Information Sciences 151 (2003) 263–282 www.elsevier.com/locate/ins Generalized fuzzy rough sets Wei-Zhi Wu a a,b,* , Ju-Sheng Mi a, Wen-Xiu Zhang a Faculty of Science, Institute for Information and System Sciences, XiÕan Jiaotong University, XiÕan, ShaanÕxi 710049, PR China b Information College, Zhejiang Ocean University, Zhoushan, Zhejiang 316004, PR China Received 30 December 2000; received in revised form 19 May 2002; accepted 3 July 2002 Abstract This paper presents a general framework for the study of fuzzy rough sets in which both constructive and axiomatic approaches are used. In constructive approach, a pair of lower and upper generalized approximation operators is defined. The connections between fuzzy relations and fuzzy rough approximation operators are examined. In axiomatic approach, various classes of fuzzy rough approximation operators are characterized by different sets of axioms. Axioms of fuzzy approximation operators guarantee the existence of certain types of fuzzy relations producing the same operators. Ó 2002 Published by Elsevier Science Inc. Keywords: Approximation operators; Fuzzy relations; Fuzzy rough sets; Rough sets 1. Introduction The theory of rough sets, proposed by Pawlak [18], is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. Using the concepts of lower and upper approximations * Corresponding author. Address: Faculty of Science, Institute for Information and System Sciences, XiÕan Jiaotong University, XiÕan, ShaanÕxi 710049, PR China. E-mail addresses: wuwz8681@sina.com (W.-Z. Wu), mijsh@sina.com (J.-S. Mi), wxzhang@ xjtu.edu.cn (W.-X. Zhang). 0020-0255/02/$ - see front matter Ó 2002 Published by Elsevier Science Inc. doi:10.1016/S0020-0255(02)00379-1 264 W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 in rough set theory, knowledge hidden in information systems may be unravelled and expressed in the form of decision rules [5,6,12,19,23,31]. The basic operators in rough set theory are approximations. There are at least two approaches for the development of the rough set theory, the constructive and axiomatic approaches. In constructive approach, binary relations on the universe, partitions of the universe, neighborhood systems, and Boolean algebras are all the primitive notions. The lower and upper approximation operators are constructed by means of these notions [5,6,8,9,11,12,16,18–31]. The constructive approach is suitable for practical applications of rough sets. On the other hand, the axiomatic approach, which is appropriate for studying the structures of rough set algebras, takes the lower and upper approximation operators as primitive notions. In this approach, a set of axioms is used to characterize approximation operators that are the same as the ones produced by using constructive approach [10,27,28]. The initiations and majority of studies on rough sets have been concentrated on constructive approaches. In PawlakÕs rough set model [19], an equivalence relation is a key and primitive notion. This equivalence relation, however, seems to be a very stringent condition that may limit the application domain of the rough set model. To solve this problem, several authors have generalized the notion of approximation operators by using nonequivalence binary relations [6,26,28,29]. This has lead to various other approximation operators [8– 12,16,21,22,25–29,31]. On the other hand, by using an equivalence relation on U , one can introduce lower and upper approximations in fuzzy set theory to obtain an extended notion called rough fuzzy set [2,3]. Alternatively, a fuzzy similarity relation can be used to replace an equivalence relation. The result is a deviation of rough set theory called fuzzy rough sets [3,4,14]. More general frameworks can be obtained which involve the approximations of fuzzy sets based on fuzzy T -similarity relations [13], fuzzy similarity relations [30], weak fuzzy partitions on U [1,7], and Boolean subalgebras of PðU Þ [15] etc. Comparing with the studies on constructive approach, there is less effort needed on axiomatic approach. The most important axiomatic studies for crisp rough sets were made by Yao [27,28], Yao and Lin [26] in which various classes of rough set algebras are characterized by different sets of axioms. Moris and Yakout [13] studied a set of axioms on fuzzy rough set, but their studies were restricted to fuzzy T -rough sets defined by fuzzy T -similarity relations which were ordinary (crisp) equivalence relations when they degenerated into crisp ones. So far, however, the axiomatic approach for the study of generalized fuzzy rough set is blank. The present paper studies generalized fuzzy rough sets in which both the constructive and axiomatic approaches are used. In the constructive approach, based on an arbitrary fuzzy relation, a pair of dual generalized fuzzy approximation operators is defined. The connections between fuzzy binary relations and fuzzy approximation operators are examined. The resulting fuzzy W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 265 rough sets are proper generalizations of generalized rough sets [26,28], rough fuzzy sets [3,17], and fuzzy rough set [3,4,30]. In the axiomatic approach, various classes of fuzzy rough sets are characterized by different sets of axioms, axioms of fuzzy approximation operators guarantee the existence of certain types of fuzzy relations producing the same operators. 2. Preliminaries Let X be a finite and nonempty set called the universe. The class of all subsets (respectively, fuzzy subsets) of X will be denoted by PðX Þ (respectively, by FðX Þ). For any A 2 FðX Þ, the a-level and the strong a-level of A will be denoted by Aa and Aaþ , respectively, that is, Aa ¼ fx 2 X : AðxÞ P ag and Aaþ ¼ fx 2 X : AðxÞ > ag, where a 2 I ¼ ½0; 1 , the unit interval, A0 ¼ X , and A1þ ¼ ;. Let U and W be two finite and nonempty universes. A fuzzy subset R 2 FðU W Þ is referred to as a fuzzy binary relation from U to W , Rðx; yÞ is the degree of relation between x and y, where ðx; yÞ 2 U W ; If U ¼ W , then R is referred to as a fuzzy relation on U . Let R be a fuzzy relation from U to W , if for each x 2 U , there exists y 2 W such that Rðx; yÞ ¼ 1, then R is referred to as a serial fuzzy relation from U to W. Definition 2.1. Let R be a fuzzy binary relation on U . R is referred to as a reflexive fuzzy relation if Rðx; xÞ ¼ 1 for all x 2 U ; R is referred to as a symmetric fuzzy relation if Rðx; yÞ ¼ Rðy; xÞ for all x, y 2 U ; R is referred to as a transitive fuzzy relation if Rðx; zÞ P _y2U ðRðx; yÞ ^ Rðy; zÞÞ for all x, z 2 U ; R is referred to as a similarity fuzzy relation if R is a reflexive, symmetric, and transitive fuzzy relation. It is easy to see that R is a serial fuzzy relation iff Ra is a serial ordinary binary relation for all a 2 I; R is a reflexive fuzzy relation iff Ra is a reflexive ordinary binary relation for all a 2 I; R is a symmetric fuzzy relation iff Ra is a symmetric ordinary binary relation for all a 2 I; R is a transitive fuzzy relation iff Ra is a transitive ordinary binary relation for all a 2 I; R is a similarity fuzzy relation iff Ra is an equivalence ordinary binary relation for all a 2 I. Definition 2.2. A set-valued mapping N : I ! PðU Þ is said to be nested if for all a, b 2 I, a 6 b ) N ðbÞ  N ðaÞ: 266 W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 The class of all PðU Þ-valued nested mappings on I will be denoted by NðU Þ. It is well-known that the following representation theorem holds. Theorem 2.3. Let N 2 NðU Þ. Define a function f : NðU Þ ! FðU Þ by: AðxÞ :¼ f ðN ÞðxÞ ¼ _a2I ða ^ N ðaÞðxÞÞ; x 2 U; where N ðaÞðxÞ is the characteristic function of N ðaÞ. Then f is a surjective homomorphism, and the following properties hold: (i) Aaþ TN ðaÞ  Aa , (ii) Aa ¼ Sk<a N ðkÞ, (iii) Aaþ ¼ k>a N ðkÞ, (iv) A ¼ _a2I ða ^ Aaþ Þ ¼ _a2I ða ^ Aa Þ. 3. Construction of generalized fuzzy rough approximation operators 3.1. Generalized rough sets Definition 3.1. Let U and W be two finite universes. Suppose that R is an arbitrary relation from U to W . We can define a set-valued function F : U ! PðW Þ by: F ðxÞ ¼ fy 2 W : ðx; yÞ 2 Rg; x 2 U: Obviously, any set-valued function from U to W defines a binary relation from U to W by setting R ¼ fðx; yÞ 2 U W : y 2 F ðxÞg. The triple ðU ; W ; RÞ is referred to as a generalized approximation space. For any set A  W , a pair of lower and upper approximations, RðAÞ and RðAÞ, are defined by RðAÞ ¼ fx 2 U : F ðxÞ  Ag; RðAÞ ¼ fx 2 U : F ðxÞ \ A 6¼ ;g: The pair ðRðAÞ; RðAÞÞ is referred to as a generalized rough set. From the definition, the following theorem can easily be verified [28]. Theorem 3.2. For any relation R from U to W , its lower and upper approximation operators satisfy the following properties: for all A, B 2 PðW Þ, W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 ðL1Þ 267 RðAÞ ¼ Rð AÞ; ðU1Þ RðAÞ ¼ Rð AÞ; ðL2Þ RðW Þ ¼ U ; ðU2Þ Rð;Þ ¼ ;; ðL3Þ RðA \ BÞ ¼ RðAÞ \ RðBÞ; ðU3Þ RðA [ BÞ ¼ RðAÞ [ RðBÞ; ðL4Þ A  B ) RðAÞ  RðBÞ; ðU4Þ A  B ) RðAÞ  RðBÞ; ðL5Þ RðA [ BÞ  RðAÞ [ RðBÞ; ðU5Þ RðA \ BÞ  RðAÞ \ RðBÞ; where A is the complement of A. Properties (L1) and (U1) show that the approximation operators R and R are dual to each other. Properties with the same number may be considered as dual properties. With respect to certain special types, say, serial, reflexive, symmetric, and transitive binary relation on the universe U , the approximation operators have additional properties [28,29], say, for serial relation ðL0Þ Rð;Þ ¼ ;; ðU0Þ RðU Þ ¼ U ; ðLU0Þ for reflexive relation for symmetric relation RðAÞ  RðAÞ; ðL6Þ RðAÞ  A; ðU6Þ A  RðAÞ; ðL7Þ A  RðRðAÞÞ; ðU7Þ for transitive relation ðL8Þ RðRðAÞÞ  A; RðAÞ  RðRðAÞÞ; ðU8Þ RðRðAÞÞ  RðAÞ: If R is an equivalence relation on U , then the pair ðU ; RÞ is the Pawlak approximation space and more interesting properties of lower and upper approximation operators can be derived [19,28,29]. 3.2. Generalized fuzzy rough sets Let R be an arbitrary fuzzy relation from U to W . Define the mapping F : U ! FðW Þ by: F ðxÞðyÞ ¼ Rðx; yÞ; ðx; yÞ 2 U W: 268 W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 For any a 2 I, we further define Fa : U ! PðW Þ by: Fa ðxÞ ¼ fy 2 W : F ðxÞðyÞ P ag; x 2 U: Also for any X 2 PðW Þ, the lower and upper approximations of X with respect to the approximation space ðU ; W ; Fa Þ are defined as follows: F a ðX Þ ¼ fx 2 U : Fa ðxÞ  X g; F a ðX Þ ¼ fx 2 U : Fa ðxÞ \ X 6¼ ;g: Lemma 3.3. If R is an arbitrary fuzzy relation from U to W and A 2 FðW Þ, let N ðaÞ ¼ F 1a ðAaþ Þ and H ðaÞ ¼ F a ðAa Þ, a 2 I. Then N and H are nested. Proof. Let 0 6 b 6 a 6 1, it is easy to see that Abþ  Aaþ , Ab  Aa , Fb ðxÞ  Fa ðxÞ, and F1a ðxÞ  F1b ðxÞ. (a) If x 2 N ðaÞ, then by the definition of N ðaÞ we have F1a ðxÞ  Aaþ . Hence F1b ðxÞ  F1a ðxÞ  Aaþ  Abþ : Thus x 2 F 1b ðAbþ Þ. Therefore N ðaÞ ¼ F 1a ðAaþ Þ  F 1b ðAbþ Þ ¼ N ðbÞ: It follows that N is nested. (b) If y 2 H ðaÞ, then by the definition of H ðaÞ we have Fa ðyÞ \ Aa 6¼ ;. Since Fa ðyÞ \ Aa  Fb ðyÞ \ Ab , we obtain Fb ðyÞ \ Ab 6¼ ;. Thus y 2 F b ðAb Þ. Therefore H ðaÞ  H ðbÞ. It follows that H is nested.  Based on Lemma 3.3 and Theorem 2.3, we can formulate the notion of generalized fuzzy rough set as follows. Definition 3.4. Let R be an arbitrary fuzzy relation from U to W . The triple ðU ; W ; RÞ is referred to as a generalized fuzzy approximation space. We define the lower and upper generalized fuzzy approximation operators F and F with respect to ðU ; W ; RÞ by: F ðAÞ ¼ _a2I ða ^ F 1a ðAaþ ÞÞ; F ðAÞ ¼ _a2I ða ^ F a ðAa ÞÞ; A 2 FðW Þ; A 2 FðW Þ: The pair ðF ðAÞ; F ðAÞÞ is referred to as a generalized fuzzy rough set. Remark. From Definition 3.4 and Theorem 2.3 we can immediately conclude that the lower approximation operator F satisfies: ðF ðAÞÞaþ  F 1a ðAaþ Þ  ðF ðAÞÞa ; and furthermore, W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 ðF ðAÞÞa ¼ \ 269 F 1k ðAkþ Þ; k<a ðF ðAÞÞaþ ¼ [ F 1k ðAkþ Þ: k>a Likewise the upper approximation operator F satisfies: ðF ðAÞÞaþ  F a ðAa Þ  ðF ðAÞÞa ; and also ðF ðAÞÞa ¼ \ F k ðAk Þ; k<a ðF ðAÞÞaþ ¼ [ F k ðAk Þ: k>a Proposition 3.5. Assume that R is an arbitrary fuzzy relation from U to W , let A 2 FðW Þ, (i) If 0 6 b 6 1=2; then F 1b ðAbþ Þ  F b ðAbþ Þ [ F 1b ðAð1bÞþ Þ: (ii) If 1=2 6 b 6 1, then F 1b ðAbþ Þ  F 1b ðAð1bÞþ Þ \ F b ðAbþ Þ: Proof (i) When 0 6 b 6 1=2, for any y 2 U , we have Fb ðyÞ  F1b ðyÞ; and Abþ  Að1bÞþ : If y 2 F b ðAbþ Þ, by the definition of lower approximation we have Fb ðyÞ  Abþ , then F1b ðyÞ  Fb ðyÞ  Abþ , in turn, y 2 F 1b ðAbþ Þ, which implies that F b ðAbþ Þ  F 1b ðAbþ Þ: If y 2 F 1b ðAð1bÞþ Þ, then by the definition of lower approximation we have F1b ðyÞ  Að1bÞþ  Abþ , that is, y 2 F 1b ðAbþ Þ, which then implies F 1b ðAð1bÞþ Þ  F 1b ðAbþ Þ: Hence F 1b ðAbþ Þ  F b ðAbþ Þ [ F 1b ðAð1bÞþ Þ: (ii) It is similar to the proof of (i).  270 W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 Theorem 3.6. If R is an arbitrary fuzzy relation from U to W . Then the pair of fuzzy approximation operators satisfies the following properties: for all A, B 2 FðW Þ, and for all a 2 I, ðFL1Þ F ðAÞ ¼ ðF ð AÞÞ; ðFU1Þ F ðAÞ ¼ ðF ð AÞÞ; F ðA _ a^Þ ¼ F ðAÞ _ a^; ðFL2Þ ðFU2Þ F ðA ^ a^Þ ¼ F ðAÞ ^ a^; ðFL3Þ F ðA ^ BÞ ¼ F ðAÞ ^ F ðBÞ; ðFU3Þ F ðA _ BÞ ¼ F ðAÞ _ F ðBÞ; ðFL4Þ A  B ) F ðAÞ  F ðBÞ; ðFU4Þ A  B ) F ðAÞ  F ðBÞ; ðFL5Þ F ðA _ BÞ  F ðAÞ _ F ðBÞ; ðFU5Þ F ðA ^ BÞ  F ðAÞ ^ F ðBÞ; where a^ is the constant fuzzy set: a^ðxÞ ¼ a, for all x 2 U and x 2 W . Proof. By Theorem 3.2 and using the fact that, for all N 2 NðU Þ, ^a2I ða _ N ðaÞÞ ¼ _a2I ða ^ N ðaÞÞ; we have  F ð AÞ ¼ _a2I ða ^ ðF a ð AÞa ÞÞ ¼ _a2I ða ^ F a ð Að1aÞþ ÞÞ ¼ _a2I ða ^ ð F a ðAð1aÞþ ÞÞÞ ¼ ^a2I ð1  a ^ ð F a ðAð1aÞþ ÞÞÞ ¼ ^a2I ðð1  aÞ _ F a ðAð1aÞþ ÞÞ ¼ ^a2I ða _ F 1a ðAaþ ÞÞ ¼ _a2I ða ^ F 1a ðAaþ ÞÞ ¼ F ðAÞ; from which (FL1) follows. (FU1) can be directly induced by (FL1). For any x 2 U , F ðA ^ a^ÞðxÞ ¼ _b2I ðb ^ F b ðA ^ a^ÞÞðxÞ ¼ supfb 2 I : Fb ðxÞ \ ðA ^ a^Þb 6¼ ;g ¼ supfb 2 I : 9y 2 W ½Rðx; yÞ P b; AðyÞ P b; a P b g ¼ a ^ supfb 2 I : 9y 2 W ½Rðx; yÞ P b; AðyÞ P b g ¼ ð^ a ^ F ðAÞÞðxÞ; which implies (FU2). Similarly we can justify (FL2). 271 W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 By (U3), we have F ðA _ BÞ ¼ _a2I ða ^ F a ðA _ BÞa Þ ¼ _a2I ða ^ F a ðAa [ Ba ÞÞ ¼ _a2I ða ^ ðF a ðAa Þ [ F a ðBa ÞÞÞ ¼ _a2I ðða ^ F a ðAa ÞÞ _ ða ^ F a ðBa ÞÞÞ ¼ ð_a2I ða ^ F a ðAa ÞÞÞ _ ð_a2I ða ^ F a ðBa ÞÞÞ ¼ F ðAÞ _ F ðBÞ: This implies (FU3). Combining (FU3) and the dual properties (FL1) and (FU1), we can easily conclude (FL3). Furthermore, since A  B ) Aa  B a ; 8a 2 I ) a ^ F a ðAa Þ  a ^ F a ðBa Þ; 8a 2 I; by the definition and the duality we conclude that (FU4) and (FL4) hold. Properties (FL5) and (FU5) follow directly from the properties (FL4) and (FU4) respectively.  Just in the case of generalized approximation operators, properties (FL1) and (FU1) show that fuzzy approximation operators F and F are dual to each other. Properties with the same number may be regarded as dual properties. The first three properties are independent. It can be checked that they imply the remaining properties. Properties (FL2) and (FU2) imply the following properties (FL2)0 and (FU2)0 which are formally similar to (L2) and (U2). ðFL2Þ0 ðFU2Þ 0 F ðW Þ ¼ U ; F ð;Þ ¼ ;: Properties (FL4) and (FU4) show that the fuzzy approximation operators are monotonic with respect to (fuzzy) set inclusion. Additional properties of fuzzy approximation operators will be given in the next subsection. 3.3. Connections between special fuzzy relations and approximation operators In this subsection, we show that some special fuzzy relations could be characterized by fuzzy approximation operators. Proposition 3.7. If R is an arbitrary fuzzy relation from U to W , then (i) F ð1y ÞðxÞ ¼ Rðx; yÞ; 8ðx; yÞ 2 U W , (ii) F ð1W nfyg ÞðxÞ ¼ 1  Rðx; yÞ; 8ðx; yÞ 2 U W ; 272 W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 (iii) F ð1X ÞðxÞ ¼ maxfRðx; yÞ : y 2 X g; 8x 2 U ; X 2 PðW Þ, (iv) F ð1X ÞðxÞ ¼ minf1  Rðx; yÞ : y 62 X g; 8x 2 U ; X 2 PðW Þ, where 1y denotes the fuzzy singleton with value 1 at y and 0 elsewhere; 1X denotes the characteristic function of X . Proof (i) When a > 0, it is clear that F a ð1y Þa ¼ F a ðfygÞ ¼ fu 2 U : Fa ðuÞ \ fyg 6¼ ;g ¼ fu 2 U : y 2 Fa ðuÞg ¼ fu 2 U : Rðu; yÞ P ag. Then F a ð1y Þa ðxÞ ¼ 1 when Rðx; yÞ P a, otherwise, F a ð1y Þa ðxÞ ¼ 0. Hence, F ð1y ÞðxÞ ¼ _a2I ða ^ F a ð1y Þa ÞðxÞ ¼ supfa 2 I : Rðx; yÞ P ag ¼ Rðx; yÞ. (ii) Follows immediately from (i) and the duality. (iii) Let F ð1X ÞðxÞ ¼ a and maxfRðx; yÞ : y 2 X g ¼ b. Then a ¼ _a2I ða ^ F a ð1X Þa ÞðxÞ ¼ supfa 2 I : x 2 F a ðX Þg ¼ supfa 2 I : Fa ðxÞ \ X 6¼ ;g ¼ supfa 2 I : fz 2 W : Rðx; zÞ P aÞg \ X 6¼ ;g: For any k < a, there exists a 2 ðk; aÞ such that fy 2 W : Rðx; yÞ P ag \ X 6¼ ;. Hence b P a > k. Therefore a 6 b. On the other hand, for any k > a, we have fy 2 W : Rðx; yÞ P kg \ X ¼ ;, that is, Rðx; yÞ < k for all y 2 X . Hence b < k. Therefore b 6 a. It follows that a ¼ b. (iv) Follows immediately from (iii) and the duality.  Theorem 3.8. If R is an arbitrary fuzzy relation from U to W , then R is serial iff one of the following properties holds: ðFL0Þ F ð;Þ ¼ ;; ðFU0Þ F ðW Þ ¼ U ; ðFLU0Þ F ðAÞ  F ðAÞ; 8A 2 FðW Þ: Proof. First we can deduce from the dual properties (FL1) and (FU1) that (FL0) and (FU0) are equivalent. Second we are to prove that R is serial () ðFU0Þ: In fact, we know from Proposition 3.7 that for any x 2 U , F ðW ÞðxÞ ¼ _y2W Rðx; yÞ. Then R is serial () 9y 2 W such that Rðx; yÞ ¼ 1 () _y2W Rðx; yÞ ¼ 1 () F ðW Þ ¼ U : At last we are to prove that W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 273 R is serial () ðFLU0Þ: If R is a serial fuzzy relation, from the definition of the approximation operators we only need to prove that for all a 2 I, F 1a ðAaþ Þ  F a ðAa Þ: In fact, if x 2 F 1a ðAaþ Þ, then by the definition of lower approximation we have F1a ðxÞ  Aaþ . It means that y 2 F1a ðxÞ or equivalently Rðx; yÞ P 1  a implies AðyÞ > a. Since R is serial, there exists y0 2 W such that Rðx; y0 Þ ¼ 1. It follows that Rðx; y0 Þ P a, i.e., y0 2 Fa ðxÞ. On the other hand, since Aðy0 Þ > a, we have y0 2 Aaþ  Aa , thus, y0 2 Fa ðxÞ \ Aa , then by the definition of upper approximation we have x 2 F a ðAa Þ. Thus we conclude that (FLU0) holds. Conversely, if we assume that (FLU0) holds, let A ¼ W , then by Proposition 3.7 and (FL2)0 we have _y2W Rðx; yÞ ¼ F ðW ÞðxÞ P F ðW ÞðxÞ ¼ U ðxÞ ¼ 1; which follows that R is serial.  Theorem 3.9. If R is an arbitrary fuzzy relation on U , then R is reflexive iff one of the following two properties holds: ðFL6Þ F ðAÞ  A; 8A 2 FðU Þ; ðFU6Þ A  F ðAÞ; 8A 2 FðU Þ: Proof. (FL6) and (FU6) are equivalent because of (FL1) and (FU1). We only need to prove that the reflexivity of R is equivalent to (FU6). Assume that R is reflexive. For any A 2 FðU Þ and x 2 U , let AðxÞ ¼ a, it is then clear that x 2 Aa . Since R is reflexive, Ra is an ordinary reflexive binary T relation as well. Hence x 2 Fa ðxÞ, and furthermore x 2 Fa ðxÞ Aa , that is, x 2 F a ðAa Þ. Since F a ðAa Þ  ðF ðAÞÞa , we have that x 2 ðF ðAÞÞa , that is, F ðAÞðxÞ P AðxÞ which implies (FU6). Assume that (FU6) holds. For any x 2 U , let A ¼ 1x . From Proposition 3.7 and by the assumption, we then have that 1 ¼ 1x ðxÞ 6 F ð1x ÞðxÞ ¼ Rðx; xÞ. It follows that R is reflexive.  Theorem 3.10. If R is an arbitrary fuzzy relation on U , then R is symmetric iff one of the following two properties holds: ðFL7Þ F ð1U nfyg ÞðxÞ ¼ F ð1U nfxg ÞðyÞ; ðFU7Þ F ð1x ÞðyÞ ¼ F ð1y ÞðxÞ; 8ðx; yÞ 2 U 8ðx; yÞ 2 U Proof. It is immediately from Proposition 3.7. U:  U; 274 W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 Lemma 3.11. If R is an arbitrary fuzzy relation on U , then ðF ðAÞÞaþ  F a ðAaþ Þ  ðF ðAÞÞa ; A 2 FðU Þ; a 2 I: Proof. For any k > a, if x 2 F k ðAk Þ, then by the definition of upper approximation we have Fk ðxÞ \ Ak 6¼ ;. Since Fk ðxÞ  Fa ðxÞ and Ak  Aaþ , we obtain that Fa ðxÞ \ Aaþ  Fk ðxÞ \ Ak 6¼ ;. S Hence x 2 F a ðAaþ Þ. Thus F k ðAk Þ  F a ðAaþ Þ for all k > a. Since ðF ðAÞÞaþ ¼ k>a F k ðAk Þ; we conclude that ðF ðAÞÞaþ  F a ðAaþ Þ  F a ðAa Þ  ðF ðAÞÞa .  Theorem 3.12. If R is an arbitrary fuzzy relation on U , then R is transitive iff one of the following two properties holds: ðFL8Þ F ðAÞ  F ðF ðAÞÞ; 8A 2 FðU Þ; ðFU8Þ F ðF ðAÞÞ  F ðAÞ; 8A 2 FðU Þ: Proof. (FL8) and (FU8) are equivalent because of (FL1) and (FU1). We are only to prove that the transitivity of R is equivalent to (FU8). To this end, let us assume that R is transitive. Then by Lemma 3.11 we have _a2I ða ^ ðF ðAÞÞaþ Þ  _a2I ða ^ ðF a ðAaþ ÞÞÞ  _a2I ða ^ ðF ðAÞÞa Þ: This implies that for all A 2 FðU Þ, _a2I ða ^ F a ðAaþ ÞÞ ¼ F ðAÞ: Using the above result and combining Lemma 3.11 and the property (U8), we then get F ðF ðAÞÞ ¼ _a2I ða ^ F a ððF ðAÞÞaþ ÞÞ  _a2I ða ^ F a ðF ðAaþ ÞÞÞ  _a2I ða ^ F a ðF a ðAa ÞÞÞ  _a2I ða ^ F a ðAa ÞÞ ¼ F ðAÞ: Thus (FU8) holds. Conversely, assume that (FU8) holds. Let x, y, z 2 U and k 2 ð0; 1 such that Rðx; yÞ P k and Rðy; zÞ P k. Then on one hand, F ðF ð1z ÞÞðxÞ 6 F ð1z ÞðxÞ ¼ Rðx; zÞ; and on the other hand, 275 W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 F ðF ð1z ÞÞðxÞ ¼ supfa : x 2 F a ðF ð1z ÞÞa g ¼ supfa : Fa ðxÞ \ ðF ð1z ÞÞa 6¼ ;g ¼ supfa : 9u 2 U ½Rðx; uÞ P a; ðF ð1z ÞÞðuÞ ¼ Rðu; zÞ P a g P minfRðx; yÞ; Rðy; zÞg P k; thus Rðx; zÞ P k. It follows that R is transitive.  Theorem 3.13. If R is an arbitrary fuzzy relation on U , then R is a fuzzy similarity relation iff F satisfies properties (FL6)–(FL8) or equivalently, F satisfies properties (FU6)–(FU8). Proof. It follows immediately from Theorems 3.9, 3.10 and 3.12.  Theorem 3.14. If R is a reflexive and transitive fuzzy relation on U , then the following properties hold: ðFL9Þ F ðAÞ ¼ F ðF ðAÞÞ; 8A 2 FðU Þ; ðFU9Þ F ðAÞ ¼ F ðF ðAÞÞ; 8A 2 FðU Þ: Proof. We are only to prove (FU9) because of the duality. Since R is reflexive, by Theorem 3.9 we have A  F ðAÞ. Using Theorem 3.6 we then conclude that F ðAÞ  F ðF ðAÞÞ. Thus Theorem 3.14 follows from Theorem 3.12.  Remark. Theorems 3.6, 3.8, 3.9 and 3.12 can be viewed as the counterparts of the generalized approximation operators. But when R is a symmetric fuzzy relation, the counterparts of properties (L7) and (U7) do not hold. Nevertheless, we can take properties (FL7) and (FU7) as the counterparts of properties (L7) and (U7). Example 3.15. Let U ¼ f1; 2; 3g, and let fuzzy relation R be given by the fuzzy set-valued function F defined by: F ð1Þ ¼ 0:1=1 þ 1=2 þ 0:6=3; F ð2Þ ¼ 1=1 þ 0:1=2 þ 0:4=3; F ð3Þ ¼ 0:6=1 þ 0:4=2 þ 0:1=3: It is easy to see that R is symmetric. Let A ¼ 0:1=1 þ 1=2 þ 0:9=3. Then it can be checked that F ðAÞ ¼ 1=1 þ 0:4=2 þ 0:4=3; F ðF ðAÞÞ ¼ 0:4=1 þ 0:6=2 þ 0:6=3; that is, the properties A  F ðF ðAÞÞ and F ðF ðAÞÞ  A do not hold. 276 W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 Theorem 3.16. If R is a reflexive fuzzy relation on U , then the following properties hold: ðFLU10Þ ðFL11Þ ðFU11Þ F ð^ aÞ ¼ F ð^ aÞ ¼ a^; a 2 I; inffF ðAÞðxÞ : x 2 U g ¼ inffAðxÞ : x 2 U g; supfF ðAÞðxÞ : x 2 U g ¼ supfAðxÞ : x 2 U g; A 2 FðU Þ; A 2 FðU Þ: Proof (a) It is easy to see that ð^ aÞa 6¼ ; () a P a () ð^ aÞ a ¼ U : Since R is reflexive, for all a 2 I, Ra is reflexive as well. Hence x 2 Fa ðxÞ for all x 2 U , i.e., Fa ðxÞ ¼ 6 ; for all a 2 I. Then for all x 2 U , we have F ð^ aÞðxÞ ¼ _a2I ða ^ F a ð^ aÞa ÞðxÞ ¼ supfa 2 I : x 2 F a ð^ aÞ a g ¼ supfa 2 I : Fa ðxÞ \ ð^ aÞa 6¼ ;g ¼ supfa 6 a : Fa ðxÞ ¼ 6 ;g ¼ a: This implies F ð^ aÞ ¼ a^. aÞ ¼ a^ follows from F ð^ aÞ ¼ a^ and the duality of the approxiProperty F ð^ mation operators. (b) Let inffAðxÞ : x 2 U g ¼ a. Obviously, a^  A. Since R is reflexive, we have aÞ  F ðAÞ  A: a^ ¼ F ð^ Hence a 6 inffF ðAÞðxÞ : x 2 U g 6 inffAðxÞ : x 2 U g ¼ a: Thus (FL11) holds. (c) (FU11) follows from (FL11) and the duality of the approximation operators.  Theorem 3.17. If R is an ordinary binary relation from U to W , and let F ðxÞ ¼ fy 2 W : ðx; yÞ 2 Rg. Then i(i) F ð1A Þ ¼ fx 2 U : F ðxÞ  Ag; A 2 PðW Þ, (ii) F ð1A Þ ¼ fx 2 U : F ðxÞ \ A 6¼ ;g; A 2 PðW Þ. Proof. Straightforward.  Remark. The above theorem shows that the pair of generalized fuzzy approximation operators is a generalization of generalized approximation operators. W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 277 Theorem 3.18. If R is an ordinary equivalence relation on U , and ½x is the Requivalent class of x, then (i) F ðAÞðxÞ ¼ inffAðyÞ : y 2 ½x g; 8A 2 FðU Þ, (ii) F ðAÞðxÞ ¼ supfAðyÞ : y 2 ½x g; 8A 2 FðU Þ. Proof. We only prove (ii) as an example. (i) can be justified similarly. For A 2 FðU Þ and x 2 U , let F ðAÞðxÞ ¼ a and supfAðyÞ : y 2 ½x g ¼ b. Then a ¼ _a2I ða ^ F a ðAa ÞÞðxÞ ¼ supfa 2 I : x 2 F a ðAa Þg ¼ supfa 2 I : Fa ðxÞ \ Aa 6¼ ;g ¼ supfa 2 I : ½x \ Aa 6¼ ;g: Since ½x \ Aa 6¼ ; implies that supfAðyÞ : y 2 ½x g P a, we have b P a. On the other hand, if k > a, then ½x \ Ak ¼ ;; that is, AðyÞ < k for all y 2 ½x . Hence b ¼ supfAðyÞ : y 2 ½x g < k. Therefore b 6 a. It follows that a ¼ b:  Remark. The above theorem shows that the pair of generalized fuzzy approximation operators is a generalization of rough fuzzy approximation operators [3,17]. 4. Axioms of generalized fuzzy approximation operators In an axiomatic approach, rough sets are axiomatized by abstract operators. For the case of fuzzy rough sets, the primitive notion is a system ðFðU Þ; FðW Þ; ^; _; ; L; H Þ, where L; H : FðW Þ ! FðU Þ are operators from FðW Þ to FðU Þ. In this subsection, we show that fuzzy approximation can be characterized by axioms, the results may be viewed as the generalization counterparts of Yao [27,28]. Definition 4.1. Let L; H : FðW Þ ! FðU Þ be two operators. They are referred to as dual operators if for all A 2 FðW Þ, ðfl1Þ LðAÞ ¼ H ð AÞ; ðfu1Þ H ðAÞ ¼ Lð AÞ: Theorem 4.2. Suppose that L; H : FðW Þ ! FðU Þ are two dual operators. Then there exists a fuzzy relation R from U to W such that for all A 2 FðW Þ, LðAÞ ¼ F ðAÞ and H ðAÞ ¼ F ðAÞ iff L and H satisfy the axioms: for all A; B 2 FðW Þ and a 2 I, 278 W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 ðfu2Þ H ð^ a ^ AÞ ¼ a^ ^ H ðAÞ; ðfl2Þ Lð^ a _ AÞ ¼ a^ _ LðAÞ; ðfu3Þ H ðA _ BÞ ¼ H ðAÞ _ H ðBÞ; ðfl3Þ LðA ^ BÞ ¼ LðAÞ ^ LðBÞ: Proof. \ ) " follows immediately from Theorem 3.6. \ ( " Suppose that the operator H obeys the axioms (fu2) and (fu3). Using H , we can define a fuzzy relation from U to W by: Rðx; yÞ ¼ H ð1y ÞðxÞ; ðx; yÞ 2 U W: It is evident that for all A 2 FðW Þ, d A ¼ _y2W ð1y ^ AðyÞ AðyÞÞ: For any x 2 U , by (fu2) and (fu3) we have d F ðAÞðxÞ ¼ F ð_y2W ð1y ^ AðyÞ AðyÞÞÞðxÞ d ¼ _y2W F ð1y ^ AðyÞ AðyÞÞðxÞ d ¼ _y2W ðF ð1y Þ ^ AðyÞ AðyÞÞðxÞ ¼ _y2W ðF ð1y ÞðxÞ ^ AðyÞÞ ¼ _y2W ðRðx; yÞ ^ AðyÞÞ ¼ _y2W ðH ð1y ÞðxÞ ^ AðyÞÞ d ¼ H ð_y2W ð1y ^ AðyÞ AðyÞÞÞðxÞ ¼ H ðAÞðxÞ; which implies that H ðAÞ ¼ F ðAÞ. LðAÞ ¼ F ðAÞ follows immediately from the conclusion H ðAÞ ¼ F ðAÞ and the assumption.  Definition 4.3. Let L; H : FðW Þ ! FðU Þ be a pair of dual operators. If L satisfies axioms (fl2) and (fl3) or equivalently, H satisfies axioms (fu2) and (fu3), then the system (FðU Þ, FðW Þ, ^, _, , L, H ) is referred to as a fuzzy rough set algebra, and L and H are referred to as fuzzy approximation operators. Theorem 4.4. Suppose that L; H : FðW Þ ! FðU Þ is a pair of dual fuzzy approximation operators, i.e., L satisfies axioms (fl2) and (fl3), and H satisfies (fu2) and (fu3). Then there exists a serial fuzzy relation R from U to W such that LðAÞ ¼ F ðAÞ and H ðAÞ ¼ F ðAÞ for all A 2 FðW Þ iff L and H satisfy axioms: W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 ðfl0Þ Lð;Þ ¼ ;; ðfu0Þ H ðW Þ ¼ U ; ðflu0Þ LðAÞ  H ðAÞ; 279 8A 2 FðW Þ: Proof. \ ) " follows immediately from Theorem 3.8, and \ ( " follows immediately from Theorems 4.2 and 3.8.  Theorem 4.5. Suppose that L; H : FðU Þ ! FðU Þ is a pair of dual fuzzy approximation operators, i.e., L satisfies axioms (fl2) and (fl3), and H satisfies (fu2) and (fu3). Then there exists a reflexive fuzzy relation R on U such that LðAÞ ¼ F ðAÞ and H ðAÞ ¼ F ðAÞ for all A 2 FðU Þ iff L and H satisfy axioms: ðfl4Þ LðAÞ  A; 8A 2 FðU Þ; ðfu4Þ A  H ðAÞ; 8A 2 FðU Þ: Proof. \ ) " follows immediately from Theorem 3.9, and \ ( " follows immediately from Theorems 4.2 and 3.9.  Theorem 4.6. Suppose that L; H : FðU Þ ! FðU Þ is a pair of dual fuzzy approximation operators. Then there exists a symmetric fuzzy relation R on U such that LðAÞ ¼ F ðAÞ and H ðAÞ ¼ F ðAÞ for all A 2 FðU Þ iff L and H satisfy axioms: ðfl5Þ Lð1U nfyg ÞðxÞ ¼ Lð1U nfxg ÞðyÞ; 8ðx; yÞ 2 U ðfu5Þ H ð1y ÞðxÞ ¼ H ð1x ÞðyÞ; 8ðx; yÞ 2 U U : U; Proof. \ ) " follows immediately from Theorem 3.10, and \ ( " follows immediately from Theorems 4.2 and 3.10.  Theorem 4.7. Suppose that L; H : FðU Þ ! FðU Þ is a pair of dual fuzzy approximation operators. Then there exists a transitive fuzzy relation R on U such that LðAÞ ¼ F ðAÞ and H ðAÞ ¼ F ðAÞ for all A 2 FðU Þ iff L and H satisfy axioms: ðfl6Þ LðAÞ  LðLðAÞÞ; ðfu6Þ H ðH ðAÞÞ  H ðAÞ; 8A 2 FðU Þ; 8A 2 FðU Þ: Proof. \ ) " follows immediately from Theorem 3.12, and \ ( " follows immediately from Theorems 4.2 and 3.12.  280 W.-Z. Wu et al. / Information Sciences 151 (2003) 263–282 Theorem 4.8. Suppose that L; H : FðU Þ ! FðU Þ is a pair of dual fuzzy approximation operators. Then there exists a similarity fuzzy relation R on U such that LðAÞ ¼ F ðAÞ and H ðAÞ ¼ F ðAÞ for all A 2 FðU Þ iff L satisfies axioms (fl4)– (fl6) and H satisfies axioms (fu4)–(fu6). Proof. It follows immediately from Theorems 4.5–4.7.  5. Conclusion As a suitable mathematical model to handle partial knowledge in data bases, rough set theory is emerging as a powerful theory and has been found its successive applications in the fields of artificial intelligence such as pattern recognition, machine learning, and automated knowledge acquisition. There are at least two aspects in the study of rough set theory: constructive and axiomatic approaches. In constructive approaches, the lower and upper approximation operators are defined in terms of binary relations, partitions of the universe, neighborhood systems or Boolean subalgebras of PðU Þ. The axiomatic approaches consider the reverse problem, namely, the lower and upper approximation operators are taken as primitive notions. A set of axioms is used to characterize approximation operators that are the same as those derived by using constructive approaches. 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