homomorphisms, the reducts invariance between two different information systems was described in Theorems 15 and 16. It is attractive that, based on these theorems, the structures of neighborhood-continuous functions and neighborhoodhomomorphisms could both integrate with relation mappings, besides the reduct feature of a single information system and the reduct invariance between two different information systems could be characterized by neighborhood-continuous functions and neighborhood-homomorphisms, respectively. 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