On the topological properties of fuzzy rough sets - Semantic Scholar
Fuzzy Sets and Systems 151 (2005) 601 – 613 www.elsevier.com/locate/fss
On the topological properties of fuzzy rough sets Keyun Qina,∗ , Zheng Peib a Department of Applied Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China b College of Computers and Mathematical-Physical Science, Xihua University, Chengdu, Sichuan, 610039, China
Received 21 September 2003; received in revised form 7 June 2004; accepted 31 August 2004 Available online 23 September 2004
Abstract This paper is devoted to the discussion of the relationship between fuzzy rough set models and fuzzy topologies on a finite universe. The (TC) axiom for fuzzy topology is proposed. It is proved that the set of all lower approximation sets based on a reflexive and transitive fuzzy relation consists of a fuzzy topology which satisfies (TC) axiom; and conversely, a fuzzy topology which obey (TC) axiom is just the set of all lower approximation sets under a reflexive and transitive fuzzy relation. That is to say, there exists a one-to-one correspondence between the set of all reflexive and transitive fuzzy relations and the set of all fuzzy topologies which satisfy (TC) axiom. © 2004 Elsevier B.V. All rights reserved. Keywords: Rough set; Fuzzy rough set; Fuzzy relation; (TC) Axiom; Approximation operator; Fuzzy topology
1. Introduction The theory of rough sets was firstly proposed by Pawlak [13,14]. It is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. The successful application of rough set theory in a variety of problems has amply demonstrated its usefulness. A key notion in Pawlak rough set model is equivalence relation. The equivalence classes are the building blocks for the construction of the lower and upper approximations. By replacing the equivalence relation ∗ Corresponding author. Tel.: +86-28-7602468; fax: +86-28-7600764.
E-mail address: [email protected] (K. Qin). 0165-0114/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2004.08.017
602
K. Qin, Z. Pei / Fuzzy Sets and Systems 151 (2005) 601 – 613
by an arbitrary binary relation, different kinds of generalization in Pawlak rough set models were obtained. It is proved that, under a crisp reflexive and transitive relation, the pair of lower and upper approximation operators is just a pair of interior and closure operators of a topology [6,21,22]. In [2,3,10,11,16], the concept of fuzzy rough sets was proposed by replacing crisp binary relations by fuzzy relations on the universe. Ref. [9] provides the axiomatics for fuzzy rough set model. Ref. [20] 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. In axiomatic approach, various classes of fuzzy rough approximation operators are characterized by different sets of axioms, these axioms guarantee the existence of certain types of fuzzy relations producing the same operators. This paper is devoted to the discussion of the relationship between fuzzy rough set models and fuzzy topologies on a finite universe. The (TC) axiom for fuzzy topology is proposed. It is proved that the set of all lower approximation sets of fuzzy sets based on a reflexive and transitive fuzzy relation consists of a fuzzy topology which satisfies (TC) axiom, and conversely, a fuzzy topology which satisfies (TC) axiom is just the set of all lower approximation sets under a reflexive and transitive fuzzy relation. It is important to note that in this paper we only consider finite universe. Results obtained for finite universe may not necessarily hold if the universe is infinite.
2. Preliminaries This section presents a review of some fundamental notions of fuzzy rough set and fuzzy topology. We refer to [2,4,7,8,15,17,19] for details. In what follows, we denote by F (U ) the set of all fuzzy subsets of U. Definition 1. Let U be nonempty finite set, R be a fuzzy relation on U. (U, R) is called an approximation space, and ∀A ∈ F (U ), R(A) and R(A), which are called upper approximation and lower approximation of A about (U, R), respectively, are fuzzy sets defined as following: R(A)(x) = ∨u∈U (R(x, u) ∧ A(u)), R(A)(x) = ∧u∈U ((1 − R(x, u)) ∨ A(u)),
x ∈ U, x ∈ U.
(1) (2)
The upper- and lower approximation can be defined based on general t-norms and fuzzy negations. In this paper, the standard operations only were considered. Pawlak rough sets is a special case of Definition 1, that is, if R is a crisp equivalence relation and A ⊆ U , then R(A) = {x ∈ U |[x]R ∩ A = ∅},
(3)
R(A) = {x ∈ U |[x]R ⊆ A},
(4)
where [x]R is an equivalence class of x about R.
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Theorem 2 (Thiele [17]). For any R ∈ F (U × U ), A, B ∈ F (U ), (1) R(U ) = U , (2) R(A ∩ B) = R(A) ∩ R(B), (3) R(∅) = ∅, (4) R(A ∪ B) = R(A) ∪ R(B), (5) if A ⊆ B, then R(A) ⊆ R(B), R(A) ⊆ R(B), (6) the pair (R, R) is dual, i.e., R(A) =∼ R(∼ A), where ∼ A is the complement of A, that is, ∀x ∈ U , ∼ A(x) = 1 − A(x). y ∈ F (U ) is called a fuzzy point if y ∈ U , ∈ [0, 1], and if x = y, y (x) = 0 otherwise
(5)
∀ ∈ [0, 1], a fuzzy set that takes constant is denoted by , that is, ∀x ∈ U , (x) = . T ⊆ F (U ) is called a fuzzy topology on U [8,12] if T satisfies (1) U, ∅ ∈ T ; (2) if A, B ∈ T , then A ∩ B ∈ T ; (3) if T1 ⊆ T , then ∪{A|A ∈ T1 } ∈ T ; (4) ∀ ∈ [0, 1], ∈ T . If T is a fuzzy topology on U, then A ∈ T is called an open set of T. If ∼ A ∈ T , then A is called a closed set of T. Remark 3. As for the definition of fuzzy topology, sometimes only (1)–(3) are demanded to be satisfied [8]. No matter which definition is adopted, they are generalization of classical topology concept.
3. Approximation space based on reflexive and transitive fuzzy relation versus fuzzy topology Let U be a nonempty finite set, R ∈ F (U × U ) a fuzzy relation on U. R is called reflexive if ∀x ∈ U , R(x, x) = 1. R is called transitive if ∀x, y, z ∈ U , R(x, y) ∧ R(y, z) R(x, z). The notion of transitivity of fuzzy relation is usually defined with respect to a t-norm. In this paper, sup-min transitive fuzzy relations are considered, together with the standard t-norm. Theorem 4 (Jusheng et al. [5], Thiele [18], and Weizhi Wu et al. [20]). If R is an arbitrary relation on U, then R is reflexive if and only if ∀A ∈ F (U ), (1) R(A) ⊆ A. (2) A ⊆ R(A). Theorem 5 (Jusheng et al. [5], Thiele [18], and Weizhi Wu et al. [20]). If R is an arbitrary relation on U, then R is transitive if and only if ∀A ∈ F (U ), (1) R(R(A)) ⊇ R(A). (2) R(R(A)) ⊆ R(A).
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Theorem 6 (Weizhi Wu et al. [20]). If R is a reflexive and transitive fuzzy relation on U, then the following properties hold: (1) R(R(A)) = R(A). (2) R(R(A)) = R(A). 3.1. Fuzzy topologies generated by approximation spaces In this section, we suppose that U is a nonempty finite set, R : U × U −→ [0, 1] is a reflexive and transitive binary fuzzy relation on U. Theorem 7. ∀Aj ∈ F (U ), j ∈ I (I is an index set), R R Aj = R(Aj ). j ∈I
(6)
j ∈I
Proof. ∀j ∈ I , ∼ ∪j ∈I R(Aj ) = ∩j ∈I ∼ R(Aj ) ⊆∼ R(Aj ), so, R(∼ ∪j ∈I R(Aj )) ⊆ R(∼ R(Aj )), and hence R(∼ ∪j ∈I R(Aj )) ⊆ ∩ j ∈I R(∼ R(Aj )) = ∩ j ∈I ∼ R(∼∼ R(Aj )) = ∩ j ∈I (∼ R(Aj )) =∼ ∪j ∈I R(Aj ). By Theorem 4 (2), we obtain R(∼ ∪j ∈I R(Aj )) =∼ ∪j ∈I R(Aj ), that is, ∪j ∈I R(Aj ) =∼ R(∼ ∪j ∈I R(Aj )) = R(∪j ∈I R(Aj )).
Theorem 8. TR = {R(A)|A ∈ F (U )} is a fuzzy topology on U. Proof. (1) By Theorems 2 (1) and 4 (1), we obtain U, ∅ ∈ TR . (2) If A, B ∈ TR , by Theorem 2 (2), we know A ∩ B ∈ TR . (3) If Aj ∈ TR , j ∈ I , by Theorem 7, ∪j ∈I Aj ∈ TR . (4) ∀ ∈ F (U ), we notice 1 − R(x, x) = 0, so, ∧u∈U (1 − R(x, u)) = 0, and R()(x) = ∧ u∈U ((1 − R(x, u)) ∨ (u)) = ∧ u∈U ((1 − R(x, u)) ∨ ) = (∧u∈U (1 − R(x, u))) ∨ = , that is, R() = , so, ∈ TR . According to (1)–(4), we obtain that TR is a fuzzy topology on U. If R is a crisp binary reflexive and transitive relation on U, then (R) = {R(A)|A ∈ F (U )} is a fuzzy topology on U and all − level topologies of (R) coincide [1]. These observations can be applied to the study of fuzzy automata. In what follows, we call TR the fuzzy topology generated by approximation space (U, R).
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Theorem 9. Let TR be the fuzzy topology as defined in Theorem 8. ∀A ∈ F (U ), (1) R(A) = i(A) = ∪{R(B)|R(B) ⊆ A}, (2) R(A) = c(A) = ∩{∼ R(B)| ∼ R(B) ⊇ A}, where i and c are interior operator and closure operator of TR , respectively. Proof. (1) By ∪{R(B)|R(B) ⊆ A} ⊆ A, we obtain R(∪{R(B)|R(B) ⊆ A}) ⊆ R(A). According to Theorem 7, R(∪{R(B)|R(B) ⊆ A}) = ∪{R(B)|R(B) ⊆ A}, so, ∪{R(B)|R(B) ⊆ A} ⊆ R(A). On the other hand, by R(A) ⊆ A, we obtain R(A) ⊆ ∪{R(B)|R(B) ⊆ A}, so, ∪{R(B)|R(B) ⊆ A} = R(A). (2) By the duality of R and R and (1), we have R(A) =∼ R(∼ A) =∼ (∪{R(B)|R(B) ⊆∼ A}) = ∩{∼ R(B)| ∼ R(B) ⊇ A}.
Theorem 10. Let TR be the fuzzy topology as defined in Theorem 8. ∀x, y ∈ U , B(x), R(x, y) = B∈C(x,y)
where C(x, y) = {B| ∼ B ∈ TR , B(x) < B(y)}. Proof. ∀x, y ∈ U , if B ∈ C(x, y), then ∼ B ∈ TR and B(x) < B(y). By R(B) = c(B) = B, we obtain B(x) = R(B)(x) = ∨u∈U (R(x, u) ∧ B(u)) R(x, y) ∧ B(y). As B(y) > B(x), so, R(x, y) B(x), and hence R(x, y) ∧B∈C(x,y) B(x). On the other hand, by R(y1 ) = c(y1 ) and R(y1 )(x) = c(y1 )(x) = = =
∨ u∈U (R(x, u) ∧ y1 (u)) = R(x, y), ∧ {D(x)| ∼ D ∈ TR , D ⊇ y1 } ∧ {D(x)| ∼ D ∈ TR , D(y) = 1} ∧ {D(x)| ∼ D ∈ TR , D(y) = 1, D(x) < 1},
we obtain R(x, y) = ∧ {D(x)| ∼ D ∈ TR , D(y) = 1, D(x) < 1} ∧ {B(x)| ∼ B ∈ TR , B(x) < B(y)} = ∧ B∈C(x,y) B(x). So, R(x, y) = ∧B∈C(x,y) B(x) holds.
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Remark 11. In approximation space (U, R), the upper approximation and lower approximation operators can be defined generally based on T-norm and fuzzy negation as follows: for each A ∈ F (U ), R(A)(x) = ∨u∈U (R(x, u)A(u)),
R(A)(x) = ∧u∈U (R (x, u)⊥A(u)),
x ∈ U,
(7)
x ∈ U,
(8)
in which is a T-norm, ⊥ is a T-conorm such that a⊥b = (a b ) hold for every a, b ∈ [0, 1]. Clearly, the Definition 1 is a special case of Eqs. (7) and (8), in which = ∧, ⊥ = ∨ and a = 1 − a. If we take Eqs. (7) and (8) as the upper and lower approximation operators, then the conclusions of Theorems 7–9 also hold. However, the conclusion of Theorem 10 may not be true. The following condition for fuzzy topology T is called (TC) axiom, • (TC) axiom: ∀x, y ∈ U , if there exists open set A ∈ T such that A(x) > A(y), then there exists open set A∗ ∈ T such that A∗ (y) = 0 and A∗ (x) A(x). Theorem 12. TR satisfies (TC) axiom. Proof. ∀x, y ∈ U , assume that there exists open set A ∈ TR such that A(x) > A(y). Let A∗ =∼ c(y1 ), then A∗ ∈ TR and c(y1 ) = ∩{D| ∼ D ∈ T , D ⊇ y1 } = ∩{D| ∼ D ∈ T , D(y) = 1}, so, we obtain A∗ (y) = 1 − c(y1 )(y) = 0, A∗ (x) = 1 − c(y1 )(x) = 1 − R(y1 )(x) = 1 − ∨u∈U (R(x, u) ∧ y1 (u)) = 1 − R(x, y) = 1 − ∧B∈C(x,y) B(x). Notice that ∼ A is a closed set, and ∼ A(x) A(y), then there exists an open set A∗ ∈ TR such that A∗ (y) = 0 and A∗ (x) = 1. Proof. For x, y ∈ U , assume that there exists an open set A ∈ TR such that A(x) > A(y), by Theorem 23, (x, y)∈R. Let B ∈ F (U ) be defined as B(u) =
1 0
if u = y, if u = y.
(12)
By R(B)(x) = c(B)(x), we obtain c(B)(x) = sup{B(v)|v ∈ SR (x)} = 0. On the other hand, c(B) = ∩{D| ∼ D ∈ TR , D ⊇ B} = ∩{D| ∼ D ∈ TR , D(y) = 1}, so, C(B)(y) = ∧{D(y)| ∼ D ∈ TR , D(y) = 1} = 1. Let A∗ =∼ c(B), then A∗ ∈ TR , and A∗ (y) = 0, A∗ (x) = 1.
4.2. Approximation spaces generated by fuzzy topologies Let T be a fuzzy topology on U, and T satisfy the following condition: • (TC ) axiom: ∀x, y ∈ U , if there exists an open set A ∈ T such that A(x) > A(y), then there exists an open set A∗ ∈ T such that A∗ (y) = 0 and A∗ (x) = 1. Theorem 25. If RT is the fuzzy relation defined by Eq. (9), then RT is a crisp binary relation, and ∀x, y ∈ U , (x, y) ∈ RT if and only if ∀A ∈ T , A(x) A(y). Proof. (1) For x, y ∈ U , if ∀A ∈ T , A(x) A(y), then for any closed set B, B(x) B(y), so, C(x, y) = ∅, and hence RT (x, y) = ∧B∈C(x,y) B(x) = 1. (2) If there exists an open set A ∈ T such that A(x) > A(y), by (TC ) axiom, there exists an open set A∗ ∈ T such that A∗ (y) = 0 and A∗ (x) = 1. Notice that ∼ A∗ ∈ C(x, y) and ∼ A∗ (x) = 0, so, RT (x, y) = ∧B∈C(x,y) B(x) = 0. According to (1) and (2), we obtain that RT is a crisp binary relation, and ∀x, y ∈ U , (x, y) ∈ RT if and only if ∀A ∈ T , A(x) A(y).
Theorem 26. If RT is the fuzzy relation defined by Eq. (9), then RT is reflexive and transitive. Proof. ∀x, y, z ∈ U , obviously, (x, x) ∈ RT , that is, R is reflexive. If (x, y) ∈ RT and (y, z) ∈ RT , then ∀A ∈ T , A(x) A(y) and A(y) A(z), so, A(x) A(z), that is, (x, z) ∈ RT . By Theorem 19, we have the following theorem,
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Theorem 27. ∀A ∈ F (U ), (1) RT (A) = c(A); (2) RT (A) = i(A). Remark 28. If the fuzzy topology T does not satisfy (TC ) axiom, then RT (A) = i(A) may not necessarily hold. Example 29. Let U = {x, y, z}, and A ∈ F (U ) be defined as A(x) = 0.5, A(y) = A(z) = 0. Assume T is the fuzzy topology generated by {A, U, ∅} ∪ {| ∈ [0, 1]}, and B ∈ F (U ) is defined as B(y) = 0.25, B(x) = B(z) = 1. As A is an open set and A(x) > A(y), A(x) > A(z), it follows that (x, y)∈RT and (x, z)∈RT and hence RT (B)(x) = inf{B(u)|u ∈ SR (x)} = B(x) = 1; i(B)(x) = sup{C(x)|C ∈ T , C ⊆ T } = 0.5, that is RT (B) = i(B). 5. Conclusion In this paper, we have discussed the relationship between approximation spaces and fuzzy topologies, proved that there exists one-to-one correspondences between the set of all fuzzy reflexive and transitive relations on a finite universe and the set of all fuzzy topologies on the same universe which satisfies (TC) axiom. Acknowledgements The authors are grateful to the referees for their valuable comments and suggestions. This work has been supported by the National Natural Science Foundation of China (Grant No. 60074014). References [1] K.S. Arun, S.P. Tiwari, On relationships among fuzzy approximation operators, and fuzzy topology, and fuzzy automata, Fuzzy Sets and Systems 138 (2003) 197–204. [2] D. Dubois, H. Prade, Rough fuzzy set and fuzzy rough sets, Internat. J. Gen. Systems 17 (1990) 191–209. [3] D. Dubois, H. Prade, Putting fuzzy sets and rough sets together, in: R. Slowinski (Ed.), Intelligent Decision Support, Kluwer Academic, Dordrecht, 1992, pp. 203–232. [4] W. Gahler, A.S. Abb-Allah, A. Kandil, On extended fuzzy topologies, Fuzzy Sets and Systems 109 (2000) 149–172. [5] Jusheng Mi, Weizhi Wu, Wenxiu Zhang, Constructive and axiomatic approaches for the study of the theory of rough sets, Pattern Recognition Artificial Intelligence 15 (2002) 280–284. [6] J. Kortelainen, On the relationship between modified sets, topological spaces and rough sets, Fuzzy Sets and Systems 61 (1994) 91–95. [7] L.I. Kuncheva, Fuzzy rough sets: application to feature selection, Fuzzy Sets and Systems 51 (1992) 147–153.
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On the topological properties of fuzzy rough sets Keyun Qina,∗ , Zheng Peib a Department of Applied Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China b College of Computers and Mathematical-Physical Science, Xihua University, Chengdu, Sichuan, 610039, China
Received 21 September 2003; received in revised form 7 June 2004; accepted 31 August 2004 Available online 23 September 2004
Abstract This paper is devoted to the discussion of the relationship between fuzzy rough set models and fuzzy topologies on a finite universe. The (TC) axiom for fuzzy topology is proposed. It is proved that the set of all lower approximation sets based on a reflexive and transitive fuzzy relation consists of a fuzzy topology which satisfies (TC) axiom; and conversely, a fuzzy topology which obey (TC) axiom is just the set of all lower approximation sets under a reflexive and transitive fuzzy relation. That is to say, there exists a one-to-one correspondence between the set of all reflexive and transitive fuzzy relations and the set of all fuzzy topologies which satisfy (TC) axiom. © 2004 Elsevier B.V. All rights reserved. Keywords: Rough set; Fuzzy rough set; Fuzzy relation; (TC) Axiom; Approximation operator; Fuzzy topology
1. Introduction The theory of rough sets was firstly proposed by Pawlak [13,14]. It is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. The successful application of rough set theory in a variety of problems has amply demonstrated its usefulness. A key notion in Pawlak rough set model is equivalence relation. The equivalence classes are the building blocks for the construction of the lower and upper approximations. By replacing the equivalence relation ∗ Corresponding author. Tel.: +86-28-7602468; fax: +86-28-7600764.
E-mail address: [email protected] (K. Qin). 0165-0114/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2004.08.017
602
K. Qin, Z. Pei / Fuzzy Sets and Systems 151 (2005) 601 – 613
by an arbitrary binary relation, different kinds of generalization in Pawlak rough set models were obtained. It is proved that, under a crisp reflexive and transitive relation, the pair of lower and upper approximation operators is just a pair of interior and closure operators of a topology [6,21,22]. In [2,3,10,11,16], the concept of fuzzy rough sets was proposed by replacing crisp binary relations by fuzzy relations on the universe. Ref. [9] provides the axiomatics for fuzzy rough set model. Ref. [20] 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. In axiomatic approach, various classes of fuzzy rough approximation operators are characterized by different sets of axioms, these axioms guarantee the existence of certain types of fuzzy relations producing the same operators. This paper is devoted to the discussion of the relationship between fuzzy rough set models and fuzzy topologies on a finite universe. The (TC) axiom for fuzzy topology is proposed. It is proved that the set of all lower approximation sets of fuzzy sets based on a reflexive and transitive fuzzy relation consists of a fuzzy topology which satisfies (TC) axiom, and conversely, a fuzzy topology which satisfies (TC) axiom is just the set of all lower approximation sets under a reflexive and transitive fuzzy relation. It is important to note that in this paper we only consider finite universe. Results obtained for finite universe may not necessarily hold if the universe is infinite.
2. Preliminaries This section presents a review of some fundamental notions of fuzzy rough set and fuzzy topology. We refer to [2,4,7,8,15,17,19] for details. In what follows, we denote by F (U ) the set of all fuzzy subsets of U. Definition 1. Let U be nonempty finite set, R be a fuzzy relation on U. (U, R) is called an approximation space, and ∀A ∈ F (U ), R(A) and R(A), which are called upper approximation and lower approximation of A about (U, R), respectively, are fuzzy sets defined as following: R(A)(x) = ∨u∈U (R(x, u) ∧ A(u)), R(A)(x) = ∧u∈U ((1 − R(x, u)) ∨ A(u)),
x ∈ U, x ∈ U.
(1) (2)
The upper- and lower approximation can be defined based on general t-norms and fuzzy negations. In this paper, the standard operations only were considered. Pawlak rough sets is a special case of Definition 1, that is, if R is a crisp equivalence relation and A ⊆ U , then R(A) = {x ∈ U |[x]R ∩ A = ∅},
(3)
R(A) = {x ∈ U |[x]R ⊆ A},
(4)
where [x]R is an equivalence class of x about R.
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603
Theorem 2 (Thiele [17]). For any R ∈ F (U × U ), A, B ∈ F (U ), (1) R(U ) = U , (2) R(A ∩ B) = R(A) ∩ R(B), (3) R(∅) = ∅, (4) R(A ∪ B) = R(A) ∪ R(B), (5) if A ⊆ B, then R(A) ⊆ R(B), R(A) ⊆ R(B), (6) the pair (R, R) is dual, i.e., R(A) =∼ R(∼ A), where ∼ A is the complement of A, that is, ∀x ∈ U , ∼ A(x) = 1 − A(x). y ∈ F (U ) is called a fuzzy point if y ∈ U , ∈ [0, 1], and if x = y, y (x) = 0 otherwise
(5)
∀ ∈ [0, 1], a fuzzy set that takes constant is denoted by , that is, ∀x ∈ U , (x) = . T ⊆ F (U ) is called a fuzzy topology on U [8,12] if T satisfies (1) U, ∅ ∈ T ; (2) if A, B ∈ T , then A ∩ B ∈ T ; (3) if T1 ⊆ T , then ∪{A|A ∈ T1 } ∈ T ; (4) ∀ ∈ [0, 1], ∈ T . If T is a fuzzy topology on U, then A ∈ T is called an open set of T. If ∼ A ∈ T , then A is called a closed set of T. Remark 3. As for the definition of fuzzy topology, sometimes only (1)–(3) are demanded to be satisfied [8]. No matter which definition is adopted, they are generalization of classical topology concept.
3. Approximation space based on reflexive and transitive fuzzy relation versus fuzzy topology Let U be a nonempty finite set, R ∈ F (U × U ) a fuzzy relation on U. R is called reflexive if ∀x ∈ U , R(x, x) = 1. R is called transitive if ∀x, y, z ∈ U , R(x, y) ∧ R(y, z) R(x, z). The notion of transitivity of fuzzy relation is usually defined with respect to a t-norm. In this paper, sup-min transitive fuzzy relations are considered, together with the standard t-norm. Theorem 4 (Jusheng et al. [5], Thiele [18], and Weizhi Wu et al. [20]). If R is an arbitrary relation on U, then R is reflexive if and only if ∀A ∈ F (U ), (1) R(A) ⊆ A. (2) A ⊆ R(A). Theorem 5 (Jusheng et al. [5], Thiele [18], and Weizhi Wu et al. [20]). If R is an arbitrary relation on U, then R is transitive if and only if ∀A ∈ F (U ), (1) R(R(A)) ⊇ R(A). (2) R(R(A)) ⊆ R(A).
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Theorem 6 (Weizhi Wu et al. [20]). If R is a reflexive and transitive fuzzy relation on U, then the following properties hold: (1) R(R(A)) = R(A). (2) R(R(A)) = R(A). 3.1. Fuzzy topologies generated by approximation spaces In this section, we suppose that U is a nonempty finite set, R : U × U −→ [0, 1] is a reflexive and transitive binary fuzzy relation on U. Theorem 7. ∀Aj ∈ F (U ), j ∈ I (I is an index set), R R Aj = R(Aj ). j ∈I
(6)
j ∈I
Proof. ∀j ∈ I , ∼ ∪j ∈I R(Aj ) = ∩j ∈I ∼ R(Aj ) ⊆∼ R(Aj ), so, R(∼ ∪j ∈I R(Aj )) ⊆ R(∼ R(Aj )), and hence R(∼ ∪j ∈I R(Aj )) ⊆ ∩ j ∈I R(∼ R(Aj )) = ∩ j ∈I ∼ R(∼∼ R(Aj )) = ∩ j ∈I (∼ R(Aj )) =∼ ∪j ∈I R(Aj ). By Theorem 4 (2), we obtain R(∼ ∪j ∈I R(Aj )) =∼ ∪j ∈I R(Aj ), that is, ∪j ∈I R(Aj ) =∼ R(∼ ∪j ∈I R(Aj )) = R(∪j ∈I R(Aj )).
Theorem 8. TR = {R(A)|A ∈ F (U )} is a fuzzy topology on U. Proof. (1) By Theorems 2 (1) and 4 (1), we obtain U, ∅ ∈ TR . (2) If A, B ∈ TR , by Theorem 2 (2), we know A ∩ B ∈ TR . (3) If Aj ∈ TR , j ∈ I , by Theorem 7, ∪j ∈I Aj ∈ TR . (4) ∀ ∈ F (U ), we notice 1 − R(x, x) = 0, so, ∧u∈U (1 − R(x, u)) = 0, and R()(x) = ∧ u∈U ((1 − R(x, u)) ∨ (u)) = ∧ u∈U ((1 − R(x, u)) ∨ ) = (∧u∈U (1 − R(x, u))) ∨ = , that is, R() = , so, ∈ TR . According to (1)–(4), we obtain that TR is a fuzzy topology on U. If R is a crisp binary reflexive and transitive relation on U, then (R) = {R(A)|A ∈ F (U )} is a fuzzy topology on U and all − level topologies of (R) coincide [1]. These observations can be applied to the study of fuzzy automata. In what follows, we call TR the fuzzy topology generated by approximation space (U, R).
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Theorem 9. Let TR be the fuzzy topology as defined in Theorem 8. ∀A ∈ F (U ), (1) R(A) = i(A) = ∪{R(B)|R(B) ⊆ A}, (2) R(A) = c(A) = ∩{∼ R(B)| ∼ R(B) ⊇ A}, where i and c are interior operator and closure operator of TR , respectively. Proof. (1) By ∪{R(B)|R(B) ⊆ A} ⊆ A, we obtain R(∪{R(B)|R(B) ⊆ A}) ⊆ R(A). According to Theorem 7, R(∪{R(B)|R(B) ⊆ A}) = ∪{R(B)|R(B) ⊆ A}, so, ∪{R(B)|R(B) ⊆ A} ⊆ R(A). On the other hand, by R(A) ⊆ A, we obtain R(A) ⊆ ∪{R(B)|R(B) ⊆ A}, so, ∪{R(B)|R(B) ⊆ A} = R(A). (2) By the duality of R and R and (1), we have R(A) =∼ R(∼ A) =∼ (∪{R(B)|R(B) ⊆∼ A}) = ∩{∼ R(B)| ∼ R(B) ⊇ A}.
Theorem 10. Let TR be the fuzzy topology as defined in Theorem 8. ∀x, y ∈ U , B(x), R(x, y) = B∈C(x,y)
where C(x, y) = {B| ∼ B ∈ TR , B(x) < B(y)}. Proof. ∀x, y ∈ U , if B ∈ C(x, y), then ∼ B ∈ TR and B(x) < B(y). By R(B) = c(B) = B, we obtain B(x) = R(B)(x) = ∨u∈U (R(x, u) ∧ B(u)) R(x, y) ∧ B(y). As B(y) > B(x), so, R(x, y) B(x), and hence R(x, y) ∧B∈C(x,y) B(x). On the other hand, by R(y1 ) = c(y1 ) and R(y1 )(x) = c(y1 )(x) = = =
∨ u∈U (R(x, u) ∧ y1 (u)) = R(x, y), ∧ {D(x)| ∼ D ∈ TR , D ⊇ y1 } ∧ {D(x)| ∼ D ∈ TR , D(y) = 1} ∧ {D(x)| ∼ D ∈ TR , D(y) = 1, D(x) < 1},
we obtain R(x, y) = ∧ {D(x)| ∼ D ∈ TR , D(y) = 1, D(x) < 1} ∧ {B(x)| ∼ B ∈ TR , B(x) < B(y)} = ∧ B∈C(x,y) B(x). So, R(x, y) = ∧B∈C(x,y) B(x) holds.
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Remark 11. In approximation space (U, R), the upper approximation and lower approximation operators can be defined generally based on T-norm and fuzzy negation as follows: for each A ∈ F (U ), R(A)(x) = ∨u∈U (R(x, u)A(u)),
R(A)(x) = ∧u∈U (R (x, u)⊥A(u)),
x ∈ U,
(7)
x ∈ U,
(8)
in which is a T-norm, ⊥ is a T-conorm such that a⊥b = (a b ) hold for every a, b ∈ [0, 1]. Clearly, the Definition 1 is a special case of Eqs. (7) and (8), in which = ∧, ⊥ = ∨ and a = 1 − a. If we take Eqs. (7) and (8) as the upper and lower approximation operators, then the conclusions of Theorems 7–9 also hold. However, the conclusion of Theorem 10 may not be true. The following condition for fuzzy topology T is called (TC) axiom, • (TC) axiom: ∀x, y ∈ U , if there exists open set A ∈ T such that A(x) > A(y), then there exists open set A∗ ∈ T such that A∗ (y) = 0 and A∗ (x) A(x). Theorem 12. TR satisfies (TC) axiom. Proof. ∀x, y ∈ U , assume that there exists open set A ∈ TR such that A(x) > A(y). Let A∗ =∼ c(y1 ), then A∗ ∈ TR and c(y1 ) = ∩{D| ∼ D ∈ T , D ⊇ y1 } = ∩{D| ∼ D ∈ T , D(y) = 1}, so, we obtain A∗ (y) = 1 − c(y1 )(y) = 0, A∗ (x) = 1 − c(y1 )(x) = 1 − R(y1 )(x) = 1 − ∨u∈U (R(x, u) ∧ y1 (u)) = 1 − R(x, y) = 1 − ∧B∈C(x,y) B(x). Notice that ∼ A is a closed set, and ∼ A(x) A(y), then there exists an open set A∗ ∈ TR such that A∗ (y) = 0 and A∗ (x) = 1. Proof. For x, y ∈ U , assume that there exists an open set A ∈ TR such that A(x) > A(y), by Theorem 23, (x, y)∈R. Let B ∈ F (U ) be defined as B(u) =
1 0
if u = y, if u = y.
(12)
By R(B)(x) = c(B)(x), we obtain c(B)(x) = sup{B(v)|v ∈ SR (x)} = 0. On the other hand, c(B) = ∩{D| ∼ D ∈ TR , D ⊇ B} = ∩{D| ∼ D ∈ TR , D(y) = 1}, so, C(B)(y) = ∧{D(y)| ∼ D ∈ TR , D(y) = 1} = 1. Let A∗ =∼ c(B), then A∗ ∈ TR , and A∗ (y) = 0, A∗ (x) = 1.
4.2. Approximation spaces generated by fuzzy topologies Let T be a fuzzy topology on U, and T satisfy the following condition: • (TC ) axiom: ∀x, y ∈ U , if there exists an open set A ∈ T such that A(x) > A(y), then there exists an open set A∗ ∈ T such that A∗ (y) = 0 and A∗ (x) = 1. Theorem 25. If RT is the fuzzy relation defined by Eq. (9), then RT is a crisp binary relation, and ∀x, y ∈ U , (x, y) ∈ RT if and only if ∀A ∈ T , A(x) A(y). Proof. (1) For x, y ∈ U , if ∀A ∈ T , A(x) A(y), then for any closed set B, B(x) B(y), so, C(x, y) = ∅, and hence RT (x, y) = ∧B∈C(x,y) B(x) = 1. (2) If there exists an open set A ∈ T such that A(x) > A(y), by (TC ) axiom, there exists an open set A∗ ∈ T such that A∗ (y) = 0 and A∗ (x) = 1. Notice that ∼ A∗ ∈ C(x, y) and ∼ A∗ (x) = 0, so, RT (x, y) = ∧B∈C(x,y) B(x) = 0. According to (1) and (2), we obtain that RT is a crisp binary relation, and ∀x, y ∈ U , (x, y) ∈ RT if and only if ∀A ∈ T , A(x) A(y).
Theorem 26. If RT is the fuzzy relation defined by Eq. (9), then RT is reflexive and transitive. Proof. ∀x, y, z ∈ U , obviously, (x, x) ∈ RT , that is, R is reflexive. If (x, y) ∈ RT and (y, z) ∈ RT , then ∀A ∈ T , A(x) A(y) and A(y) A(z), so, A(x) A(z), that is, (x, z) ∈ RT . By Theorem 19, we have the following theorem,
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Theorem 27. ∀A ∈ F (U ), (1) RT (A) = c(A); (2) RT (A) = i(A). Remark 28. If the fuzzy topology T does not satisfy (TC ) axiom, then RT (A) = i(A) may not necessarily hold. Example 29. Let U = {x, y, z}, and A ∈ F (U ) be defined as A(x) = 0.5, A(y) = A(z) = 0. Assume T is the fuzzy topology generated by {A, U, ∅} ∪ {| ∈ [0, 1]}, and B ∈ F (U ) is defined as B(y) = 0.25, B(x) = B(z) = 1. As A is an open set and A(x) > A(y), A(x) > A(z), it follows that (x, y)∈RT and (x, z)∈RT and hence RT (B)(x) = inf{B(u)|u ∈ SR (x)} = B(x) = 1; i(B)(x) = sup{C(x)|C ∈ T , C ⊆ T } = 0.5, that is RT (B) = i(B). 5. Conclusion In this paper, we have discussed the relationship between approximation spaces and fuzzy topologies, proved that there exists one-to-one correspondences between the set of all fuzzy reflexive and transitive relations on a finite universe and the set of all fuzzy topologies on the same universe which satisfies (TC) axiom. Acknowledgements The authors are grateful to the referees for their valuable comments and suggestions. This work has been supported by the National Natural Science Foundation of China (Grant No. 60074014). References [1] K.S. Arun, S.P. Tiwari, On relationships among fuzzy approximation operators, and fuzzy topology, and fuzzy automata, Fuzzy Sets and Systems 138 (2003) 197–204. [2] D. Dubois, H. Prade, Rough fuzzy set and fuzzy rough sets, Internat. J. Gen. Systems 17 (1990) 191–209. [3] D. Dubois, H. Prade, Putting fuzzy sets and rough sets together, in: R. Slowinski (Ed.), Intelligent Decision Support, Kluwer Academic, Dordrecht, 1992, pp. 203–232. [4] W. Gahler, A.S. Abb-Allah, A. Kandil, On extended fuzzy topologies, Fuzzy Sets and Systems 109 (2000) 149–172. [5] Jusheng Mi, Weizhi Wu, Wenxiu Zhang, Constructive and axiomatic approaches for the study of the theory of rough sets, Pattern Recognition Artificial Intelligence 15 (2002) 280–284. [6] J. Kortelainen, On the relationship between modified sets, topological spaces and rough sets, Fuzzy Sets and Systems 61 (1994) 91–95. [7] L.I. Kuncheva, Fuzzy rough sets: application to feature selection, Fuzzy Sets and Systems 51 (1992) 147–153.
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