On the generalization of fuzzy rough sets

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 13, NO. 3, JUNE 2005 343 On the Generalization of Fuzzy Rough Sets Daniel S. Yeung, Fellow, IEEE, Degang Chen, Eric C. C. Tsang, John W. T. Lee, Member, IEEE, and Wang Xizhao, Senior Member, IEEE Abstract—Rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty, it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets. The existing generalizations of fuzzy rough sets are all based on special fuzzy relations (fuzzy similarity relations, -similarity relations), it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches. In this paper, from the viewpoint of constructive approach, we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them, the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined. In axiomatic approach, we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms. The lattice and topological structures of fuzzy rough sets are also proposed. In order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets, a special lower approximation operator is applied to a fuzzy reasoning system, which coincides with the Mamdani algorithm. Index Terms—Approximation operators, completely distributive lattice, fuzzy rough sets, fuzzy topology, rough sets. I. INTRODUCTION HE concept of rough set was originally proposed by Pawlak [1] as a mathematical approach to handle imprecision, vagueness, and uncertainty in data analysis. This theory has amply been demonstrated to have its usefulness and versatility by successful applications in a variety of problems [6]–[8]. The theory of rough sets deals with the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximations. By using the concepts of lower and upper approximations in rough set theory, knowledge hidden in information systems may be unraveled and expressed in the form of decision rules [2]–[5]. Another particular use of rough set theory is that of attribute reduction in databases. Given a dataset with discretized attribute values, it is possible to find a subset of the T Manuscript received March 24, 2003; revised December 11, 2003 and August 16, 2004. The work of D. Chen was supported by Tianyuan Mathematics under Grant A0324613 and by the Liaoning Education Department under Grant 20161049. This work was also supported by the Hong Kong Research Grant Council under Grant B-Q826. D. S. Yeung, E. C. C. Tsang, and J. W. T. Lee are with the Department of Computing, The Hong Kong Polytechnic University, Hung Hom, Hong Kong. D. Chen is with the Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, P. R. China. W. Xizhao is with the Faculty of Mathematics and Computer Science, Hebei University, Baoding, China. Digital Object Identifier 10.1109/TFUZZ.2004.841734 original attributes that are the most informative. This leads to the concept of attributes reduction which can be viewed as the strongest and most characteristic results in rough set theory to distinguish itself from other theories. However, as mentioned in [32], in the existing databases the values of attributes could be both of symbolic and real-valued. The traditional rough set (TRS) theory will have difficulty in handling such values. There is a need for some methods which have the capability of utilizing set approximations and attributes reduction for crisp and real-values attributed datasets, and making use of the degree of similarity of values. This could be accomplished by combining fuzzy sets and rough sets, i.e., fuzzy rough sets [10]. Theories of fuzzy sets and rough sets are generalization of classical set theory for modeling vagueness and fuzziness respectively, it is generally accepted that these two theories are related but distinct and complementary with each other [9]–[12], [33]–[35]. Fuzzy rough sets encapsulate the related but distinct concepts of fuzziness and indiscernibility, both of which occur as a result of uncertainty in knowledge or data, thus a method employing fuzzy rough sets should be adopted to handle this uncertainty. There are at least two approaches for the development of the fuzzy rough set theory, the constructive and axiomatic approaches. In constructive approach, fuzzy relations on the universe is the primitive notion, the lower and upper approximation operators are constructed by means of this notion. Dubois and Prade [10] was one of the first researchers to propose the concept of fuzzy rough sets from the constructive approach, they constructed a pair of upper and lower approximation operators of fuzzy sets with respect to a fuzzy similarity relation by using the -norm Min and its dual conorm Max. Noticed that Min and Max are special -norm and conorm, Radzikowska and Kerre [13] presented a more general approach to the fuzzification of rough sets. Specifically, they defined a broad family of fuzzy rough sets with respect to a fuzzy similarity relation, each one of which is determined by an implicator and a -norm. On the other hand, the axiomatic approach takes the lower and upper approximation operators as primitive notions. In this approach, a set of axioms is used to characterize approximation operators. Moris and Yakout [14] studied a set of axioms on fuzzy rough sets. However, their works were restricted to fuzzy -rough set defined by fuzzy -similarity relations. The same approximation operators were also studied in [15]. By comparing with the constructive approach, the axiomatic approach aims to investigate the mathematical characters of fuzzy rough sets rather than to develop methods for applications. Another valuable generalization of rough set theory to fuzzy case is that the fuzzy neighborhood system can be viewed as a generalized approximation theory of fuzzy sets [36], [37]. 1063-6706/$20.00 © 2005 IEEE 344 Fuzzy rough sets have been applied to solve practicial problems such as being used in neural networks [26]–[28], medical time series [29], case generation [30], mining stock price [31], and descriptive dimensionality reduction [32]. For the purpose of making fuzzy rough set theory complete and further exploring its applications, it is necessary to conclude what have been mentioned in the literatures and present a unified framework for it. This will be presented in the first part of this paper which includes Sections III–V. As we mentioned before, the most important concept of rough set theory is the attribute reduction in databases. In the fuzzy rough set theory, less efforts have been put on the attribute reduction in fuzzy databases. In the crisp rough set theory [1], all definable (or observable) sets form a Boolean algebra of a partition, which is a “trivial” kind of -algebra, this statement is the theoretical foundation of the attribute reduction in databases. However, for the fuzzy rough set one may notice that the Boolean algebra would not be suitable since a fuzzy (not and . crisp) set does not satisfy On the other hand the fuzzy set theory always deals with infinite cases while the crisp rough set theory deals with finite cases. Here we suggest the completely distributive lattice to replace the Boolean algebra for the definable fuzzy sets. The second part of this paper, found in Section VI, is to study the lattice structure of fuzzy rough sets and to set up a theoretical foundation for our future work of developing algorithms for attributes reduction in fuzzy databases. The relationship between fuzzy rough set and fuzzy topology was firstly studied by Boixader in [15], they proved that the lower and upper approximation operators with respect to a fuzzy -similarity relation were fuzzy interior operator and fuzzy closure operator respectively. In [18], the fuzzy topology defined by a special approximation operator of fuzzy sets were studied and applied to fuzzy automata. For the fuzzy rough sets with respect to arbitrary fuzzy relations, it is worth investigating the sufficient and necessary conditions that the lower and upper approximation operators could be fuzzy interior operators [19] and fuzzy closure operators [19], respectively. This will be presented in the third part of this paper, found in Section VII. When the theories of our proposed fuzzy rough sets mentioned in the previous three parts have been established, we can present a unified framework for fuzzy rough sets theory and set up its mathematical foundation for extending its applications. In Section VIII, we will apply a special lower approximation operator to fuzzy reasoning. It is well known that the existing fuzzy reasoning algorithms were all based on Zadeh’s CRI rule, in which the Mamdamni algorithm was the most popular one. Our algorithm can be shown to be just equal to the Mamdamni algorithm for single input and single output fuzzy control systems. Our algorithm could also handle multiple inputs and single output fuzzy control systems. In the same section, we will also discuss other possible applications of generalized fuzzy rough sets. This paper is organized as follows. In Section II, first we recall basic notions of crisp rough sets; then we give definitions and properties of fuzzy logical operators; some former works on fuzzy rough sets are also listed and compared. In Section III, we define two upper and two lower approximation operators with IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 13, NO. 3, JUNE 2005 respect to an arbitrary fuzzy relation and study their properties. In Section IV, the relations between special fuzzy relations and fuzzy approximation operators are examined. In Section V, various classes of fuzzy approximation operators are characterized by different sets of axioms. In Section VI, we study the lattice structure of fuzzy rough sets. In Section VII, we study the fuzzy topological structure of fuzzy rough sets. In Section VIII, we apply a special lower approximation operator to a fuzzy reasoning system and discribe other possible applications. Finally, a conclusion is given. II. PRELIMINARIES A. Rough Approximations and Rough Sets Let denote a finite and nonempty set called the universe. is an equivalence relation on , i.e., is Suppose reflexive, symmetric, and transitive. The equivalence relation partitions the set into disjoint subsets. Elements in the same equivalence class are said to be indistinguishable. Equivalence classes of are called elementary sets. Every union of elementary sets is called a definable set [1]. The empty set is considered to be a definable set, thus all the definable sets form an Boolean is called an approximation space. Given an arbialgebra. , one can characterize by a pair of lower and trary set is the upper approximations. The lower approximation greatest definable set contained in , and the upper approximais the least definable set containing . They can tion be computed by two equivalent formulas The lower approximation and upper approximation satisfy the following properties: P1) P2) P3) P4) P5) P6) . From these six properties, one can obtain many properties of rough sets, we only list these six properties because they can be treated as axiomatic characteristics of rough sets. This was pointed out in [16] and [17], and the operator-oriented approach to rough sets was also proposed. In [16], some axioms were applied to present the axiomatic rough set theory when the universe is a general set. They had proved that if a pair of set operators satisfy their axioms (1 –6 ) which were adopted from the axioms of Kuratowski’s closure operator 1 ) ; 2 ) ; 3 ) ; YEUNG et al.: ON THE GENERALIZATION OF FUZZY ROUGH SETS 4 ) 5 ) 6 ) ; ; ; such that then there is an equivalence relation . Similar results were also obtained for neighborhood systems (a generalized rough set theory), so the results in [16] can be viewed as the beginning of an axiomatic rough set theory. The axiomatic rough sets were considered in more detail in [17] when the universe was finite. Supis a finite universe, an arbitrary binary relation on pose , then for every , the with respect to general lower and upper approximations of are defined as follows: It is pointed in [17] that if a pair of dual set operators satisfied then there exists a binary relation on . Furthermore, if such that satisfied , respectively, then there on exists a reflexive, symmetric, and transitive relation such that respectively. It can be summarized that P1), P2), and P3) are elementary for rough sets and P4), P5), and P6) correspond to the reflexivity, symmetry, and transitivity of relation , respectively. On the other hand, it is worth mentioning that there are more general approximation theories called neighborhood systems than the above mentioned generalized rough set theory. For details of neighborhood systems theory, we refer the readers to [36] and [37]. 345 It is easy to prove that if a -norm is lower semi-continuous, such that . A trianthen there exists gular conorm (shortly -conorm) is an increasing, associative, and commutative mapping that sat. Three isfies the boundary condition well-known continuous -conorms are • • • the standard max operator (the smallest -conorm [13]); the probabilistic sum the bounded sum ; . It is easy to prove that if a -conorm is upper semi-continsuch that .A uous, then there exists is a decreasing mapping satisfying negator and . The negator is usually referred to as the standard negator. A negator is called for all , every involuinvolutive iff tive negator is continuous and strictly decreasing [20]. Given a negator , a -norm and a -conorm are dual with respect iff De Morgan laws are satisfied, i.e., to . For every , where is the fuzzy power set will be used to denote fuzzy comon , the symbol determined by a negator , i.e., for every plement of . Given a triangular norm , the binary operation on , is is lower semicontincalled a -implicator based on . If uous, then is called the residuation implication of , or the -residuated implication. The properties of -residuated are listed as follows [14] ( is simplified as implication ). For all , we have and is monotone in the right argument; is antimonotone in the left argument; iff B. Fuzzy Logical Operators This subsection summarizes fuzzy logical operators found in [13], [14], [20], and [25]. A triangular norm, or shortly -norm, is an increasing, associative and commutative mapping that . The satisfies the boundary condition most popular continuous -norms are • • • (the the standard min operator largest -norm [13]); ; the algebraic product the bold intersection (also called the Lukasiewicz -norm) ; . For a -conorm , an operator is defined as . If and are dual with respect to an involutive negator , then and are dual with respect to the involutive negator , i.e., . 346 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 13, NO. 3, JUNE 2005 If is lower semicontinuous, then uous, we can get the dual properties of as follows: is upper semi-continby the properties of ; is monotone in the right argument; is antimonotone in the left argument; ; ; satplicator they mean a function , and isfying for every . Their lower and upper approximation operators were defined as for every They also refer to three special lower approximation operators with respect to three special border implicators called -implicator, -implicator and -implicator. The composition, duality, and interactions with union and intersections of fuzzy rough set were examined. In [14] the -similarity relation was used to define fuzzy rough sets. Suppose is a nonempty universe. By a -simthey mean a fuzzy relation on which ilarity relation , is , for every and . If is the -residuated implication of a lower semi-continuous -norm , then the lower and upper approximation operators were defined as for every ; ; ; ; ; ; ; . C. Fuzzy Rough Sets and Fuzzy Rough Approximations In Pawlak rough set theory [1], an equivalence relation is a key and primitive notion. For fuzzy rough sets, a fuzzy similarity relation is used to replace an equivalence relation. Let be a nonempty universe. A fuzzy binary relation on is called a fuzzy similarity relation if is reflexive , symand sup-min transitive ( metric , the similarity class with is a fuzzy set on defined by for all . The concept of fuzzy rough set was first proposed by Dubois and Prade [10], their idea was as follows. Let be a nonempty universe and a fuzzy binary relation on the fuzzy power set of . A fuzzy rough set is a pair of fuzzy sets on such that for every It was proved that the approximation operators the following properties: FP1) ; FP2) FP3) ; FP4) FP5) FP6) here and have ; In [13] the above Dubois and Prade fuzzy rough sets was generalized from Max, Min to a border implicator and a -norm with respect to a fuzzy similarity relation. By a border im- can be equivalently characterized by axioms ; ; ; ; . can be equivalently characterized by axioms ; ; ; ; . These axioms are not distinguishable in terms of their degree of importance to fuzzy rough sets. Some of them are not independent, i.e., some of them can not be independently applied to characterize the basic properties of the fuzzy relation . For example, can not characterize the -transitivity of , it as an additionally condition. Without need alone can not characterize the -transitivity of . Without and , both and can not ensure a fuzzy relation such that . These have been clearly indicated in the Proof of Theorem 4.5 found in [14]. The fuzzy rough sets presented in [13] and [14] are closely related and they produce similar upper and lower approximation operators. The difference is that in [13] for every -norm they all use the same fuzzy similarity relation while in [14] they match a -similarity relation for every -norm . The abovementioned generalizations of fuzzy rough sets can be summarized by the following three characteristics: 1) They are defined by means of different special fuzzy relations (fuzzy similarity relation or fuzzy -similarity relation) respectively, so a unified framework for fuzzy rough sets has not been developed. 2) Their methods to define the upper approximation operator are similar, roughly speaking, there is only one kind of upper approximation operator. YEUNG et al.: ON THE GENERALIZATION OF FUZZY ROUGH SETS 3) In [14] axioms of fuzzy approximation operators guarantee the existence of fuzzy -similarity relations that produce the same operators, these axioms are not distinguishable in terms of their degree of importance to fuzzy rough sets and some are not independent. According to 1), a natural extension of the existing approaches is to consider fuzzy rough sets which are defined relatively to arbitrary fuzzy binary relations. In the crisp case, this problem was broadly discussed in the literature [17]. By 2) it leads us to consider other kind of upper approximation operators. From 3) we should distinguish which axioms are primitive for the approximation operators and which axioms guarantee the existence of special fuzzy relations. The main purpose of the present paper is to construct two pairs of lower approximation operators and upper approximation operators respectively by the constructive approach and characterize them by some axioms using the axiomatic approach, thus we can set up a unified framework for fuzzy rough sets theory which is of both theoretical and practical importance. For example, the open problem concerning a complete operator-oriented characterization of Lukasiewicz fuzzy rough sets proposed in [13] will be solved completely by our axiomatic approach, and a special lower approximation operator can be used to develop a fuzzy reasoning algorithm. III. APPROXIMATION OPERATORS WITH RESPECT TO AN ARBITRARY FUZZY RELATION In this section we assume and to be a lower semi-continuous -norm and an upper semi-continuous -conorm respectively and they are dual with respect to an involutive negator . and are defined as in Section II-B. The main content of this section is to define approximation operators of fuzzy sets with respect to the above logic operations, study the relations among them and investgate their basic properties such as the distributive properties. Suppose is a nonempty universe (may not be finite), an arbitrary fuzzy relation on , we define the following approximation operators for every fuzzy set , 1) -upper approximation operator: . 2) -lower approximation operator: . 3) -upper approximation operator: . 4) -lower approximation operator: . and are the generalizations of approximation Obviously operators in [14]. In [13] for every -norm they all use the same fuzzy similarity relation, here is an arbitrary fuzzy relation, so by this means is the generalization of lower approximation operator with respect to a -implicator in [13], is the generalization of lower approximation operator with respect to a -implicator in [13], is the generalization of upper approximation operator in [13]. is a new definition. First we study the relations among them. 347 Proposition 3.1: For every ments hold. 1) ; 2) Proof: 1) For any , the following state- . It is similar to 1) since and are dual with respect to . 2) The above proposition shows that and and are dual with respect to the involutive negator . Generally and and are not dual with respect to the involutive negator , but they satisfy the following proposition. For every , we denote to be the fuzzy sets of given by to be the fuzzy sets of given by to be the fuzzy set of given by to be the fuzzy set of given by , and to be the fuzzy set . of given by Proposition 3.2: For every and , the following statements hold: 1) ; 2) . Proof: For every 1) 348 2) IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 13, NO. 3, JUNE 2005 For every Proposition 3.3: For every following statements hold: 1) and , the 2) By these three propositions, the connections among , and are clear. In the following we study their basic properties. , Proposition 3.4: For any have the following properties: and 1) ; 2) ; Proof: 1) For each 2) Hence, we complete the proof. By Propositions 3.1 and 3.2, we have the following obvious and results which present the connections between and . For each YEUNG et al.: ON THE GENERALIZATION OF FUZZY ROUGH SETS 349 properties by these basic properties such as the monotone propand erties. It is worth noting that the dualities of and are only used in the Proofs of Propositions 3.1–3.3, this means that Propositions 3.4 and 3.5 have no relation with the dualities and . Propositions 3.1–3.5 will be used as basic of and axioms of approximation operators in Section 5. Another thing we should mention is that some properties of this section such as the distributive properties with respect to union and intersection have also been studied in [11], [35], [36], [39] under a general framework of neighborhood systems. Proposition 3.4 indicates that these approximation operaters are all distributive. Proposition 3.5: For every , the following statements hold, ; 1) 2) . 1) Proof: For each IV. CONNECTIONS BETWEEN APPROXIMATION OPERATORS AND SPECIAL FUZZY RELATIONS The fuzzy rough sets with respect to fuzzy similarity relation [13] or -similarity relation [14] had been proved to have many properties, but these properties were not distinguishable in terms of their degree of importance to the fuzzy rough sets. As what we have studied in the previous section, some of them are basic while others may be relative to special fuzzy relations. The main purposes of including this section are to examine the relationships between special properties and special fuzzy relations. In this section we also assume and to be a lower semi-continuous -norm and an upper semi-continuous -conorm respectively and they are dual with respect to an involutive negator . and are defined as in Section 2.2. To begin with, we first introduce a useful lemma. Lemma 4.1: Suppose is a fuzzy relation on , then for every 1) ; 2) Proof: 1) For 2) 2) For each every . The left part follows 1) of Proposition 3.2 For every . The left part follows 2) of Proposition 3.2. The following Theorem 4.1–4.3 present the relationships between special properties of approximation operators and reflexivity, symmetry and transitivity of fuzzy relation respectively. Theorem 4.1: Suppose is a fuzzy relation on , then the following statements are equivalent. for ; 3) ; 4) ; 1) is reflexive; 2) 5) . Proof: If is reflexive, then for every . We have If for holds, for every , then we have The above Propositions 3.1–3.5 are the basic properties of our four approximation operators. Certainly we can get more Hence 1) 2). let 350 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 13, NO. 3, JUNE 2005 If 2) holds, by 1) of Proposition 3.2 we have Proof: Suppose is -transitive. For each , we have If 3) holds, by 1) of Proposition 3.2 we have Hence 2) 3). 4), 3) 5) follow from Proposition 3.1. 2) is reflexive, then for every , each pair of If , and will be called a , and fuzzy rough set respectively. Theorem 4.2: Suppose is a fuzzy relation on , then for the following statements are equivalent. is symmetric; 1) 2) ; 3) ; ; 4) 5) . Proof: It follows from Lemma 4.1. Lemma 4.2: Suppose is a fuzzy relation on , then , the following statements hold: for 1) ; and 2) . Proof: 1) For each 2) Suppose 2) holds. For each we have 1) , let , . Hence, 2) holds. Suppose 3) holds. By 1) of Proposition 3.2, we have Hence, 3) 2) 1) holds. Suppose 1) holds. For every For each Theorem 4.3: Suppose is a fuzzy relation on , then for the following statements are equivalent: 1) is -transitive; 2) ; 3) ; 4) ; and 5) . Hence, 1) 3) holds. 2) 4), 3) 5) follows from the dualand , respectively. ities of and First it should be pointed out that 1) of Lemma 4.1 and Lemma 4.2 have been proved in [14] when is a -similarity relation. and are not the key Another thing is the dualities of and property for Theorems 4.1–4.3. If we do not use their dualities we can also prove these theorems. When we consider the approximations of fuzzy sets with respect to a norm , we match the same norm to characterize the -transitivity of , this is different from the ones in [13] where they match the same Minsimilarity relation for every . Theorems 4.1–4.3 propose the deep connections among special properties of approximation operators and special fuzzy relations. By Theorems 4.1–4.3, we have the following theorem. YEUNG et al.: ON THE GENERALIZATION OF FUZZY ROUGH SETS Theorem 4.4: Suppose is a fuzzy relation on lowing statements are equivalent. is a -similarity relation; 1) 2) ; 3) 351 . The fol- Suppose the operator define a fuzzy relation as then if , then we have satisfies 1) and 2). With . For each , , , so ; 4) ; 5) . Now, we know that the properties in Propositions 3.1–3.5 are basic for the approximation operators, and the properties of approximation operators in Theorems 4.1–4.3 correspond to special fuzzy relations. By combining them together, we can get other properties of approximation operators with respect to a -similarity relation, we list them as follows. is a -similarity relation, then Theorem 4.5: Suppose , and have the the approximation operators following properties. 1) . . 2) , and are monotone. 3) All of 4) . 5) . The statements with respect to and are proved in [14], and the statements with respect to and can be proved by the -dualities of and and . Hence, For every . , we have which implies . Theorem 5.2: Let be a lower semicontinuous -norm, and be a fuzzy set operator, then there exists if and only if a binary fuzzy relation such that satisfies Proof: By Propositions 3.4 and 3.5 is clear. Suppose satisfies 1) and 2). By using define a fuzzy relation on as . For each , if we have , we V. AXIOMATIC APPROACHES OF FUZZY ROUGH SETS In crisp rough set theory, the axiomatic approaches of approximation operators had been studied in details. However, in fuzzy rough set theory, less efforts have been put on studying the axiomatic approaches. In [14], some axioms were proposed to characterize upper and lower approximation operators of fuzzy sets with respect to a -similarity relation, but they are not distinguishable in terms of their degree of importance and are not independent. This section focuses on the axiomatic characterizations of , and by some independent axioms. In Theorems 5.1–5.4, first we present the axioms for each approximation operator that guarantee the existence of a fuzzy relation which produces the same operator. Theorem 5.1: Let be an upper semi-continuous -conorm, an involutive negator, and be a fuzzy set operator, then there exists a fuzzy binary relation such that if and only if satisfies Proof: By Propositions 3.4 and 3.5 is clear. so , hence For . every , we have which implies . Theorem 5.3: Let be a lower semicontinuous -norm, the -residuated implication, and be a fuzzy set operator, then there exists a binary fuzzy relation such that if and only if satisfies Proof: By Propositions 3.4 and 3.5 is clear. Suppose satisfies 1) and 2). By using , we define a fuzzy relation on as 352 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 13, NO. 3, JUNE 2005 , then by 2) we for any . , if , we have , and have For each so . For every , we have which implies . Theorem 5.4: Suppose is an upper semicontinuous -conorm, an involutive negator, and is defined as in be a fuzzy set operator, Section II-B. Let then there exists a binary fuzzy relation such that if satisfies and only if . Proof: By Propositions 3.4 and 3.5 is clear. Suppose satisfies 1) and 2). By using , we define a fuzzy relation on as , then , for any . , if , we have For each Proof: and 5.2. Suppose have is clear by Proposition 3.1, and Theorems 5.1 and satisfies 1). Let us define , and , then by Theorems 5.1 and 5.2 we and . By 1) we have , and . Let . This hence, completes the proof. Similarly, we can get the proof when and satisfy 2). Theorem 5.6: Suppose is a lower semicontinuous -norm, is an upper semicontinuous -conorm and are dual with respect to an involutive negator , and and are defined as in be fuzzy set operators. Section II. Let satisfies 1) and 2) and satisfies 1) and 2), If then there exists a binary fuzzy relation such that and if and only if and satisfies one of Proof: and 5.4. Suppose is clear by Proposition 3.1 and Theorems 5.3 and 5.3 and 5.4, we have and Proposition 3.1 we have so and . For every , we have which implies . Using Theorems 5.1–5.4, we have proposed the axiomatic approaches for the lower and upper approximation operators respectively and every approximation operator is characterized by two axioms. In Theorems 5.5–5.8, we set up the connections among these approximation operators. First, we provide a useful lemma. Lemma 5.1: Let be two fuzzy relations on , then if and only if for any , one of the statements ; 2) ; 3) ; and 4) holds: 1) . The proof follows from Lemma 4.1. Theorem 5.5: Suppose is a lower semicontinuous -norm, is an upper semicontinuous -conorm and are dual with respect to an involutive negator . Let be fuzzy set operators. If satisfies 1) and 2), and satisfies 1) and 2), then there exists a binary fuzzy relation such that and if and only if and satisfy one of the following statements: satisfies and 1). We define and , then by Theorems . By 1) , hence and . Let . This completes the proof. Similarly, we can get the proof when and satisfy 2). Theorem 5.7: Let be an upper semicontinuous -conorm, an involutive negator, be fuzzy set satisfies 1) and 2), and satisfies 1) operators. If and 2), then there exists a binary fuzzy relation such that and if and only if and satisfy one of the following statements: Proof: is clear by Proposition 3.2 and Theorems 5.1 and 5.4. Suppose and satisfy 1). We define , then by Theorems 5.1 and 5.4 we have and . By 1) and Proposition 3.2 we have , hence and . Let . This completes the proof. Similarly, we can get the proof when and satisfy 2). Theorem 5.8: Let be a lower semicontinuous -norm and its residuation implication, be fuzzy YEUNG et al.: ON THE GENERALIZATION OF FUZZY ROUGH SETS set operators. If satisfies 1) and 2), and satisfies 1) and 2), then there exists a binary fuzzy relation such and if and only if and satisfy that one of the following statements: Proof: and 5.3. Suppose is clear by Proposition 3.2 and Theorems 5.2 and satisfies 1). We define then by Theorems 5.2 and 5.3 we have and . By 1) and Proposition 3.2, we have , hence . Let . This completes the proof. Similarly, we can and satisfy 2). prove that By Theorems 5.5–5.8, we have the following conclusion. Theorem 5.9: Suppose is a lower semicontinuous -norm, is an upper semicontinuous -conorm and they are dual with respect to an involutive negator , and are defined as in be fuzzy Section II-B. Let satisfies 1) and 2), satisfies 1) set operators, If and 2), satisfies 1) and 2) and satisfies 1) 2), then there exists a binary fuzzy relation such that and and if and only if one of the following two statements hold. 1) . 2) . The following Theorem 5.10–5.13 present the axiomatic characterizations of approximation operators with respect to special fuzzy relations. Theorem 5.10: Suppose is a lower semicontinuous -norm, is an upper semicontinuous -conorm and they are dual with respect to an involutive negator . Let be a fuzzy set operator satisfying 1) and 2), then the following statements hold. 1) There exists a reflective fuzzy relation such that if and only if . 2) There exists a symmetric fuzzy relation such that if and only if . 3) There exists a -transitive fuzzy relation such that if and only if . The proof follows immediately from Theorems 5.1, 4.1–4.3. Theorem 5.11: Let be a lower semicontinuous -norm, and be a fuzzy set operator satisfying 1) 2), then the following statements hold. and 1) There exists a reflective fuzzy relation such that if and only if . 353 such that There exists a symmetric fuzzy relation if and only if . 3) There exists a -transitive fuzzy relation such that if and only if . The proof follows immediately from Theorems 5.2 and 4.1–4.3. Theorem 5.12: Let be a lower semicontinuous -norm, the -residuated implication, and be a fuzzy set operator satisfying 1) and 2), then the following statements hold. such that 1) There exists a reflective fuzzy relation if and only if . such that 2) There exists a symmetric fuzzy relation if and only if . 3) There exists a -transitive fuzzy relation such that if and only if . The proof follows immediately from Theorems 5.3 and 4.1–4.3. is a lower semicontinuous Theorem 5.13: Suppose -norm, is an upper semi-continuous -conorm and they are dual with respect to an involutive negator is defined as in be a fuzzy set operator Section II-B. Let satisfying 1) and 2), then the following statements hold. 1) There exists a reflective fuzzy relation such that if and only if . 2) There exists a symmetric fuzzy relation such that if and only if 2) . There exists a -transitive fuzzy relation such that if and only if . The proof follows immediately from Theorems 5.4 and 4.1–4.3. In the concluding remarks of [13], the authors proposed an open problem concerning a complete operator-oriented characterization of Lukasiewicz fuzzy rough approximations deterwhere (the mined by is the Lukasiewicz -norm) and . It was also pointed out in [13] residuation implication of that was an -implicator based on and , which means . If is a fuzzy similarity relathat tion on , then the upper approximation operator with respect by the definition in [13] is to 3) By the definition in [13], the lower approximation operator with is respect to An example was constructed in [13] to indicate that it was by possible not to produce a pair of fuzzy set operators any fuzzy relation even they satisfy Lin and Liu’s axioms [16]. By the axiomatic approach in this section, we have the operatororiented characterization of Lukasiewicz fuzzy rough approxas the following imations operators determined by theorem. are two Theorem 5.14: Suppose fuzzy set operators, then there exists a fuzzy similarity relation 354 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 13, NO. 3, JUNE 2005 such that and satisfy the following axioms. 1) . 2) if and only if . and Example 5.2: Let be defined as We , so be an infinite universe, also have , and . . 3) 4) . . 5) then it is clear . By Theorems 5.2, 5.3, 5.11, and 5.12, we know there exsuch that ists a reflexive and symmetric fuzzy relation and if and only if and satisfy is defined as axioms 1)–4), where . It is clear that axiom 5) is equivalent to the sup-min transitivity of , thus we have the operator-oriented characterization of Lukasiewicz fuzzy rough . It should approximations operators determined by and satisfy axioms 1)–4) and axiom be noticed that if and , then the above defined is a -transitive relation and not a sup-min transitive cannot be characterized by the relation, so axioms proposed in [16] and [17]. In the crisp rough set theory [16], [17] the lower approximation operator and the upper approximation operator are just required to satisfy and , respectively. But for the fuzzy case, is not equivalent to and is not equivalent to even the universe is finite. So, if a fuzzy set operator just satisfies finite distributive property it is possible that this operator cannot be produced by a fuzzy relation. This implies the necessarity of and in the definition of our lower approximation operator and the upper approximation operator . Here we do not mean to attribute the and ’s distributive properties with respect to union and intersection (respectively) to the finite-ness of the universe, but we only want to show the difference between finite and infinite distributive properties of and in the fuzzy case and give some examples of fuzzy set operators which can not be produced by fuzzy relations. Let us observe the following examples. be Example 5.1: Let defined as and satisfy . 2) just Recall in the proof of Theorem 5.1 axiom needs to hold for every . So, for the operator in this example we have since , and we also have since . Thus, satisfies 2) for every . At the mean time, in the proof of Theorem 5.2 2) just needs to hold for every . Similarly, to the axiom, in this example satisfy case of we can have the operator 2) for every . For every , we have since . We also have which implies , hence we have and satisfy 2) in Theorem 5.3. Similarly, to the case of we can prove that satisfies 2) in Theorem 5.4. On the other hand, we have and , so . We also have and , so . , an involuFor every fuzzy binary relation and an upper semicontinuous -conorm tive negator , suppose , then . 1) If and there exists such that , then . If we take , then . We have 2) , so Suppose . and for every . Since is upper semisuch that continuous, there exists . Let us take such , then and that , so . 3) then it is clear and satisfy On the other hand, , and , so If we have , then , for any , , so . Hence, for every fuzzy binary relation , every involutive negator and every upper semicontinuous -conorm holds. For every residuation implication , suppose , let us take such that , then YEUNG et al.: ON THE GENERALIZATION OF FUZZY ROUGH SETS and , so . For every fuzzy binary relation and a lower semicon. tinuous -norm , suppose 1) and there exists Suppose such that . If we take , then . We have , so . 2) Suppose and for every . Since is lower semicontinuous, there exists such that . Let us take such that , then and , so . 3) If , for any , we have , . so and lower semiHence, for every fuzzy binary relation . For every with continuous -norm , we have and involurespect to an upper semicontinuous -conorm , if we take tive negator , suppose such that , then and , so . The previous example proposes a pair of operators which can not be produced by a fuzzy relation. In the following example, we will provide another example which can not be produced by a fuzzy relation. Example 5.3: Suppose is a nonempty universe. For every , if , we define . If , then . For any , we define define , then and can not be produced by a fuzzy relation. Otherwise, we assume that there exists a fuzzy relation and a lower semi-continuous -norm such that , since for every , we know is reflexive, so for any . and , then , so the above But if assumption could not be true. VI. LATTICE STRUCTURES OF FUZZY ROUGH SETS For the preliminaries of lattice theory, we refer the readers to the Appendix. The main purpose of this section is to determine which kind of fuzzy sets are elementary to approximate other fuzzy sets and offer a lattice structure for these elementary fuzzy sets. In this section, we always assume and to be a lower semicontinuous -norm and an upper semicontinuous -conorm, respectively, and they are dual with respect to an involutive negator . and are defined as in Section II-B, is a -similarity relation. By Theorem 4.5, we know and . Let , we have the following theorem. and are CCD lattices and for Theorem 6.1: any if and only if . 355 and are subsets of and Proof: Since is a CCD lattice, to prove and are CCD lattices we only need to prove they are complete. If , then and hold by Proposition 3.4. This completes the proof of complete. The completeness of can be proved simness of ilarly. For any , so if and . only if Corollary 6.1: By Theorem 6.1, we know the invariant fuzzy sets with reand and possess the same lattice strucspect to ture, respectively. In the following, we study the structures of and . First, we begin with . It is well known that for any and , we have . Let , then for any is the join of some ele, we have the following theorem. ments in Theorem 6.2: Every element in is a join-irreducible el. ement of , if Proof: For every , then we have and . Since , we have . , then , thus , Assume and is a join-irreducible element of hence . It is naturally desirable to have every join-irreducible element belong to , but this is not really true. Indeed there of may exist a join-irreducible element of which is not in . Let us observe an example after presenting a lemma. Lemma 6.1: For every , we have: ; and 2) if and only if 1) . Proof: 1) . , then we have 2) If . , then for every If . Similarly we can prove . Example 6.1: Let be an infinite by set and a fuzzy relation defined on for every and . It is clear that is a fuzzy Min-similarity relation. by . Then, for every , Let us denote and by Lemma 6.1, we have if since , so and . Thus, we have a chain . Let , it is clear and for every . For every , suppose , if , so ; if , then then and , hence . 356 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 13, NO. 3, JUNE 2005 If and , suppose . Since and , there exists such that when holds. Otherwise, there exists an infinite such that which implies subsequence . If , , then there exists such that suppose holds. If we assume , then when when which implies , it is a contradiction, hence holds which implies is a join-irreducible element. Theorem 6.3: If is a finite set, then every join-irreducible . element belongs to , let Proof: For every join-irreducible element , then there exists such that . Let then then , and is a finite set by Lemma 6.1 we have . Since , and which implies . , let , similarly For to we can get every element in to be the union and every element in to be a joinof some elements in irreducible element. For two different -similarity relations and , we have the following theorem. and are two different -simiTheorem 6.4: Suppose larity relations, then the following statements are equivalent. ; 2) ; and 3) for each 1) . and , then for every Proof: If so , so which implies , for every , since If is equivalent to 2). , for each If If for each for each , we have and . , we have , so for every . Hence, 1) , we have , then , so . Hence, 2) is equivalent to 3). Similarly, we have the following theorem. and are two different -simiTheorem 6.5: Suppose larity relations, then the following statements are equivalent. 1) ; 2) ; and 3) for each . By Theorems 6.4 and 6.5, we know for two different -simand , if , then a -definable ilarity relations and is also a -de( -definable) fuzzy set with respect to . This statefinable ( -definable) fuzzy set with respect to ment implies that a smaller fuzzy relation may approximate the fuzzy sets more precisely, so when consider the aggression of -fuzzy relations the aggression operator Min may be a reasonable choice. In the crisp rough set theory [1], a set is called definable if its lower and upper approximations are equal. A set is definable if and only if its complement is definable. Every equivalence class is definable. All the definable sets form a Boolean algebra which is generated by all the equivalence classes, and this is the foundation of the attribute reduction of databases. For the fuzzy case, it is quite different. In our study, we can define two kind of definable fuzzy sets, one is called -definable and their collection is , while the other is called -defin. A fuzzy set is -definable if able and their collection is and only if its dual with respect to is -definable. It is clear that the Boolean algebra is not able to characterize the structures of -definable sets and -definable sets. So we propose the CCD lattice in this section for this purpose. It should be mentioned that the Boolean algebra is also a special CCD lattice. In rough set theory as well known, attributes reductions of information systems keep every definable set invariant while relative reductions of decision systems keep the lower approximations of equivalence classes of the decision attributes invariant. In this section, we present which kind of fuzzy sets can be applied to approximate other fuzzy sets as elementary granules and offer a suitable algebra structure for them, thus results in this section is the mathematical foundation to develop algorithms for the reduction of fuzzy databases which is our future work. VII. RELATIONSHIPS BETWEEN APPROXIMATION OPERATORS AND FUZZY TOPOLOGIES In the crisp rough set theory, the relationships between rough sets and topological space have been studied in detail. Suppose is a universe, an arbitrary binary relation on , in [22] and [23] it is proven that is reflexive is and transitive if and only if a Kuratowski saturated close operator on (a Kuratowski closure operator on is called saturated if the usual is replaced by requirement , for ), thus the crisp rough set can introduce a special topological space. For the fuzzy rough sets, in [14] it was pointed out that the upper approximation operbelongs to a very special subclass of the fuzzy closure ator operators of the class of fuzzy topological spaces called “fuzzy -neighborhood spaces [24],” and the lower approximation operator belongs to a very special subclass of the fuzzy interior operators of the class of fuzzy topological spaces called “fuzzy -locality spaces [25],” here is a lower semi-continuous triangular norm and is a -similarity relation. It is also YEUNG et al.: ON THE GENERALIZATION OF FUZZY ROUGH SETS pointed out in [15] that the above and were fuzzy closure operator and fuzzy interior operator in Lowen’s sense [19] and define two respectively when is continuous and different fuzzy topologies. But the fuzzy interior operator with and the fuzzy closure operator with respect to respect to are not presented, the converse problem, i.e., give an arbitrary fuzzy topological space, under what conditions that this fuzzy topology can be induced by approximation operators, are also not studied. Another thing we should point out here is that in is [18], a fuzzy approximation operator defined as , here is an arbitary crisp relation on . It is easy to prove that is a special when is a crisp relation and . It is case of our is a fuzzy Kuratowski saturated close opproven in [18] that erator if and only if is reflexive and transitive, and the fuzzy topology defined by have been applied to fuzzy automata. So the study on the relationship between approximation operators and fuzzy topologies is both of theoretical and practical importance. The purpose of this section is to study the relationships between approximation operators and fuzzy topologies in detail by using our constructive and axiomatic approaches of fuzzy rough sets. It will be shown in this section that for a fuzzy relation being reflexive and transitive is enough to ensure the approximation operators to be fuzzy closure operators and fuzzy interior operators respectively. For the preliminaries of fuzzy topology theory we refer the readers to the Appendix. By our constructive approaches of approximation operators of fuzzy sets we can have the following theorems. Theorem 7.1: Let be a lower semicontinuous -norm, a fuzzy relation on , then the following statements are equivalent. is a fuzzy closure 1) is reflexive and -transitive; 2) is a fuzzy interior operator. operator; and 3) Theorem 7.2: Let be an upper semicontinuous -conorm, an involutive negator, is the dual -norm of with respect a fuzzy relation on , then the following statements are to equivalent. 1) is reflexive and -transitive; 2) is a fuzzy interior operator; and 3) is a fuzzy closure operator. For the proof of the previous theorems it is only necessary to point out that the reflexity of is enough to prove . Thus, by and we can define two fuzzy topologies, one is , and the other is . By and we can also define two fuzzy topologies, one is , and the other is . Both and are fuzzy Kuratowski saturated closure operators. The fuzzy closure operators with respect to and are also fuzzy Kuratowski saturated closure operators, so all of , and are special fuzzy topologies, i.e., they are closed under the operation of infinite intersection of fuzzy sets. Generally these fuzzy topologies are not equal to each other. If and are dual to , then the fuzzy closure operator with respect to is , the fuzzy closure operator with is , thus we have and . respect to 357 Furthermore, if is a -similarity relation, then and , thus and are dual to , i.e., every open fuzzy set in one fuzzy topology is a closed set with respect to another fuzzy topology. On the other hand, the axioms of fuzzy interior operator and fuzzy closure operator can not guarantee the existence of a reflexive and transitive fuzzy relation that produces the same operators since the fuzzy topologies defined by fuzzy interior operator and fuzzy closure operator are just required to be closed under the operation of finite intersection of fuzzy sets. By our study on the axiomatic approaches of approximation operators of fuzzy sets, we have obtained the following theorems. Theorem 7.3: Let be a fuzzy interior operator, an upper semicontinuous -conorm, is the dual -norm of with respect to , then there exists a reflexive, and -transitive fuzzy relation such that if and only if satisfies. ; (2) (1) . Theorem 7.4: Let be a fuzzy interior operator, a lower semicontinuous -norm, then there exists a reflexive, and -transuch that if and only if sitive fuzzy relation satisfies. ; (2) (1) . Theorem 7.5: Let be a fuzzy closure operator, an is the dual -norm of upper semicontinuous -conorm, with respect to , then there exists a reflexive, and -transuch that if and only if sitive fuzzy relation satisfies: 1) ; and 2) . Theorem 7.6: Let be a fuzzy closure operator, a lower semicontinuous -norm, then there exists a reflexive and -tranif and only if satsitive fuzzy relation such that ; and 2) isfies: 1) . VIII. APPLICATION TO FUZZY REASONING In this section, we use a special lower approximation operator to develop a fuzzy reasoning algorithm for a single input and single output fuzzy control system and compare it with the well-known Mamdani algorithm. At the end of this section, we discuss possible applications of our generalized fuzzy rough sets to some practial problems. The purpose of including this section is to demonstrate that our proposed fuzzy rough set theory, which generalizes the fuzzy similarity relation in the existing fuzzy rough sets to an arbitrary fuzzy relation, can have wider range of applications than the existing fuzzy rough sets. First, we have to define the approximation operators between two different universes and . Suppose the fuzzy relation is defined on and are a lower semicontinuous -norm and an upper semicontinuous -conorm, respectively, and they are dual with respect to an involutive negator . and are defined as in Section II-B, for every and , we have the following. 1) -upper approximation operator: . 358 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 13, NO. 3, JUNE 2005 Fig. 1.Membership functions of A . B Fig. 2. Membership functions of Fig. 3. Membership function of A . . -lower approximation operator: . -upper approximation operator: 3) . 4) -lower approximation operator: . , let , then the aforementioned If approximation operators are just the ones defined in Section III . with respect to . Suppose and are defined as shown in is , they form inferFigs. 1 and 2, the center point of ence rules: if is then is ( may not be equal to ). By Mamdani algorithm the fuzzy implication relation is , here is a general fuzzy relation dewithout any special properties. If a fuzzy rough fined on set is developed based on a fuzzy similarity relation, then it can have limited applications in fuzzy reasoning systems, e.g., since the previous implication relation is not a fuzzy similary relation, the existing fuzzy rough sets cannot handle Mamdani algorithm with this fuzzy implication relation . However, our proposed fuzzy rough sets can handle this kind of Mamdani algorithm. This is one of the important reasons why we generalize the fuzzy similarity relation to an arbitrary fuzzy relation in this paper in order for it to have wider applications. For any fuzzy , the inference result can be comset 2) according to puted by the CRI rule, this is just the Min-upper approximation operator with respect to . For an input , its usual fuzzification is If otherwise as shown in Fig. 3, by the centroid defuzzifizer its output is . If we fuzzify to the triangular fuzzy number , , then we can compute as shown in Fig. 4, and let the lower approximation of by Max-lower approximation operator YEUNG et al.: ON THE GENERALIZATION OF FUZZY ROUGH SETS Fig. 4. 359 Membership function of A . Let defuzzifizer, we obtain the output of , by using the centroid as It is easy to prove that , so , i.e., our fuzzy reasoning we have algorithm produces the same results as those produced by Mamdani algorithm. In the Mamdani algorithm, if we fuzzify to the triangular fuzzy number , then the output is , here , . In this section, we only apply the and generally -conorm Max. Certainly other logical operators can also be used for fuzzy reasoning and more new algorithms for fuzzy control systems could be developed by using our proposed fuzzy rough sets theory. As mentioned before, the generalized fuzzy rough sets can easily be generalized to the case of two different universes, so it may have applications to this case not just limited to fuzzy reasoning. On the other hand, when dealing with similar degree between objects, we require the fuzzy similarity relation ( -similarity relation) to have three basic properties: reflexity, symmetry, and transitivity ( -transitivity). However, in some real world situations the propagation of similarity does not hold and the transitivity property is not required. For example, to a certain degree a Sphinx is half similar to a human being and half similar to a lion, but a human being is not similar to a lion to any degree in the mammal world. Furthermore, as mentioned in [38], similarity is often considered as similarity from a reference object, with symmetry not being essential. In [38], even though only the crisp similarity relation is mentioned, we think this arugment could be extended to the fuzzy case. For example, it is generally assumed that North Korea is politically similar to China, but not so often to say that China is politically similar to North Korea. So, our generalized fuzzy rough sets may have applications to deal with these kinds of problems since we relex the fuzzy similarity relation ( -similarity relation) to an arbitrary fuzzy relation. IX. CONCLUSION Rough set theory and fuzzy set theory are two mathematical tools to deal with uncertainty. Combing them together is of both theoretical and practical importance. This paper studies fuzzy rough sets and develop a unified framework by constructive and axiomatic approaches. The connections with lattice theory and fuzzy topology are also examined. Thus, a mathematical foundation is set up for the further application of fuzzy rough sets. As an application to fuzzy reasoning, it is pointed out that the CRI rule of fuzzy reasoning is a special upper approximation operator and it is possible to apply lower approximation operators to develop algorithms for fuzzy controller. The future work will be concentrated on the knowledge discovery methods in fuzzy information systems. APPENDIX SOME DEFINITIONS AND RESULTS ABOUT LATTICE THEORY AND FUZZY TOPOLOGY First, we review some basic notions and results of the lattice theory. A lattice is a partially ordered set in which any two elements have a least upper bound and a greatest lower bound. We and the greatest denote the least upper bound of and by . A lattice is said to be complete if any (filower bound by has a least upper bound (sup) nite or infinite) subset and a greatest lower bound (inf) . An element in a lattice is said to be join-irreducible if and imply that or . If is a nonzero join-irreducible element in , then call a molecule of . A lattice is called completely distributive if it satisfies the following conditions: where and are nonempty index sets and . In this section a complete completely distributive lattice will be denoted as a CCD lattice. It is well known that each element of a CCD lattice is a join of join-irreducible elements (i.e., molecules) [21]. is a CCD lattice. The collecFor example, tion of all join-irreducible elements of is . Now we recall some basic concepts in fuzzy topological theory. is a fuzzy topology Definition A [19]: A subset T of on , if and only if, it satisfies the following. 1) If , then . , then . 2) If 3) For every . 360 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 13, NO. 3, JUNE 2005 Definition B [19]: A mapping is a fuzzy it satisfies interior operator, if and only if, for all the following. ; 2) ; 3) 1) ; and 4) . is a fuzzy Definition C [19]: A mapping it satisfies: closure operator, if and only if, for all 1) ; 2) ; 3) ; and 4) . The elements of a fuzzy topology are called open fuzzy sets, and it is easy to prove that a fuzzy interior operator defines a fuzzy topology (so, the are the fixed points of ). A fuzzy cloopen fuzzy sets of sure operator defines a fuzzy topology (so, the closed sets with respect to are the fixed points of ). ACKNOWLEDGMENT The authors would like to thank the referees for their valuable suggestions in improving this paper. REFERENCES [1] Z. Pawlak, “Rough sets,” Int. J. Comput. Inf. Sci., vol. 11, no. 5, pp. 341–356, 1982. [2] I. Jagielska, C. Matthews, and T. Whitfort, “An investigation into the application of neural networks, fuzzy logic, genetic algorithms, and rough sets to automated knowledge acquisition for classification problems,” Neurocomput., vol. 24, pp. 37–54, 1999. [3] M. Kryszkiewicz, “Rough set approach to incomplete information systems,” Inf. Sci., vol. 112, pp. 39–49, 1998. [4] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data. Norwell, MA: Kluwer, 1991. [5] S. Tsumoto, “Automated extraction of medical expert system rules from clinical databases based on rough set theory,” Inf. Sci., vol. 112, pp. 67–84, 1998. [6] A. Skowron and L. Polkowski, Rough Sets in Knowledge Discovery. Berlin, Germany: Springer-Verlag, 1998, vol. 1, 2. [7] Intelligent Decision Support: Handbook of Applications and Advances of the Rough Sets Theory, R. Slowinski, Ed., Kluwer, Norwell, MA, 1992. [8] “Rough sets, fuzzy sets and knowledge discovery,” in Workshop in Computing, W. P. Ziarko, Ed.. London, U. K., 1994. [9] S. Chanas and D. Kuchta, “Further remarks on the relation between rough and fuzzy sets,” Fuzzy Sets Syst., vol. 47, pp. 391–394, 1992. [10] D. Dubois and H. Prade, “Rough fuzzy sets and fuzzy rough sets,” Int. J. Gen. Syst., vol. 17, no. 2–3, pp. 191–209, 1990. [11] Int. Dec. Sut: Handbook of Applications and Advances of the Rough Sets Theory, R. Slowinski, Ed., Kluwer, Norwell, MA, 1992, pp. 287–304. [12] Y. Y. Yao, “A comparative study of fuzzy sets and rough sets,” J. Inf. Sci., vol. 109, pp. 227–242, 1998. [13] A. M. Radzikowska and E. E. Kerre, “A comparative study of fuzzy rough sets,” Fuzzy Sets Syst., vol. 126, pp. 137–155, 2002. [14] N. N. Morsi and M. M. Yakout, “Axiomatics for fuzzy rough set,” Fuzzy Sets Syst., vol. 100, pp. 327–342, 1998. [15] D. Boixader, J. Jacas, and J. Recasens, “Upper and lower approximations of fuzzy sets,” J. Gen. Syst., vol. 29, no. 4, pp. 555–568, 1999. [16] T. Y. Lin and Q. Liu, “Rough approximate operators: Axiomatic rough set theory,” in Rough Sets, Fuzzy Sets, and Knowledge Discovery, W. P. Ziarko, Ed. Berlin, Germany: Springer-Verlag, 1994. [17] Y. Y. Yao, “Constructive and Algebraic methods of the theory of rough sets,” Inf. Sci., vol. 109, pp. 21–47, 1998. [18] K. A. Srivastava and S. P. Tiwari, “On relationships among fuzzy approximation operators, fuzzy topology, and fuzzy automata,” Fuzzy Sets Syst., 2005. [19] R. Lowen, “Fuzzy topological spaces and fuzzy compactness,” J. Math. Anal. Appl., vol. 56, pp. 621–633, 1976. [20] G. J. Klir and B. Yuan, Fuzzy Logic: Theory and Applications. Upper Saddle River, NJ: Prentice-Hall, 1995. [21] G. J. Wang, “Theory of topological molecular lattices,” Fuzzy Sets Syst., vol. 47, pp. 351–376, 1992. [22] J. Kortelainen, “On relationships between modified sets, topological spaces, and rough sets,” Fuzzy Sets Syst., vol. 61, pp. 91–95, 1994. [23] Y. Y. Yao, “Two views of the theory of rough sets in finite universes,” Int. J. Approx. Reason., vol. 15, pp. 291–317, 1996. [24] N. N. Morsi, “Dual fuzzy neighborhood spaces II,” J. Fuzzy Math., vol. 3, pp. 29–67, 1995. [25] , “Fuzzy T -locality spaces,” Fuzzy Sets Syst., vol. 69, pp. 193–219, 1995. [26] M. Sarkar and B. Yegnanarayana, “Fuzzy-rough neural networks for vowel classification,” in Proc. 1998 IEEE Int. Conf. Systems, Man, and Cybernetics, vol. 5, Oct. 11–14, 1998, pp. 4160–4165. , “Application of fuzzy-rough sets in modular neural networks,” in [27] Neural Networks Proc. IEEE Int. Conf. Computational Intelligence, vol. 1, May 4–8, 1998, pp. 741–746. [28] P. Lingras, “Fuzzy-rough and rough-fuzzy serial combinations in neurocomputing,” Neurocomput., vol. 36, no. 1–4, pp. 29–44, Feb. 2001. [29] M. Sarkar, “Measures of ruggedness using fuzzy-rough sets and fractals: Applications in medical time series,” in Proc. 2001 IEEE Int. Conf. Systems, Man, and Cybernetics, vol. 3, 2001, pp. 1514–1519. [30] S. K. Pal and P. Mitra, “Case generation using rough sets with fuzzy representation,” IEEE Trans. Knowledge Data Eng., 2005. [31] Y.-F. Wang, “Mining stock price using fuzzy rough set system,” Exp. Syst. Appl., 2005. [32] R. Jensen and Q. Shen, “Fuzzy-rough sets for descriptive dimensionality reduction,” in Proc. 2002 IEEE Int. Conf. Fuzzy Systems, vol. 1, 2002, pp. 29–34. [33] S. Miyamoto, “Fuzzy multisets with infinite collections of memberships,” in Proc. 7th Int. Fuzzy Systems Assoc. World Congr. (IFSA’97). Durham, NC, June 25–30, 1997, pp. 17–33. [34] T. Y. Lin, “Sets with partial memberships: A rough sets view of fuzzy sets,” in Proc. 1998 World Congr. Computational Intelligence (WCCI98), Anchorage, AK, May 4–9, 1998, pp. 785–790. , “Granular fuzzy sets: A view from rough sets and probability the[35] ories,” Int. J. Fuzzy Syst., vol. 13, no. 2, pp. 373–381, Jun. 2002. , “Granular computing on binary relations I: Data mining and [36] neighborhood systems,” in Rough Sets in Knowledge Discovery, A. Skowron and L. Polkowski, Eds. Berlin, Germany: Physica-Verlag, 1998, pp. 107–121. , “Granular computing on binary relations II: Rough set represen[37] tations and belief functions,” in Rough Sets in Knowledge Discovery, A. Skowron and L. Polkowski, Eds. Berlin, Germany: Physica-Verlag, 1998, pp. 122–140. [38] R. Slowinski and D. Vanderpooten, “A generalized definition of rough approximation based on similarity,” IEEE Trans. Knowledge Data Eng., vol. 12, pp. 331–336, 2000. [39] T. Y. Lin, “Granular computing: Fuzzy logic and rough sets,” in Computing with Words in Information/Intelligent Systems, L. A. Zadeh and J. Kacprzyk, Eds. Berlin, Germany: Physica-Verlag, 1999, pp. 183–200. Daniel S. Yeung (M’92–SM’99–F’04) received the Ph.D. degree in applied mathematics from Case Western Reserve University, Cleveland, OH, in 1974. He has worked as an Assistant Professor of Mathematics and Computer Science at Rochester Institute of Technology, Rochester, NY, as a Research Scientist in the General Electric Corporate Research Center Schennectady, NY, and as a System Integration Engineer at TRW, San Diego, CA. He was the Chairman of the Department of Computing, The Hong Kong Polytechnic University, Hong Kong for seven years, where he is now a Chair Professor. His current research interests include neural-network sensitivity analysis, data mining, Chinese computing, and fuzzy systems. Dr. Yeung was the President of IEEE Hong Kong Computer Chapter (1991 and 1992), an Associate Editor for both the IEEE TRANSACTIONS ON NEURAL NETWORKS and the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS B. He is also the Coordinating Chair (Cybernetics) for the IEEE SMC Society, and the General Co-Chair of the 2002–2003 International Conference on Machine Learning and Cybernetics. He has recently been elected by the IEEE SMC Society a Vice President for Technical Activities. YEUNG et al.: ON THE GENERALIZATION OF FUZZY ROUGH SETS Degang Chen was born in 1969 in Tonghua, Jilin Provence, China. He received the M.S. degree from Northeast Normal University, Changchun, Jilin, China, in 1994, and the Ph.D. degree from Harbin Institute of Technology, Harbin, China, in 2000. He has worked as a Postdoctoral Fellow with Xi’an Jiaotong University, Xi’an, China, from 2000 to 2002 and with Tsinghua University, Tsinghua, China, from 2002 to 2004. Since 1994, he has been a Teacher at Bohai University, Jinzhou, Liaoning, China. His research interests include fuzzy group, fuzzy analysis, rough sets, and SVM. Eric C. C. Tsang received the B.Sc. degree in computer studies from the City University of Hong Kong, Kowloon, Hong, Kong, in 1990, and the Ph.D. degree in computing from the Hong Kong Polytechnic University, Hong, Kong, in 1996. He is an Assistant Professor with the Department of Computing, the Hong Kong Polytechnic University. His main research interests are in the area of fuzzy expert systems, fuzzy neural networks, machine learning, genetic algorithms, and fuzzy support vector machine. 361 John W. T. Lee (M’88) received the B.Sc. degree from the University of Hong Kong, Hong Kong, the M.B.A. degree from the Australian Graduate School of Management, New South Wales University, New South Wales, Australia, and the Ph.D. degree from the University of Sunderland, Sunderland, U. K., in 1972, 1983, and 2000 respectively. He spent 12 years in the computer industry, working for a number of companies including NCR, Hong Kong and Esso, Australia, before joining The Hong Kong Polytechnic University in 1986. He is currently an Associate Professor in the Department of Computing, The Hong Kong Polytechnic University. His major research interests include fuzzy sets and systems, knowledge discovery and data mining, and applications of artificial intelligence. Dr. Lee currently chairs the Technical Committee on Expert and Knowledge Based Systems in the IEEE Systems, Man, and Cybernetics Society. Xizhao Wang (M’99–SM’03) received the B.Sc. and M.Sc. degrees in mathematics from Hebei University, Baoding, China, in 1983 and 1992, respectively, and the Ph.D. degree in computer science from Harbin Institute of Technology, Harbin, China, in 1998. From 1983 to 1998, he worked as a Lecturer, an Associate Professor, and a Full Professor in the Department of Mathematics, Hebei University, Hebei, China. From 1998 to 2001, he worked as a Research Fellow with the Department of Computing, Hong Kong Polytechnic University, Kowloon, Hong Kong. Since 2001, he has been the Dean and Professor of the Faculty of Mathematics and Computer Science, Hebei University. His main research interests include inductive learning with fuzzy representation, fuzzy measures and integrals, neuro-fuzzy systems and genetic algorithms, feature extraction, multiclassifier fusion, and applications of machine learning. Dr. Wang is the General Co-Chair of the 2002 and 2003 International Conference on Machine Learning and Cybernetics, cosponsored by the IEEE SMC Society.