Abstract
Fuzzy Inf. Eng. (2012) 3: 249-259 DOI 10.1007/s12543-012-0114-0 ORIGINAL ARTICLE Wen-Sheng Du· Bao-Qing Hu · Yan Zhao Received: 25 March 2011/ Revised: 23 June 2012/ Accepted: 25 July 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In the present paper, we mainly discuss the (⊥,)-generalized fuzzy rough sets introduced by B.Q. Hu and Z.H. Huang with both constructive approach and ax- iomatic approach considered. In the former, we started from the investigation of the properties of the ⊥-upper and -lower approximation operators generated by binary fuzzy relations. In the latter, by deﬁning a pair of fuzzy set-theoretic operators, we show (⊥,)-fuzzy rough approximation operators can be characterized by diﬀerent sets of axioms. Keywords (⊥,)-fuzzy rough sets · (⊥,)-generalized fuzzy rough sets · Fuzzy relation · Fuzzy sets 1. Introduction The rough set theory, introduced by Pawlak in 1982 [1], serves as a useful tool to study intelligent systems characterized by insuﬃcient and incomplete information [2]. A key and primitive notion in Pawlak’s rough set model is an equivalence rela- tion [1, 2]. However, the requirement of an equivalence relation seems to be a very restrictive condition in the applications of rough set theory in practice. To address this issue, many authors have generalized the notion of approximation operators by non-equivalence binary relations from both theoretic and practical needs. Almost at the same time, Dubois and Prade proposed the concept of rough fuzzy sets and fuzzy rough sets [3, 4] as generalizations of rough sets by taking fuzzy en- vironment into consideration. The approximation operators have been generalized into Zadeh’s fuzzy sets and their extensions in the literature [5-13]. Hu and Huang Wen-Sheng Du ()· Bao-Qing Hu () School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P.R.China email: wsdu@whu.edu.cn bqhu@whu.edu.cn Yan Zhao Department of Mathematics, College of Information Science and Technology, Jinan University, Guangzhou 510632, P.R.China 250 Wen-Sheng Du · Bao-Qing Hu · Yan Zhao (2012) presented the (⊥,)-generalized fuzzy rough sets [14], among others, which will turn out to be much more useful in some practical applications. There are at least two approaches for the development of the fuzzy rough set the- ory, namely, the constructive and the axiomatic approaches [5]. In the former, the relation to the universe of discourse is the primitive notion. The lower and upper approximation operators are constructed by this notion. On the other hand, the latter takes the lower and upper approximation operators as the primitive notion. In this approach, a set of axioms is used to characterize approximation operators, and these axioms guarantee the existence of certain types of fuzzy relations producing the same operators. A general framework for the study of rough set and fuzzy rough set was estab- lished by Yao [5, 15, 16] and Wu et al [17, 18], respectively. The (⊥,)-generalized fuzzy rough sets are intuitionistic generalizations of generalized fuzzy rough sets. Nevertheless, the (⊥,)-generalized fuzzy rough approximation operators have not been studied systemically, or more precisely, they were just considered in the part of constructive approach. The other parts of this paper are organized as follows. We recall brieﬂy basic concepts of fuzzy sets and generalized fuzzy rough sets that will be needed in the following sections in Section 2. In Section 3, we mainly study the construction of (⊥,)-generalized fuzzy rough sets, in which the properties of the approximation operators generated by some binary fuzzy relations are investigated. Next, in Section 4, the axiomatic characterization of (⊥,)-fuzzy rough approximation operators are in discussion. Finally, some concluding remarks and suggestions for future work are given in Section 5. 2. Preliminaries 2.1. Fuzzy Logical Operators and Fuzzy Relations A binary operation (⊥) on the closed unit interval [0,1] is a t-norm (t-conorm) if it satisﬁes (T1) (xy)z = x(yz), (T2) xy = yx, (T3) y ≤ z ⇒ xy ≤ xz, (T4) x1 = x (x⊥0 = x). If t-norm and t-conorm⊥ satisfy 1− x⊥y = (1− x)(1− y) for all x, y ∈ [0, 1], then they are called dual. And in what follows, we always assume that t-norm and t-conorm ⊥ are dual. We deﬁne ⊥-union and -intersection pointwise by the formulas (A∪ B)(x) = A(x)⊥B(x) and (A∩ B)(x) = A(x)B(x), respectively. Let X be a ﬁnite and nonempty universe of discourse. The class of all subsets (respectively, fuzzy subsets) of X will be denoted by P (X) (respectively, by F (X)). Given two fuzzy sets A, B ∈ F (X), we say that A is a subset of B and write A ⊆ B iﬀ A(x) ≤ B(x) for all x ∈ X. Union, intersection and complement of fuzzy sets are respectively denoted by ∪, ∩, . A fuzzy subset R ∈ F (X × Y ) is referred to as a fuzzy binary relation from X −1 −1 to Y . The inverse relation of R is R ∈ F (Y × X) with R(x, y) = R (y, x) for all (x, y) ∈ X× Y . Let R ∈ F (X× Y ) and S ∈ F (Y× Z). Then the max- composition of R and S is R◦ S (x, z) = ∨ {R(x, y)S (y, z)} and the min-⊥ composition of R and y∈Y Fuzzy Inf. Eng. (2012) 3: 249-259 251 S is R ◦ ˆ S (x, z) = ∧ {R(x, y)⊥S (y, z)} for all x ∈ X, z ∈ Z. Specially, if = ∧ and ⊥ y∈Y ⊥ = ∨, then◦ and◦ ˆ are written as ◦ and◦ ˆ , respectively. Let R ∈ F (X × Y ), R is referred to as a ﬁrst (second) serial, or shortly serial fuzzy relation from X to Y . If for each ﬁrst variable x ∈ X (second variable y ∈ Y ), there exists second variable y ∈ Y (ﬁrst variable x ∈ X) such that R(x, y) = 1. Furthermore, if X = Y , then R is referred to as a fuzzy relation on X. A fuzzy binary relation R on X is called a -equivalence relation iﬀ R is • Reﬂexive: R(x, x) = 1 for all x ∈ X. • Symmetric: R(x, y) = R(y, x) for all x, y ∈ X. • -transitive: R ⊇ R◦ R, i.e., R(x, z) ∨ {R(x, y)R(y, z)} for all x, z ∈ X. y∈X A fuzzy binary relation R on X is referred to as a -Euclidean fuzzy relation iﬀ −1 R ⊇ R ◦ R, i.e., R(y, z) ∨ {R(x, y)R(x, z)} for all y, z ∈ X. Then there are two x∈X equivalent deﬁnitions of -equivalence relation. • R is called a -equivalence relation iﬀ R is reﬂexive and -Euclidean. • R is called a-equivalence relation iﬀ R is serial, symmetric and-Euclidean. 2.2. Generalized Fuzzy Rough Sets The following deﬁnition and theorems on generalized fuzzy rough sets can be found in [17, 18]. Deﬁnition 1 Let X and Y be two ﬁnite and nonempty universes of discourse and R a fuzzy relation from X to Y. For an arbitrary fuzzy set A ∈ F (Y ), the lower and upper approximation of A, denoted respectively by R(A) and R(A), with respect to the approximation space (X, Y, R) are fuzzy sets of X with the following memberships: R(A)(x) = ∧ {A(y)∨ (1− R(x, y))}, y∈Y R(A)(x) = ∨ {A(y)∧ R(x, y)}. y∈Y The pair (R(A), R(A)) is referred to as a generalized fuzzy rough set, and R and R are referred to as lower and upper generalized fuzzy rough approximation operator, respectively. From the deﬁnition, the following theorem can be easily derived. Theorem 1 For any fuzzy relation R from X to Y, its lower and upper generalized fuzzy rough approximation operators satisfy the following properties: For any A, B ∈ F (Y ), c c c c (FL1) R(A) = (R(A )) , (FU 1) R(A) = (R(A )) ; (FL2) R(A∪α ˆ ) = R(A)∪α ˆ , (FU 2) R(A∩α ˆ ) = R(A)∩α ˆ ; (FL3) R(A∩ B) = R(A)∩ R(B), (FU 3) R(A∪ B) = R(A)∪ R(B); (FL4) A ⊆ B ⇒ R(A) ⊆ R(B), (FU 4) A ⊆ B ⇒ R(A) ⊆ R(B); (FL5) R(A∪ B) ⊇ R(A)∪ R(B), (FU 5) R(A∩ B) ⊆ R(A)∩ R(B). 252 Wen-Sheng Du · Bao-Qing Hu · Yan Zhao (2012) With respect to certain special types, say, serial, reﬂexive, symmetric, and transi- tive fuzzy relation on the universe of discourse X, the approximation operators have additional properties. Theorem 2 If R is an arbitrary fuzzy relation from X to Y, R and R are the fuzzy rough approximation operators, then R is serial iﬀ one of the following properties: For any A, B ∈ F (Y ), ∀α ∈ [0, 1], (FLU 0) R(A) ⊆ R(A); (FL0) R(∅) = ∅, (FU 0) R(Y ) = X; (FL0) R(ˆα) = α ˆ , (FU 0) R(ˆα) = α ˆ . Theorem 3 Let R be an arbitrary fuzzy relation on X, R and R the lower and up- per generalized fuzzy rough approximation operators. Then we have the following equivalencies: (A) ⊆ A,∀A ∈ F (X), (1) R is reﬂexive ⇐⇒ R ⇐⇒ A ⊆ R(A),∀A ∈ F (X); (2) R is symmetric ⇐⇒ R(1 )(x) = R(1 )(y),∀x, y ∈ X, X\{y} X\{x} ⇐⇒ R(1 )(x) = R(1 )(y),∀x, y ∈ X; {y} {x} (3) R is transitive ⇐⇒ R(A) ⊆ R(R(A)),∀A ∈ F (X), ⇐⇒ R(R(A)) ⊆ R(A),∀A ∈ F (X). What properties do the Euclidean fuzzy rough approximation operators possess? We will give the answer to it in the following section. 3. Construction of (⊥,)-generalized Fuzzy Rough Sets If R ∈ F (X × Y ) and A ∈ F (Y ), then we deﬁne R ◦ A(x) = ∨ {R(x, y)A(y)} y∈Y and R ◦ ˆ A(x) = ∧ {R(x, y)⊥A(y)} for all x ∈ X. The concept of (⊥,)-generalized ⊥ y∈Y fuzzy rough sets determined by a t-norm and its dual t-conorm ⊥ was introduced by Hu and Huang in [14] as follows. Deﬁnition 2 Let X and Y be two ﬁnite and nonempty universes of discourse and R a fuzzy relation from X to Y. For a fuzzy set A ∈ F (Y ), the ⊥-lower and -upper approximation of A, denoted respectively by R (A) and R (A), are fuzzy sets of X with the following memberships: R (A)(x) = ∧ {A(y)⊥(1− R(x, y))}, y∈Y R (A)(x) = ∨ {A(y)R(x, y)}. y∈Y The triple (X, Y, R) is referred to as a (⊥,)-generalized fuzzy rough approximation space and the pair of fuzzy sets (R (A), R (A)) is called a (⊥,)-generalized fuzzy rough set of A. Specially, if ⊥ = ∨ and = ∧,(R (A), R (A)) is written as (R(A), R(A)), which is a generalized fuzzy rough set [18] (see also [19]). If X = Y , then (R (A), R (A)) is called a (⊥,)-fuzzy rough set. Fuzzy Inf. Eng. (2012) 3: 249-259 253 −1 −1 Similarly, we may also deﬁne an analogous pair (R (A), R (A)) for A ∈ F (Y ) as follows (∀x ∈ X): −1 R (A)(x) = ∧ {A(y)⊥(1− R(y, x))}, y∈Y −1 R (A)(x) = ∨ {A(y)R(y, x)}. y∈Y Theorem 4 Suppose that (R (A), R (A)) is an arbitrary (⊥,)-generalized fuzzy rough set with respect to the (⊥,)-generalized fuzzy rough approximation space (X, Y, R). For any A, B ∈ F (Y ),α ∈ [0, 1], we have c c c c (FL1) R (A) = (R (A )) , (FU 1) R (A) = (R (A )) ; ⊥ ⊥ (FL2) R (A∪ α ˆ ) = R (A)∪ α ˆ , (FU 2) R (A∩ α ˆ ) = R (A)∩ α ˆ ; ⊥ ⊥ ⊥ ⊥ (FL3) R (A∩ B) = R (A)∩ R (B), (FU 3) R (A∪ B) = R (A)∪ R (B); ⊥ ⊥ ⊥ (FL4) A ⊆ B ⇒ R (A) ⊆ R (B), (FU 4) A ⊆ B ⇒ R (A) ⊆ R (B); ⊥ ⊥ (FL5) R (A∪ B) ⊇ R (A)∪ R (B), (FU 5) R (A∩ B) ⊆ R (A)∩ R (B). ⊥ ⊥ ⊥ Proof For any x ∈ X,wehave (FL1) R (A)(x) = ∧ {A(y)⊥(1− R(x, y))} y∈Y = 1− [1−∧ {A(y)⊥(1− R(x, y))}] y∈Y = 1− [∨ {(1− A(y))R(x, y)}] y∈Y = 1− R (A )(x). (FL2) R (A∪ α ˆ )(x) = ∧ {(A∪ α ˆ )(y)⊥(1− R(x, y))} ⊥ y∈Y ⊥ = ∧ {A(y)⊥(1− R(x, y))}⊥α y∈Y = R (A)(x)⊥α = (R (A)∪ α ˆ )(x). (FL3) R (A∩ B)(x) = ∧ {(A∩ B)(y)⊥(1− R(x, y))} y∈Y = ∧ {(A(y)⊥(1− R(x, y))∧ (B(y)⊥(1− R(x, y))} y∈Y = ∧ {A(y)⊥(1− R(x, y))}∧∧ {B(y)⊥(1− R(x, y))} y∈Y y∈Y = R (A)(x)∧ R (B)(x). ⊥ ⊥ (FL4) and (FL5) can be derived from [14]. The others can be proved in a similar way. The proof ends. In the case of (⊥,)-generalized fuzzy rough approximation operators, properties (FL1) and (FU1) show that R and R are dual to each other. Properties with the same number may be viewed as dual properties. In addition, properties (FL2) and (FU2) imply R (Y ) = X and R (∅) = ∅, respectively. Theorem 5 [14] Let R ∈ F (X × Y ). Then R is serial ⇐⇒ R (Y ) = X ⇐⇒ R (A) ⊆ R (A),∀A ∈ F (Y ). Corollary 1 Let R ∈ F (X × Y ). Then R is serial ⇐⇒ R (ˆα) = α ˆ ⇐⇒ R (ˆα) = α ˆ ⇐⇒ R (∅) = ∅,∀α ∈ [0, 1]. Proof It is well known that R (Y ) = X ⇐⇒ R (∅) = ∅ and R (ˆα) = α ˆ ⇐⇒ R (ˆα) = α. ˆ And by Theorem 5, it suﬃces to prove R (ˆα) = α ˆ ⇐⇒ R (∅) = ∅. ⊥ ⊥ ⊥ It is clear that R (ˆα) = α ˆ implies R (∅) = ∅. ⊥ ⊥ Conversely, R (ˆα) = R (ˆα∪∅) = R (∅)∪α ˆ = α ˆ . The proof is complete. ⊥ ⊥ ⊥ 254 Wen-Sheng Du · Bao-Qing Hu · Yan Zhao (2012) Theorem 6 [14] Let (R (A), R (A)) be an arbitrary (⊥,)-fuzzy rough set with re- spect to the (⊥,)-fuzzy rough approximation space (X, R). Then we have the follow- ing equivalencies: (1) R is reﬂexive ⇐⇒ R (A) ⊆ A,∀A ∈ F (X), ⇐⇒ A ⊆ R (A),∀A ∈ F (X); (2) R is symmetric ⇐⇒ R (1 )(x) = R (1 )(y),∀x, y ∈ X, X\{y} X\{x} ⊥ ⊥ ⇐⇒ R (1 )(x) = R (1 )(y),∀x, y ∈ X; {y} {x} (3) Ris -transitive ⇐⇒ R (A) ⊆ R (R (A)),∀A ∈ F (X), ⊥ ⊥ ⊥ ⇐⇒ R (R (A)) ⊆ R (A),∀A ∈ F (X). Corollary 2 If R is a reﬂexive fuzzy relation on X, then for all A ∈ F (X), we have R (R (A)) ⊆ R (A) ⊆ R (R (A)) ⊆ R (A) ⊆ R (R (A)). ⊥ ⊥ ⊥ ⊥ Proof If R is reﬂexive, by Theorem 6 we have R (A) ⊆ A ⊆ R (A),∀A ∈ F (X). Furthermore, in terms of the monotonicity of R and replace A with R (A), we can get R (R (A)) ⊆ R (A) ⊆ R (R (A)) and R (R (A)) ⊆ R (A) ⊆ R (R (A)), ⊥ ⊥ ⊥ ⊥ ⊥ respectively. This concludes the proof that R (R (A)) ⊆ R (A) ⊆ R (R (A)) ⊆ R (A) ⊆ ⊥ ⊥ ⊥ ⊥ R (R (A)). Corollary 3 If R is a reﬂexive and -transitive fuzzy relation on X, then for all A ∈ F (X), we have R (A) = R (R (A)), R (R (A)) = R (A). ⊥ ⊥ ⊥ Proof Since R is reﬂexive, R (A) ⊆ A holds for all A ∈ F (X). By substituting A for R (A), we have R (R (A)) ⊆ R (A). ⊥ ⊥ ⊥ ⊥ On the other hand, since R is -transitive, according to Theorem 6, the following holds R (A) ⊆ R (R (A)). ⊥ ⊥ ⊥ Thus R (A) = R (R (A)). Likewise, we can conclude that R (R (A)) = R (A). ⊥ ⊥ ⊥ Theorem 7 If R is a reﬂexive or second serial fuzzy relation on X, then for all A ∈ F (X), R (A)(x) = A(x) , x∈X x∈X R (A)(x) = A(x) . x∈X x∈X Proof The assumption that R is reﬂexive or second serial implies ∨ {R(x, y)} = x∈X Fuzzy Inf. Eng. (2012) 3: 249-259 255 1,∀y ∈ X. Thus R (A)(x) = A(y)R(x, y) = A(y)R(x, y) x∈X x∈X y∈X y∈X x∈X = A(y) R(x, y) = A(y) y∈X x∈X y∈X = A(x) . x∈X Since the other one can be proved in an analogous way, this completes the proof. It has been proved that R is Euclidean iﬀ R(R(A)) ⊆ R(A)iﬀ R(A) ⊆ R(R(A)) in rough fuzzy approximation spaces for all A ∈ F (X). It wonders whether these properties still hold for fuzzy rough approximation oper- ators or (⊥,)-fuzzy rough approximation operators in general. Example 1 Let X = {1, 2, 3} and R be a fuzzy relation on X deﬁned by the fuzzy matrix as follows ⎡ ⎤ ⎢ ⎥ 0.11.00.6 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ R = 0.11.00.6 . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0.10.60.7 Routine calculation shows that R is Euclidean. Since ∧ is the biggest t-norm, R is - Euclidean for any t-norm. Let A = 0.8/1+ 0.3/2+ 0.5/3. Then it can be calculated that R(R(A)) = 0.4/1+ 0.4/2+ 0.4/3 and R(A) = 0.3/1+ 0.3/2+ 0.4/3. Unluckily, as can be seen from the previous example, the counterparts of properties for Euclidean fuzzy rough approximation operators do not hold, let alone (⊥,)- fuzzy rough approximation operators. However, we have the following theorem. Theorem 8 [19] Let (R (A), R (A)) be an arbitrary (⊥,)-fuzzy rough set with re- spect to the (⊥,)-fuzzy rough approximation space (X, R). Then we have the follow- ing characterizations: −1 Ris -Euclidean ⇐⇒ R (R (A)) ⊇ R (A),∀A ∈ F (X), ⊥ ⊥ ⊥ −1 ⇐⇒ R (R (A)) ⊆ R (A),∀A ∈ F (X). In consequence, we can conclude that, for arbitrary fuzzy rough set (R(A), R(A)), −1 −1 R is Euclidean iﬀ R (R(A)) ⊇ R(A)iﬀ R (R(A)) ⊆ R(A),∀A ∈ F (X). Example 2 (Continued From Example 1) By computing, the considering results are summarized in Table 1. Corollary 4 If R is a reﬂexive and -Euclidean fuzzy relation on X, then for all A ∈ F (X), we have −1 −1 R (R (A)) = R (A), R (R (A)) = R (A). ⊥ ⊥ ⊥ 256 Wen-Sheng Du · Bao-Qing Hu · Yan Zhao (2012) Table 1: Discussion on the properties of-Euclidean fuzzy rough sets. −1 −1 R (A) R (A) R (R (A)) R (R (A)) R (R (A)) ⊥ ⊥ ⊥ ⊥ = ∧ (0.5,0.5,0.5) (0.3,0.3,0.4) (0.1,0.5,0.5) (0.9,0.3,0.4) (0.4,0.4,0.4) = (0.3,0.3,0.0) (0.3,0.3,1.0) (0.0,0.3,0.0) (1.0,0.3,1.0) (0.6,0.6,0.7) = (0.3,0.3,0.0) (0.3,0.3,1.0) (0.0,0.3,0.0) (1.0,0.3,1.0) (0.6,0.6,0.7) = ˆ· (0.3,0.3,0.35) (0.3,0.3,0.58) (0.04,0.3,0.25) (0.93,0.3,0.58) (0.35,0.35,0.40) Note: Drastic product : ab = min{a, b},if a = 1or b = 1, otherwise 0. Łukasiewicz product : a b = max{0, a+ b− 1}. algebraic product ˆ· : aˆ·b = ab. Proof The proof is similar to the proof of Corollary 2. 4. Axiomatic Characterization of (⊥,)-fuzzy Rough Approximation Operators In this section, we will study the axiomatic characterization of (⊥,)-fuzzy rough sets by deﬁning a pair of fuzzy set-theoretic operators. Deﬁnition 3 [17, 18] Let L, H be two fuzzy set-theoretic operators from F (Y ) to F (X). They are referred to as dual operators if for all A ∈ F (Y ), L and H sat- isfy the axiom ( fl1), or equivalently, the axiom ( fu1). c c c c ( fl1) L(A) = (H(A )) , ( fu1) H(A) = (L(A )) . Lemma 1 [14] Let (R (A), R (A)) be an arbitrary (⊥,)-generalized fuzzy rough set with respect to the (⊥,)-generalized fuzzy rough approximation space (X, Y, R).For all (x, y) ∈ X × Y, we have R (1 )(x) = R(x, y), {y} R (1 )(x) = 1− R(x, y). Y\{y} Theorem 9 Suppose that L, H : F (Y ) → F (X) are two dual operators. Then there exists a fuzzy relation R from X to Y such that for all A ∈ F (Y ),L(A) = R (A) , H(A) = R (A) iﬀ L satisﬁes the axioms ( fl2) and ( fl3), or equivalently, H satisﬁes the axioms ( fu2) and ( fu3), for all A, B ∈ F (Y ),α ∈ [0, 1]. ( fl2) L(ˆα∪ A) = α ˆ ∪ L(A), ( fu2) H(ˆα∩ A) = α ˆ ∩ H(A); ⊥ ⊥ ( fl3) L(A∩ B) = L(A)∩ L(B), ( fu3) H(A∪ B) = H(A)∪ H(B). Proof “⇐” Let H obey (fu2) and (fu3). We can deﬁne a fuzzy relation R from X to Y as follows: R(x, y) = H(1 )(x), ∀(x, y) ∈ X × Y . {y} It should be noted that A = ∪ (1 ∩ A(y)),∀A ∈ F (Y ). Then for∀x ∈ X, y∈Y {y} R (A)(x) = R (1 ∩ A(y)) (x) = R (1 ∩ A(y)) (x) {y} {y} y∈Y y∈Y = R (1 )∩ A(y) (x) = (R (1 )(x)A(y) {y} {y} y∈Y y∈Y Fuzzy Inf. Eng. (2012) 3: 249-259 257 = R(x, y)A(y) = H(1 )(x)A(y) {y} y∈Y y∈Y = H(1 )∩ A(y) (x) = H 1 ∩ A(y) (x) {y} {y} y∈Y y∈Y = H(A)(x), which implies that H(A) = R (A). The proof of the other one is analogous. “⇒” It follows immediately from Theorem 4. Deﬁnition 4 Let L, H : F (Y ) → F (X) be a pair of dual operators. If L satisﬁes ax- ioms (ﬂ2) and (ﬂ3) or equivalently H satisﬁes axioms (fu2) and (fu3), then the system (F (X), F (Y ), ∩, ∪, , L, H) is referred to as a (⊥,)-fuzzy rough set algebra, and L and H are referred to as (⊥,)-fuzzy rough approximation operators. Theorem 10 Let L, H : F (Y ) → F (X) be a pair of dual (⊥,)-fuzzy rough approx- imation operators. Then there exists a serial fuzzy relation R from X to Y such that L(A) = R (A),H(A) = R (A), for all A ∈ F (Y ) iﬀ L satisﬁes the axiom ( fl0) or ⊥ ⊥ ( fl4), or equivalently, H satisﬁes the axiom ( fu0) or ( fu4), or L and H satisfy the axiom ( flu0). ( flu0) L(A) ⊆ H(A), ( fl0) L(ˆα) = α ˆ , ( fu0) H(ˆα) = α ˆ , ( fl4) L(∅) = ∅, ( fu4) H(Y ) = X, where∀α ∈ [0, 1]. Proof “⇒” It follows immediately from Theorem 5 and Corollary 1. “⇐” It follows immediately from Theorems 5 and 9, Corollary 1. Theorem 11 Let L, H : F (X) → F (X) be a pair of dual (⊥,)-fuzzy rough approxi- mation operators. Then there exists a reﬂexive (symmetric,-transitive, respectively) fuzzy relation R on X such that L(A) = R (A),H(A) = R (A), for all A ∈ F (X), ⊥ ⊥ ∀x, y ∈ Xiﬀ L satisﬁes the axiom ( fl5) (( fl6), ( fl7), respectively), or equivalently, H satisﬁes the axiom ( fu5) (( fu6), ( fu7), respectively). ( fl5) L(A) ⊆ A; ( fl6) L(1 )(x) = L(1 )(y); ( fu6) H(1 )(x) = H(1 )(y). X\{y} X\{x} {y} {x} ( fl7) L(A) ⊆ L(L(A)); ( fu7) H(H(A)) ⊆ H(A). Proof “⇒” It follows immediately from Theorem 6. “⇐” It follows immediately from Theorems 6 and 9. 5. Conclusion Rough set theory can be developed in at least two ways, the constructive and the axiomatic approaches. In this paper, we have developed a general framework for the study of (⊥,)-generalized fuzzy rough sets, which can be regarded as the completion of the work by Hu and Huang [14]. 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Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Sep 1, 2012
Keywords: (⊥, ⊤)-fuzzy rough sets; (⊥, ⊤)-generalized fuzzy rough sets; Fuzzy relation; Fuzzy sets