Axiomatic characterizations of (S, T)-fuzzy rough approximation operators

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Axiomatic characterizations of (S, T)-fuzzy rough approximation operators Wei-Zhi Wu a,b,∗, You-Hong Xu a,b, Ming-Wen Shao c, Guoyin Wang d

Q1

a

School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan, Zhejiang 316022, PR China Key Laboratory of Oceanographic Big Data Mining & Application of Zhejiang Province, Zhejiang Ocean University, Zhoushan, Zhejiang 316022, PR China c College of Computer & Communication Engineering, Chinese University of Petroleum, Qingdao, Shandong 266580, PR China d Chongqing Key Laboratory of Computational Intelligence, Chongqing University of Posts and Telecommunications, Chongqing 400065, PR China b

a r t i c l e

i n f o

Article history: Received 15 September 2014 Revised 22 September 2015 Accepted 12 November 2015 Available online xxx Keywords: Approximation operators Fuzzy relations Fuzzy rough sets Rough sets Triangular norms

a b s t r a c t Axiomatic characterizations of approximation operators are of importance in the study of rough set theory. In this paper axiomatic characterizations of relation-based (S, T)-fuzzy rough approximation operators are investigated. By employing a triangular conorm S and a triangular norm T on [0, 1], we first introduced the constructive definitions of S-lower and T-upper fuzzy rough approximation operators with their essential properties. We then propose an operator-oriented characterization of (S, T)-fuzzy rough sets, that is, fuzzy set-theoretic operators defined by axioms guarantee the existence of different types of fuzzy relations which produce the same operators. We show that the S-lower (and, respectively, T-upper) fuzzy rough approximation operators generated by a generalized fuzzy relation can be described by only one axiom. We further show that (S, T)-fuzzy rough approximation operators corresponding to special types of fuzzy relations, such as serial, reflexive, symmetric, and T-transitive ones as well as any of their compositions, can also be characterized by single axioms. © 2015 Elsevier Inc. All rights reserved.

1

1. Introduction

2

The theory of rough sets was originally proposed by Pawlak [14,15] as a formal tool for modelling and processing incomplete information. The basic structure of rough set theory is an approximation space consisting of a universe of discourse and a relation imposed on it. Based on the approximation space, the notions of lower and upper approximation operators can be constructed. Using the concepts of lower and upper approximations in the rough set theory, knowledge hidden in information tables may be unraveled and expressed in the form of decision rules. Thus, the most important concepts in original rough set theory are the lower and upper approximation operators derived from an approximation space. There are mainly two different approaches for the development of approximation operators, namely the constructive and axiomatic approaches. In the constructive approach, relations on the universe of discourse, partitions (or coverings) of the universe of discourse, neighborhood systems, and Boolean algebras are all primitive notions. The lower and upper approximation

3 4 5 6 7 8 9 10

Q2

∗ Corresponding author at:School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan, Zhejiang 316022, PR China. Tel.: +86 580 8180928; fax: +86 580 2550029. E-mail addresses: [email protected], [email protected] (W.-Z. Wu), [email protected] (Y.-H. Xu), [email protected] (M.-W. Shao), [email protected] (G. Wang).

http://dx.doi.org/10.1016/j.ins.2015.11.028 0020-0255/© 2015 Elsevier Inc. All rights reserved.

Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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operators are constructed by means of these notions. In contrast to the constructive approach, the axiomatic approach, also called algebraic approach, does not take approximation spaces as primitive notions. Rather, it regards the abstract 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 the constructive approach. Many authors explored and developed the axiomatic approach in the study of rough set theory [1,4–6,10,18,19,26,30, 32–34,36–38]. The research of the axiomatic approach has also been extended to approximation operators in the fuzzy environment [6,7,9,11–13,16,17,20–25,27–29,31,35]. To seek for minimal independent axiom sets characterizing rough set approximation operators is an important issue in the research of the axiomatic approach. Yang and Li [30] gave a completion of Yao’s study on relation-based rough approximation operators and presented independent axiomatic sets characterizing various classes of rough set approximation operators. In [29], Yang further examined that axioms in every set of axioms proposed by Wu and Zhang [28] are independent. Zhang and Luo [37] presented the minimization of axiom sets on approximation operators generated by coverings. Recently, Ma et al. [10] explored some new minimal axiom sets of the classical rough set. However, their axiomatic systems consist of at least two axioms. An interesting and important mathematical question is that whether rough set approximations can be characterized by only one axiom. As we know, in topological space theory, closure operators are characterized by the Kuratowski closure axioms [2]. And this axiomatic system, consisting of four statements, is equivalent to a single axiom [2]. Since the lower and upper approximation operators in the rough set theory are strongly related to the interior and closure operators in the topological space [23,25,26,32], recently, Liu [8] showed that we only need one axiom which is necessary and sufficient to characterize relationbased approximation operators both in crisp and fuzzy cases determined by the triangular norm T = min and the triangular conorm S = max. It is well-known that a typical family of fuzzy rough set models is determined by triangular norms, Mi et al. [11] and Wu [23] established different set of independent axiomatic sets to characterize various types of fuzzy rough approximation operators determined by a generalized triangular norm T. Along the line of Liu’s study on the fuzzy environment, the present paper will generalize his study to the relation-based lower and upper fuzzy rough approximation operators determined by arbitrary triangular conorms and triangular norms. In the next section, we review some basic notions and results of triangular conorms and triangular norms, and we then define concepts and present properties of inf S-product and sup T-product of two fuzzy sets determined by a triangular conorm S and a triangular norm T respectively. In Section 3, by using a triangular conorm S and a triangular norm T, we introduce S-lower and T-upper fuzzy rough approximation operators based on a fuzzy relation and present properties of the fuzzy rough approximation operators. In Section 4, we review the existing independent axiomatic sets to characterize various types of S-lower and T-upper fuzzy rough approximation operators, and we further explore single axioms to characterize S-lower and T-upper fuzzy rough approximation operators generated by various types of fuzzy relations. We conclude the paper with a summary in Section 5.

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2. Preliminaries

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

43

In this section we recall some basic notions and previous results which will be used in the later parts of this paper.

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2.1. Triangular norms and triangular conorms

45

A triangular norm [3], or t-norm in short, is an increasing, associative and commutative mapping T: [0, 1] × [0, 1] → [0, 1] (where [0, 1] is the unit interval) that satisfies the boundary condition: T (α , 1 ) = α for all α ∈ [0, 1]. The most popular continuous t-norms are: • the standard min operator T (α , β ) = min{α , β} (the largest t-norm [3]), M • the algebraic product TP (α , β ) = α ∗ β , • the bold intersection (also called the Łukasiewicz t-norm) TL (α , β ) = max{0, α + β − 1}. A triangular conorm [3], or t-conorm in short, is an increasing, associative and commutative mapping S: [0, 1] × [0, 1] → [0, 1] that satisfies the boundary condition: S(α , 0 ) = α for all α ∈ [0, 1]. Three well-known continuous t-conorms are: • the standard max operator SM (α , β ) = max{α , β} (the smallest t-conorm), • the probabilistic sum SP (α , β ) = α + β − α ∗ β , • the bounded sum SL (α , β ) = min{1, α + β}. A t-norm T and a t-conorm S are said to be dual with each other if and only if (iff) the De Morgan’s laws are satisfied, i.e.

46 47 48 49 50 51 52 53 54 55 56 57

58 59 60 61 62 63 64

T ( α , β ) = 1 − S ( 1 − α , 1 − β ),

α , β ∈ [0, 1],

(1)

S ( α , β ) = 1 − T ( 1 − α , 1 − β ),

α , β ∈ [0, 1].

(2)

It is well known that if T is a t-norm, then S defined as in Eq. (2) is a t-conorm, and if S is a t-conorm, then T defined as in Eq. (1) is a t-norm. Let U be a nonempty set called the universe of discourse. By a fuzzy set in U we mean a mapping F: U → [0, 1]. The class of all subsets of U (resp. all fuzzy sets of U) will be denoted by P (U ) (resp. F (U )). For y ∈ U, 1y will denote the fuzzy singleton with  will denote value 1 at y and 0 elsewhere; For M ∈ P (U ), 1M will denote the characteristic function of M, and for α ∈ [0, 1], α (x ) = α for all x ∈ U. For convenience, when there is no confusion, sometime we will use  the constant fuzzy set, i.e., α 1 or 1U to Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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stand for the universe set U and  0 for the empty set ∅. Zadeh’s fuzzy union and fuzzy intersection will be denoted by ∪ and ∩ , respectively. For any A ∈ F (U ), ∼ A will be used to denote the fuzzy complement of A ∈ F (U ) in U, i.e., (∼ A )(x ) = 1 − A(x ) for all x ∈ U. For A, B, A j ∈ F (U ), j ∈ J, where J is an index set, we denote

70

• A⊆B B(x), ∀x ∈ U,   iff A(x) ≤ • A j (x ) = A j (x ), x ∈ U,

71

•

69

j∈J 



j∈J

A j (x ) =

j∈J

72

3



A j (x ), x ∈ U.

j∈J

Given a t-norm T, a t-conorm S, and two fuzzy sets A, B ∈ F (U ), we define two fuzzy sets T(A, B) and S(A, B) as follows:









T (A, B )(x ) = T A(x ), B(x ) , x ∈ U, S(A, B )(x ) = S A(x ), B(x ) , x ∈ U. 73 74

The following two propositions give some basic properties of fuzzy set operations defined by Eq. (3). Proposition 1. Let T be a t-norm. Then for A, B, C, A j ∈ F (U ), j ∈ J, where J is an index set, we have:

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(T1) T (A, B ) = T (B, A ), (T2) A⊆B ⇒T(A,  C)⊆T(B,  C),  (T3) T A, T (B, C ) = T T (A, B ), C ,      (T4) T A j, B = T A j , B , provided that T is continuous,

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(T5) T

75 76 77

j∈J 



A j, B =

j∈J

80 81

(3)

j∈J    T A j , B , provided that T is continuous, j∈J

1, A ) = T (U, A ) = A, (T6) T ( 0, A ) = T (∅, A ) = ∅. (T7) T ( 

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Proof. They are straightforward by the definition of triangular norm.

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Proposition 2. Let S be a t-conorm. Then for A, B, C, A j ∈ F (U ), j ∈ J, where J is an index set, we have: S(A, B ) = S(B, A ), A⊆B ⇒S(A, C)⊆S(B, C), S(A, S(B, C)) = S(S(A, B), C ),   S A j, B = S A j , B , provided that S is continuous,

87

(S1) (S2) (S3) (S4)

88

(S5) S

84 85 86

j∈J 



A j, B =

j∈J

89 90 91 92

j∈J    S A j , B , provided that S is continuous, j∈J

(S6) S( 1, A ) = S(U, A ) = U, 0, A ) = S(∅, A ) = A. (S7) S( Proof. They are straightforward by the definition of triangular conorm.



Moreover, it can easily be verified that if T and S are dual with each other then

T (A, B ) = ∼ S(∼ A, ∼ B ),

S(A, B ) = ∼ T (∼ A, ∼ B ),

∀A, B ∈ F (U ).

(4)

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2.2. sup T-products and inf S-products of fuzzy sets

94

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In [7,8], by using the largest t-norm min and the smallest t-conorm max , Liu introduced notions of inner product and outer product of two fuzzy sets to study the axiomatic characterizations of relation-based fuzzy rough approximation operators. In this subsection we extend these notions to any t-norm and t-conorm in the fuzzy environment which are called sup T-product and inf S-product, respectively.

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Definition 1. Let T be a t-norm. For A, B ∈ F (U ), the sup T-product of A and B, denoted as (A, B )T , is defined as follows:

95 96

(A, B )T =







T A ( x ), B ( x ) .

(5)

x∈U

99 100 101 102 103

Proposition 3 below shows some basic properties of the sup T-product. Proposition 3. Let T be a t-norm. Then for A, B, A j ∈ F (U ), j ∈ J, where J is an index set, and for a ∈ [0, 1], we have: (I1) (A, B )T = (B, A )T ,  (I2) (∅, B )T = 0, (U, B )T = x∈U B(x ), (I3) A ⊆ B ⇒ (A, C )T ≤ (B, C )T , ∀C ∈ F (U ), Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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(I4) (I5) (I6)

107

(I7)

104 105

(A, C )T ≤ (B, C )T , ∀C ∈ F (U ) ⇒ A ⊆ B, (A, C )T = (B, C )T , ∀C ∈ F (U) ⇒ A = B, T ( a, A ), B = T a, (A, B )T ,   T   A j, B

j∈J

108 109

T

=

A j, B

j∈J

T

, provided that T is continuous.

Proof. Properties (I1)–(I3) are straightforward. (I4) For any x ∈ U, let C = 1x , then

(A, C )T =







T A ( y ), 1 x ( y ) =

 

T A ( y ), 0

111 112







∨ T A ( x ), 1 = A ( x ).

y=x

y∈U

110

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Similarly, we can conclude (B, C )T = B(x ). Consequently, by (A, C )T ≤ (B, C )T , we have A(x) ≤ B(x) for all x ∈ U. Thus A⊆B. (I5) can be immediately obtained from (I4). (I6) For a ∈ [0, 1], by the definition of t-norm, we have



T ( a , A ), B



T

=





T T ( a, A )(x ), B(x )

x∈U

=



 









T T a, A(x ) , B(x ) =

x∈U

 

T A ( x ), B ( x )

= T a,







T a, T A(x ), B(x )

x∈U







= T a, (A, B )T .

x∈U

113 114

(I7) is straightforward.  Definition 2. Let S be a t-conorm. For A, B ∈ F (U ), the inf S-product of A and B, denoted as [A, B]S , is defined as follows:

[A, B]S =







S A ( x ), B ( x ) .

(6)

x∈U

115 116

The following Proposition 4 presents some basic properties of the inf S-product. Proposition 4. Let S be a continuous t-conorm. Then for A, B, A j ∈ F (U ), j ∈ J, where J is an index set, and for a ∈ [0, 1], we have:

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[A, B]S = [B, A]S ,  [∅, B]S = x∈U B(x ), [U, B]S = 1, A ⊆ B ⇒ [A, C]S ≤ [B, C]S , ∀C ∈ F (U ), [A, C]S ≤ [B, C]S , ∀C ∈ F (U ) ⇒ A ⊆ B, [A, C]S = [B, C]S , ∀C ∈ F (U ) ⇒ A = B,    S( a, A ), B = S a, [A, B]S ,  S  (O7) A j, B = A j , B , provided that S is continuous,

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(O8) If T is the t-norm dual to S, then

117 118 119 120 121 122

(O1) (O2) (O3) (O4) (O5) (O6)

j∈J

125 126 127 128

S

j∈J

S

[A, B]S = 1 − (∼ A, ∼ B )T , (A, B )T = 1 − [∼ A, ∼ B]S . Proof. The proofs of (O1)–(O7) are similar to the ones of Proposition 3. (O8) is straightforward.  In what follows, T and S will always stand for a continuous t-norm and a continuous t-conorm, respectively.

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3. Constructive definitions of (S, T)-fuzzy rough approximation operators

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In this section we review some basic notions and properties of fuzzy rough approximation operators determined by a t-norm T and a t-conorm S. Let us start with introducing the following basic notion.

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Definition 3 ([24]). Let U and W be two nonempty universes of discourse. A fuzzy subset R ∈ F (U × W ) is referred to as a fuzzy  relation from U to W, and R(x, y) is the degree of relation from x to y, where (x, y) ∈ U × W. If for all x ∈ U, y∈W R(x, y ) = 1, then R is a serial fuzzy relation from U to W. If U = W, then R is called a fuzzy relation on U, R is a reflexive fuzzy relation if R(x, x ) = 1 for all x ∈ U; R is a symmetric fuzzy relation if R(x, y ) = R(y, x ) for all x, y ∈ U; R is a T-transitive fuzzy relation if R(x, z) ≥ ∨y ∈ U T(R(x, y), R(y, z)) for all x, z ∈ U; and R is a T-similarity fuzzy relation if it is reflexive, symmetric, and T-transitive. Now we introduce the notion of (S, T)-fuzzy rough approximation operators. Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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Definition 4 ([11,23]). Let U and W be two non-empty universes of discourse and R a fuzzy relation from U to W. Then the triple (U, W, R) is called a generalized fuzzy approximation system. For a fuzzy set A ∈ F (W ), the S-lower and T-upper approximations of A w.r.t. (U, W, R), denoted as R (A) and R(A ), respectively, are fuzzy sets in U defined by:

R(A )(x ) =







S 1 − R(x, y ), A(y ) ,

x ∈ U,

(7)

y∈W

141

R(A )(x ) =







T R(x, y ), A(y ) ,

x ∈ U.

(8)

y∈W

144

The operators R, R : F (W ) → F (U ) are referred to as S-lower and T-upper fuzzy rough approximation operators of (U, W, R) respectively, and the pair (R(A ), R(A )) is called the (S, T)-fuzzy rough set of A w.r.t. (U, W, R). In general, for simplicity, we will call R and R the (S, T)-fuzzy rough approximation operators.

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Remark 1.

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(1) In Definition 4, when T = min and S = max, then





y∈W



R(A )(x ) =



R(A )(x ) =

 (1 − R(x, y )) ∨ A(y ) , 

R(x, y ) ∧ A(y ) ,

x ∈ U,

x ∈ U.

y∈W

147

(2) If R in Definition 4 is a relation from U to W, then it can be verified that



R(A )(x ) =



y∈W



R(A )(x ) =







T R(x, y ), A(y ) =

149

A ( y ),

x ∈ U,

y∈Rs (x )



A ( y ),

x ∈ U,

y∈Rs (x )

y∈W

148



S 1 − R(x, y ), A(y ) =

where Rs (x ) = {y ∈ U |(x, y ) ∈ R}. More specially, if R is an equivalence relation on U, then

R(A )(x ) =



A ( y ),

x ∈ U,

A ( y ),

x ∈ U,

y∈[x]R

R(A )(x ) =



y∈[x]R

150 151

where [x]R is the R-equivalent class including x. For a fuzzy relation R from U to W and x ∈ U, we define a fuzzy set R(x) on W as follows:

R(x )(y ) = R(x, y ), 152 153 154 155 156 157 158 159 160 161 162

y ∈ W.

Then we can observe that R(A )(x ) = (R(x ), A )T and R(A )(x ) = [∼ R(x ), A]S . The following theorem presents some basic properties of (S, T)-fuzzy rough approximation operators. Theorem 1 ([23]). Let (U, W, R) be a generalized fuzzy approximation system. Then the S-lower and T-upper fuzzy rough approximation operators defined in Definition 4 satisfy the following properties: For A, B ∈ F (W ), A j ∈ F (W ), j ∈ J, where J is an index set, (x, y) ∈ U × W, M ∈ P (W ), and α ∈ [0, 1], (FLD ) R(A ) =∼ R(∼ A ), provided that S and T are dual with each other, (FUD ) R(A ) =∼R(∼ A ), provided that S and T are dual with each other,  (FL1 ) S R(A ),  a = R S(A,  a) ,     (FU1 ) T R(A ),  a = R T (A,  a) ,      (FL2 ) R Aj = R Aj ,

(FU2 ) R

 j∈J 



Aj =

j∈J

   R Aj ,

j∈J

j∈J

165

(FL3) A⊆B ⇒ R (A)⊆ R (B), (FU3 ) A⊆ B ⇒ R(A ) ⊆ R(B ),      (FL4 ) R Aj ⊇ R Aj ,

166

(FU4 ) R

163 164

 j∈J 

j∈J

167 168

(9)



Aj ⊆

   R Aj ,

j∈J

j∈J

(FL5 ) R(W ) = U, (FU5 ) R ∅W = ∅U ,

Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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(FL6 ) (FU6 ) (FL7 ) (FU7 ) (FL8 )

174

(FU8 )

169 170 171 172

175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198

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  R 1W −{y} (x ) = 1 − R(x, y ),   R 1y (x ) = R(x, y ),  ⊆ R (α  ), α , )⊆ α R(α    R 1M ( x ) = 1 − R(x, y ) , y∈M /    R 1M ( x ) = R(x, y ). y∈M

Properties (FL1) and (FU1) show that, when S and T are dual with each other, the S-lower and T-upper fuzzy rough approximation operators R and R are dual with each other. The following theorem indicates that the property of a serial fuzzy relation can be characterized by the properties of its induced (S, T)-fuzzy rough approximation operators [23]. Theorem 2. Let (U, W, R) be a generalized fuzzy approximation system, and R and R the S-lower and T-upper fuzzy rough approximation operators induced by (U, W, R), respectively. Then: R is serial , ∀α ∈ [0, 1], ) = α ⇐⇒ (FL0 ) R(α , ∀α ∈ [0, 1], ) = α ⇐⇒ (FU0 ) R(α  ⇐⇒ (FL0 ) R ∅W = ∅U , ⇐⇒ (FU0 )



R(W ) = U.

If R is a fuzzy relation on U, then the pair (U, R) is called a fuzzy approximation space [11,23]. The following theorem shows that some properties of (S, T)-fuzzy rough approximation operators derived from the fuzzy approximation space (U, R) can be used to characterize the special property of the fuzzy relation R. Theorem 3 ([11,23]). Let (U, R) be a fuzzy approximation space, and R and R the S-lower and T-upper fuzzy rough approximation operators induced by (U, R), respectively. Then: (1) R is reflexive ⇐⇒ (FLR ) R(A ) ⊆ A, ∀A ∈ F (U ), ⇐⇒ (FUR ) A ⊆ R(A ), ∀A ∈ F (U ). (2) R is symmetric    ⇐⇒ (FLS ) R 1U−{y} (x ) = R 1U−{x} (y ), ∀(x, y ) ∈ U × U,

 

 

⇐⇒ (FUS ) R 1y (x ) = R 1x (y ), ∀(x, y ) ∈ U × U. (3) if S and T are dual with eachother, then R is T-transitive ⇐⇒ (FLT ) R(A ) ⊆ R R(A ) , ∀A ∈ F (U ),





⇐⇒ (FUT ) R R(A ) ⊆ R(A ), ∀A ∈ F (U ).

199

4. Axiomatic characterizations of (S, T)-fuzzy rough approximation operators

200

212

In [11,23], axiomatic characterizations of various types of dual pairs of (S, T)-fuzzy rough approximation operators were proposed. However, we observe that each set of axioms in [11,23] for characterizing (S, T)-fuzzy rough approximation operator consists of at least two axioms. In this section we review the axiomatic characterizations of (S, T)-fuzzy rough approximation operators and show that we only need one axiom to characterize each S-lower (and resp. T-upper) fuzzy rough approximation operator with various situations. To give a clear picture of this section for the reader, we first present Table 1 to clarify the connections between properties of fuzzy relations such as serial, reflexive, symmetric, T-transitive, as well as their various compositions and the corresponding axioms for characterizing (S, T)-fuzzy rough approximation operators generated by the fuzzy relations. The entries “ser.”, “refl.”, “symm.”, “T-trans.”, and “T-sim.” in the first column stand for “serial”, “reflexive”, “symmetric”, “T-transitive”, and “T-similarity”, respectively. The second and the third columns present the existing axioms in [23] for characterizing the S-lower and T-upper fuzzy rough approximation operators generated by the fuzzy relation. The fourth and the fifth columns give our single axioms for characterizing the (S, T)-fuzzy rough approximation operators which are the same as ones defined by the constructive approach in the present paper.

213

4.1. Axioms for generalized (S, T)-fuzzy rough approximation operators

214

In this subsection we will explore single axioms to characterize the (S, T)-fuzzy rough approximation operators generated by arbitrary fuzzy relations. In [23], Wu gave sets of axioms to characterize generalized dual (S, T)-fuzzy rough approximation operators, Theorem 6 below presents a deviation of results in [23] in which S and T may not be dual with each other.

201 202 203 204 205 206 207 208 209 210 211

215 216 217

Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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7

Table 1 Axioms for (S, T)-fuzzy rough approximation operators. R

Axioms for L

Axioms for H

Single axiom for L

Single axiom for H

Any

(AFL1), (AFL2)

(AFU1), (AFU2)

Ser.

(AFL1), (AFL2), (AFL0) (AFL1), (AFL2), (AFLR) (AFL1), (AFL2), (AFLS) (AFL1), (AFL2), (AFLT) (AFL1), (AFL2) (AFL0), (AFLS) (AFL1), (AFL2) (AFL0), (AFLT) (AFL1), (AFL2) (AFLR), (AFLS) (AFL1), (AFL2) (AFLR), (AFLT) (AFL1), (AFL2) (AFLS), (AFLT) (AFL1), (AFL2) (AFLR), (AFLS), (AFLT)

(AFU1), (AFU2), (AFU0)) (AFU1), (AFU2), (AFUR) (AFU1), (AFU2), (AFUS) (AFU1), (AFU2), (AFUT) (AFU1), (AFU2) (AFU0)), (AFUS) (AFU1), (AFU2) (AFU0)), (AFUT) (AFU1), (AFU2) (AFUR), (AFUS) (AFU1), (AFU2) (AFUR), (AFUT) (AFU1), (AFU2) (AFUS), (AFUT) (AFU1), (AFU2) (AFUR), (AFUS), (AFUT)

(ASL) or (ASL) (ASL0)

(ATU) or (ATU) (ATU0)

(ASLR)

(ATUR)

(ASLS) or (ASLS) (ASLT)

(ATUS) or (ATUS) (ATUT)

(ASLS0)

(ATUS0)

(ASLT0)

(ATUT0)

(ASLRS) or (ASLRS) (ASLRT)

(ATURS) or (ATURS) (ATURT)

(ASLST) or (ASLST) (ASLE) or (ASLE)

(ATUST) or (ATUST) (ATUE) or (ATUE)

Refl. Symm. T-trans. Ser. & symm. Ser. & T-trans. Refl. & symm. Refl. & T-trans. Symm. & T-trans. T-sim.

218 219

Theorem 4. Let L, H : F (W ) → F (U ) be two fuzzy operators. Then: (1) there exists a fuzzy relation R from U to W such that

∀A ∈ F (W )

L ( A ) = R ( A ), 220 221 222

iff L satisfies  the following  (AFL1)  and (AFL2):   = L S(A, α ) , ∀A ∈ F (W ), ∀α ∈ [0, 1]. (AFL1) S L(A ), α      (AFL2) L Aj = L A j , ∀A j ∈ F (W ), j ∈ J, where J is any index set. j∈J

223

j∈J

(2) there exists a fuzzy relation R from U to W such that

∀A ∈ F (W )

H ( A ) = R ( A ), 224 225 226

227 229 230 231 232 233 234 235

Theorem 5. Let L : F (W ) → F (U ) be a fuzzy operator. Then there exists a fuzzy relation R from U to W such that Eq. (10) holds iff L satisfies following (ASL): (ASL) ∀A j ∈ F (W ), ∀a j ∈ [0, 1], j ∈ J, where J is any index set,

 

S aj , A j



=

j∈J

237 238

240



 

S aj , L A j

.

(12)

j∈J

Proof. “⇒” If there exists a fuzzy relation R from U to W such that Eq. (10) holds, then, by Theorem 4, L satisfies (AFL1) and (AFL2). Therefore            L S aj , A j = L S aj , A j = S aj , L A j . j∈J

239

j∈J

According to [23], (AFL1) and (AFL2) (resp. (AFU1) and (AFU2)) are independent, thus Theorem 4 indicates that {(AFL1), (AFL2)} (resp. {(AFU1), (AFU2)}) is the basic set of axioms to characterize the generalized S-lower (resp. T-upper) fuzzy rough approximation operator, i.e., axioms (AFL1) and (AFL2) (resp. (AFU1) and (AFU2)) can be used to define the abstract set-theoretic S-lower (resp. T-upper) fuzzy rough approximation operator L : F (W ) → F (U ) (resp. H : F (W ) → F (U )) which is the same as defined by the constructive approach in Definition 4. The next two theorems show that the set of axioms {(AFL1), (AFL2)} (resp. {(AFU1), (AFU2)}) can be replaced by a single axiom.

L 236

(11)

iff H satisfies  the following  (AFU1)and (AFU2):   = H T (A, α ) , ∀A ∈ F (W ), ∀α ∈ [0, 1]. (AFU1) T H (A ), α      (AFU2) (AFU2 ) H Aj = H A j , ∀A j ∈ F (W ), j ∈ J, where J is any index set. j∈J

228

(10)

j∈J

j∈J

Thus L satisfies (ASL). “⇐” Assume that L satisfies (ASL), from L we define a fuzzy relation R from U to W as follows:





R(x, y ) = 1 − L 1W −{y} (x ),

(x, y ) ∈ U × W.

(13)

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For any A ∈ F (W ), notice that

A=





 S A (y ), 1W −{y} ,

(14)

y∈W

242

then for any x ∈ U we have

   S A(y ), 1W −{y} (x ) y∈W     = S A (y ), L 1W −{y} (x ) y∈W

    = S A(y ), L 1W −{y} (x ) y∈W

  = S A(y ), 1 − R(x, y ) y∈W

  = S 1 − R(x, y ), A(y )

L(A )(x ) = L

y∈W

= R(A )(x ) 243 244 245 246

249

 

T aj , A j



251

=



by the commutativity of S by Eq. (7 ).



 

T aj , H A j

.

(15)

Proof. “⇒” If there exists a fuzzy relation R from U to W such that Eq. (11) holds, then, by Theorem 4, H satisfies (AFU1) and (AFU2). Therefore            H T aj , A j = H T aj , A j = T aj , H A j . j∈J

j∈J

Thus H satisfies (ATU). “⇐” Assume that H satisfies (ATU), from H we define a fuzzy relation R from U to W as follows:

 

R(x, y ) = H 1y (x ), 252

by Eq.(13 )

j∈J

j∈J

250

by the definition

Theorem 6. Let H : F (W ) → F (U ) be a fuzzy operator. Then there exists a fuzzy relation R from U to W such that Eq. (11) holds iff H satisfies following (ATU): (ATU) ∀A j ∈ F (W ), ∀a j ∈ [0, 1], j ∈ J, where J is any index set,

j∈J

247

by (ASL )



Thus Eq. (10) holds.

H

248

by Eq.(14 )

(x, y ) ∈ U × W.

(16)

For any A ∈ F (W ), notice that

A=



 T (A ( y ), 1 y ),

(17)

y∈W

253

then for any x ∈ U we have

     T A ( y ), 1 y ( x ) y∈W       = T A(y ), H 1y (x ) y∈W     = T A ( y ), H 1 y ( x ) y∈W   T A(y ), R(x, y ) = y∈W   = T R(x, y ), A(y )

H (A )(x ) = H

y∈W

= R(A )(x )

by Eq.(17 ) by (ATU ) by the definition by Eq.(16 ) by the commutativity of T by Eq. (8 ).

254

Thus Eq. (11) holds. 

255 257

By employing the inf S-product and sup T-product in Section 2, we can obtain another single axiom for characterizing the S-lower (and, resp., T-upper) fuzzy rough approximation operator. We first introduce the notions of S-lower inverse operator and T-upper inverse operator of a fuzzy operator.

258

Definition 5. Let U and W be two nonempty universes of discourse, for a fuzzy operator O : F (W ) → F (U ), denote

256



O−1 (A )(y ) = O(1W −{y} ), A S



S

=



 





S O 1W −{y} (x ), A(x ) , A ∈ F (U ), y ∈ W.

(18)

x∈U

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259

O−1 (A )(y ) = O 1y , A T

  



=

T



T O 1y (x ), A(x ) , A ∈ F (U ), y ∈ W.

, O−1 : F (U ) → F (W ) are referred to as the S-lower inverse operator and T-upper inverse operator of O, respectively. O−1 S T

261

Now, Theorem 7 below shows that the axiom (ASL) can be equivalently replaced by the following condition (ASL) .

263 264

265 266 267

Theorem 7. Let L : F (W ) → F (U ) be a fuzzy operator. Then there exists a fuzzy relation R from U to W such that Eq. (10) holds iff L satisfies following (ASL) : (ASL ) ∀A ∈ F (U ), ∀B ∈ F (W ),



A, L(B )







= B, L−1 (A ) . S

S

(20)

S

Proof. By Theorem 5, we only need to prove that “(ASL) ⇔(ASL)”. “⇒” If L satisfies (ASL) , then, for all A j ∈ F (W ) and all aj ∈ [0, 1], j ∈ J, where J is any index set, by properties (O1) and (O5) in Proposition 4, we only need to prove that





C, L





S(aj , A j )

j∈J

268

= C,



 



S aj , L A j

S



C, L



S(aj , A j )

=

j∈J

 S(aj , A j ), L−1 (C ) S

j∈J



S

=

S(aj , A j ), L−1 (C ) S

j∈J





=





=











=



by (O6 )

S

by (ASL ) by (O1 ) by (O6 )

S

S aj , L(A j ) , C







by (O7 )



S

S(aj , L(A j ))

j∈J

270









by (O7 )

S

S

j∈J

= C,

S



j∈J



S

S aj , L(A j ) , C



by (ASL )



S a j , L ( A j ), C

j∈J

=



S a j , C, L(A j )

j∈J

=



S a j , A j , L−1 (C ) S

j∈J

(21)

S





∀C ∈ F (U ).

,

j∈J

In fact, for any C ∈ F (U ), we have



269

(19)

x∈U

260

262

9

by (O1 ). S

Thus L satisfies (ASL). “⇐” Assume that L satisfies (ASL). For any A ∈ F (U ) and B ∈ F (W ), notice that

B=





 S B (y ), 1W −{y} ,

(22)

y∈W

271

then

[A, L(B )]S =

x∈U

=

x∈U

=

x∈U

=

x∈U



S A(x ), L(B )(x )





S A ( x ), L

 S A ( x ),

 S A ( x ),





by Eq.(6 )



y∈W



 S (B (y ), 1W −{y} ) (x )



 S (B (y ), L(1W −{y} )) (x )

y∈W









S B(y ), L 1W −{y} (x )

by Eq.(22 )

 by (ASL )





by the definition

y∈W

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=



=







































S A(x ), S B(y ), L 1W −{y} (x )

y∈W x∈U

=



S A(x ), S B(y ), L 1W −{y} (x )

x∈U y∈W



S B(y ), S A(x ), L 1W −{y} (x )

y∈W x∈U

=

y∈W

=





S B ( y ),





273 274

275 276 277

278 279 280



by Eq. (18 )



by Eq.(6 ).

S



(ATU ) ∀A ∈ F (U ), ∀B ∈ F (W ),



A, H (B )







= B, HT−1 (A ) .

T

(23)

T

Proof. By Theorem 6, we only need to prove that “(ATU) ⇔(ATU)”. “⇒” If H satisfies (ATU) , then, for all A j ∈ F (W ) and all aj ∈ [0, 1], j ∈ J, where J is any index set, by properties (I1) and (I5) in Proposition 3, we only need to prove that







T aj , A j





 =

j∈J

C,





 



T aj , H A j

T

∀C ∈ F (U ).

,

j∈J

(24)

T

In fact, for any C ∈ F (U ), we have





C, H







T aj , A j



j∈J

= T

=

 

= = = =



T aj , A j , HT−1 (C )

j∈J j∈J

j∈J





T aj , A j , HT−1 (C )









T a j , A j , HT−1 (C ) T a j , H ( A j ), C

j∈J 

T



 

j∈J 

 

T aj , H (A j ) , C

  

T aj , H (A j )

= C,

j∈J







T

T





T aj , H (A j ) , C

j∈J

283

by (S4 )

Theorem 8. Let H : F (W ) → F (U ) be a fuzzy set operator. Then there exists a fuzzy relation R from U to W such that Eq. (11) holds iff H satisfies following (ATU) :

C, H

282

by the associativity of S

Analogous to Theorem 7, by using the T-upper inverse operator and the sup T-product, we can obtain another single axiom to characterize the T-upper fuzzy rough approximation operator.



281

by (S4 )



S A(x ), L 1W −{y} (x )

S B(y ), L−1 (A )(y ) S

= B, L−1 (A ) S Thus L satisfies (ASL) .

x∈U

y∈W

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T

T

T

T



by (ATU ) by (I7 ) by (I6 )



by (ATU ) by (I6 ) by (I7 ) by (I1 ).

Thus H satisfies (ATU). “⇐” Assume that H satisfies (ATU). For any A ∈ F (U ) and B ∈ F (W ), notice that

B=







 T B ( y ), 1 y ,

(25)

y∈W

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284

then



A, H (B )

 T

=





T A(x ), H (B )(x )



by Eq. (5 )

      = T A ( x ), H T B(y ), 1y (x ) x∈U y∈W         = T A ( x ), T B ( y ), H 1 y (x ) x∈U y∈W      = T A ( x ), T B ( y ), H 1 y ( x ) x∈U  y∈W     = T A ( x ), T B ( y ), H 1 y ( x ) x∈U y∈W      = T A ( x ), T B ( y ), H 1 y ( x ) y∈W x∈U      = T B ( y ), T A ( x ), H 1 y ( x ) y∈W x∈U      = T B ( y ), T A ( x ), H 1 y ( x ) y∈W x∈U   = T B(y ), HT−1 (A )(y ) y∈W   = B, HT−1 (A ) x∈U

11



T

by Eq.(25 ) by (ATU ) by the definition by (T5 )

by the associativity of T by (T5 ) by Eq.(19 ) by Eq.(5 ).

285

Thus H satisfies (ATU) .

286

4.2. Axioms for serial (S, T)-fuzzy rough approximation operators

287

290

In this subsection we will explore single axioms to characterize the (S, T)-fuzzy rough approximation operators generated by a serial fuzzy relation. According to [23], axiomatic characterizations of serial (S, T)-fuzzy rough approximation operators are summarized as the following theorem.

291

Theorem 9 ([23]). Let L, H : F (W ) −→ F (U ) be two fuzzy operators. Then:

288 289

292 293



(1) there exists a serial fuzzy relation R from U to W such that Eq. (10) holds iff L satisfies {(AFL1), (AFL2)}, and one property in {(AFL0), (AFL0) },

(AFL0 ) (AFL0 ) 294 295

) = α L(α  , ∀α ∈ [0, 1], L ∅W = ∅U .

(2) there exists a serial fuzzy relation R from U to W such that Eq. (11) holds iff H satisfies {(AFU1), (AFU2)}, and one property in {(AFU0), (AFU0) },

) = α , H (α H (W ) = U.

(AFU0 ) (AFU0 ) 296 297 298 299 300 301 302

Remark 2. (AFL1), (AFL2) and (AFL0) (resp. (AFU1), (AFU2), and (AFU0)) in Theorem 9 are independent, i.e., any two of them cannot deduce the third one [11,23]. Thus {(AFL1), (AFL2), (AFL0)} (resp. {(AFU1), (AFU2), (AFU0)}) is the basic set of axioms to characterize the serial S-lower (resp. T-upper) fuzzy rough approximation operator. The next theorem shows that the set of axioms {(AFL1), (AFL2), (AFL0)} can be replaced by a single axiom. Theorem 10. Let L : F (W ) → F (U ) be a fuzzy operator. Then there exists a serial fuzzy relation R from U to W such that Eq. (10) holds iff L satisfies following (ASL0): (ASL0)∀A j ∈ F (W ), ∀a j ∈ [0, 1], j ∈ J, where J is an index set,





U − L ∅W



∩L

 

S aj , A j

j∈J

303 304 305 306

308



=





S aj , L(A j ) .

(26)

j∈J

Proof. “⇒” If there exists a serial fuzzy relation R from U to W such that Eq. (10) holds, then, by Theorem 2, we conclude that L(∅W ) = ∅U . Thus, by Theorem 5, we see that L satisfies (ASL0). “⇐” Assume that L satisfies (ASL0). By taking J = {1}, a1 = 1 and A1 =  0 = ∅W in (ASL0), from (S6) in Proposition 2 we have (U − L(∅W )) ∩ L(W ) = U. Hence,





L ∅W = ∅U . 307

∀α ∈ [0, 1],

(27)

Therefore, L satisfies (ASL). It follows from Theorem 5 that there exists a fuzzy relation R from U to W such that Eq. (10) holds. Moreover, by Eq. (27) and Theorem 2, we conclude that R is serial.  Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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Analogous to Theorem 10, the following theorem shows that the set of axioms {(AFU1), (AFU2), (AFU0)} for characterizing the serial T-upper fuzzy rough approximation operator can be replaced by one axiom. Theorem 11. Let H : F (W ) → F (U ) be a fuzzy operator. Then there exists a serial fuzzy relation R from U to W such that Eq. (11) holds iff H satisfies following (ATU0): (ATU0) ∀A j ∈ F (W ), ∀a j ∈ [0, 1], j ∈ J, where J is any index set,





U − H (W ) ∪ H

 

T aj , A j



=



j∈J

314 315 316 317

[m3Gsc;December 4, 2015;19:32]





T aj , H (A j ) .

(28)

j∈J

Proof. “⇒” If there exists a serial fuzzy relation R from U to W such that Eq. (11) holds, then, by Theorem 2, we conclude that H (W ) = U. Thus, by Theorem 6, we see that H satisfies (ATU0). “⇐” Assume that H satisfies (ATU0). By taking J = {1}, a1 = 0 and A1 = W in (ATU0), then from (T7) in Proposition 1 we have (U − H (W )) ∪ H (∅W ) = ∅U . Hence, U − H (W ) = ∅U . Thus

H (W ) = U.

(29)

319

Therefore, by Eq. (28), H satisfies (ATU). It follows from Theorem 6 that there exists a fuzzy relation R from U to W such that Eq. (11) holds. Moreover, by Eq. (29) and Theorem 2, we conclude that R is serial. 

320

4.3. Axioms for reflexive (S, T)-fuzzy rough approximation operators

321

324

In this subsection we will explore single axioms to characterize the (S, T)-fuzzy rough approximation operators generated by a reflexive fuzzy relation. According to [23], axiomatic characterizations of reflexive (S, T)-fuzzy rough approximation operators are summarized as the following theorem.

325

Theorem 12 ([23]). Let L, H : F (U ) −→ F (U ) be two fuzzy operators. Then:

318

322 323

326

(1) there exists a reflexive fuzzy relation R on U such that

L ( A ) = R ( A ), 327 328 329

∀A ∈ F (U )

iff L satisfies properties {(AFL1), (AFL2), (AFLR)}, where (AFLR ) L(A ) ⊆ A, ∀A ∈ F (U ). (2) there exists a reflexive fuzzy relation R on U such that

∀A ∈ F (U )

H ( A ) = R ( A ), 330 331 332 333 334 335 336 337 338

(AFUR ) A ⊆ H (A ), ∀A ∈ F (U ). Remark 3. (AFL1), (AFL2) and (AFLR) (resp. (AFU1), (AFU2), and (AFUR)) in Theorem 12 are independent, i.e., any two of them cannot deduce the third one [11,23]. Thus {(AFL1), (AFL2), (AFLR)} (resp. {(AFU1), (AFU2), (AFUR)}) is the basic set of axioms to characterize reflexive S-lower (resp. T-upper) fuzzy rough approximation operator. The next theorem shows that the set of axioms {(AFL1), (AFL2), (AFLR)} can be replaced by a single axiom. Theorem 13. Let L : F (U ) → F (U ) be a fuzzy operator. Then there exists a reflexive fuzzy relation R on U such that Eq. (30) holds iff L satisfies following (ASLR): (ASLR) ∀A j ∈ F (U ), ∀a j ∈ [0, 1], j ∈ J, where J is any index set,

 

S aj , A j

j∈J

339 341 342 343



=

 

S aj , A j

  

 

S aj , L A j

∩

j∈J

.

∀B ∈ F (U ).

(33)

That is, L satisfies (AFLR) in Theorem 12. Moreover, inclusion relation (33) implies following Eq. (34),









S b, L(B ) ∩ S( b, B ) = S  b, L(B ) , ∀B ∈ F (U ), ∀b ∈ [0, 1]. 345 346

(32)

j∈J

Proof. “⇒” If there exists a reflexive fuzzy relation R on U such that Eq. (30) holds, then, by Theorem 3, we conclude that L(B)⊆B a, L(B )) ⊆ S( a, B ) for all a ∈ [0, 1] and all B ∈ F (U ). Thus, by Theorem 5, we see that L satisfies for all B ∈ F (U ) and hence S( (ASLR). “⇐” Assume that L satisfies (ASLR). For any B ∈ F (U ), let J = {1}, by taking a1 = 0 and A1 = B in (ASLR), we have L(B ) = B ∩ L(B ). Then

L(B ) ⊆ B, 344

(31)

iff H satisfies properties {(AFU1), (AFU2), (AFUR)}, where

L

340

(30)

(34)

Hence, it is easy to observe that L satisfies (ASL). Therefore, by Theorem 5, there exists a fuzzy relation R on U such that Eq. (30) holds. Furthermore, by inclusion relation (33) and Theorem 3, we conclude that R is reflexive.  Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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347 348 349 350 351

Analogous to Theorem 13, the following theorem shows that the set of axioms {(AFU1), (AFU2), (AFUR)} for characterizing the reflexive T-upper fuzzy rough approximation operator can be replaced by one axiom. Theorem 14. Let H : F (U ) → F (U ) be a fuzzy operator. Then there exists a reflexive fuzzy relation R on U such that Eq. (31) holds iff H satisfies following (ATUR): (ATUR) ∀A j ∈ F (U ), ∀a j ∈ [0, 1], j ∈ J, where J is any index set,

H

 

T aj , A j



=

 

T aj , A j

j∈J

352 353 354 355 356

   

 

T aj , H A j

∪

j∈J

.

(35)

j∈J

Proof. “⇒” If there exists a reflexive fuzzy relation R on U such that Eq. (31) holds, then, by Theorem 3, we conclude that B⊆H(B) for all B ∈ F (U ) and hence T ( a, B ) ⊆ T ( a, H (B )) for all a ∈ [0, 1] and B ∈ F (U ). Thus, by Theorem 6, we observe that H satisfies (ATUR). “⇐” Assume that H satisfies (ATUR). For any B ∈ F (U ), let J = {1}, by taking a1 = 1, and A1 = B in (ATUR), we have H (B ) = B ∪ H (B ). Then

∀B ∈ F (U ).

B ⊆ H ( B ), 357

13

(36)

That is, H satisfies (AFUR) in Theorem 12. Moreover, inclusion relation (36) implies following Eq. (37),









T ( b, B ) ∪ T  b, H (B ) = T  b, H (B ) , ∀B ∈ F (U ), ∀b ∈ [0, 1].

(37)

359

Hence, it can be verified that H satisfies axiom (ATU). Therefore, by Theorem 6, there exists a fuzzy relation R on U such that Eq. (31) holds. Furthermore, by inclusion relation (36) and Theorem 3, we conclude that R is a reflexive fuzzy relation. 

360

4.4. Axioms for symmetric (S, T)-fuzzy rough approximation operators

361

364

In this subsection we will explore single axioms to characterize the (S, T)-fuzzy rough approximation operators generated by a symmetric fuzzy relation. According to [23], axiomatic characterizations of symmetric (S, T)-fuzzy rough approximation operators are summarized as the following theorem.

365

Theorem 15 ([23]). Let L, H : F (U ) −→ F (U ) be two fuzzy operators. Then:

358

362 363

366 367 368 369 370 371 372

(1) there exists fuzzy relation R on U such that Eq. (30) holds iff L satisfies properties {(AFL1), (AFL2), (AFLS)}, where  a symmetric    (AFLS ) L 1U−{x} (y ) = L 1U−{y} (x ), ∀(x, y ) ∈ U × U. (2) there exists  a symmetric  fuzzy  relation R on U such that Eq. (31) holds iff H satisfies properties {(AFU1), (AFU2), (AFUS)}, where (AFUS ) H 1x (y ) = H 1y (x ), ∀(x, y ) ∈ U × U. Remark 4. (AFL1), (AFL2) and (AFLS) (resp. (AFU1), (AFU2), and (AFUS)) in Theorem 15 are independent, i.e., any two of them cannot deduce the third one [11,23]. Thus {(AFL1), (AFL2), (AFLS)} (resp. {(AFU1), (AFU2), (AFUS)}) is the basic set of axioms to characterize the symmetric S-lower (resp. T-upper) fuzzy rough approximation operator.

374

We will examine that the set of axioms {(AFL1), (AFL2), (AFLS)} (resp. {(AFU1), (AFU2), (AFUS)}) can be replaced by a single axiom. To this end, we first give two auxiliary lemmas.

375

Lemma 1. Let L : F (U ) → F (U ) be a fuzzy operator. If L satisfies (ASL), then the following statements are equivalent:

373

376 377 378









(1) L 1U−{x} (y ) = L 1U−{y} (x ), ∀(x, y ) ∈ U × U. (2) L(A ) = L−1 (A ), ∀A ∈ F (U ). S

Proof. “(1)⇒(2)” For any A ∈ F (U ), notice that

A=





 S A (y ), 1U−{y} ,

(38)

y∈U

379

then for any x ∈ U we have

   S A(y ), 1U−{y} (x )

L(A )(x ) = L

by Eq.(38 )

y∈U

=

    S A(y ), L 1U−{y} (x ) y∈U

=

y∈U

=



















S A(y ), L 1U−{y} (x )

S A(y ), L 1U−{x} (y )

by (ASL ) by the definition by (1 )

y∈U

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= A, L 1U−{x}



by Eq. (6 )

S

= L−1 (A )(x ) S 380 381

[m3Gsc;December 4, 2015;19:32]

by Eq.(18 ).

L−1 (A ).

Thus we conclude L(A ) = S “(2)⇒(1)” For any (x, y) ∈ U × U, since

L(A )(y ) = L−1 (A )(y ), ∀A ∈ F (U ), S 382

(39)

by taking A = 1U−{x} in Eq. (39), we have









L 1U−{x} (y ) = L−1 S 1U−{x} (y )  = 1U−{x} , L 1U−{y}

 S   = S 1U−{x} (z ), L 1U−{y} (z ) z∈U          = S 1, L 1U−{y} (z ) ∧ S 0, L 1U−{y} (x )

{z∈U |z=x}

= L 1U−{y} (x ). 383 384 385 386 387

 Lemma 2. Let H : F (U ) → F (U ) be a fuzzy operator. If H satisfies (ATU), then the following statements are equivalent:

 

 

(1) H 1x (y ) = H 1y (x ), ∀(x, y ) ∈ U × U. (2) H (A ) = HT−1 (A ), ∀A ∈ F (U ).

Proof. “(1)⇒(2)” For any A ∈ F (U ), notice that

A=







 T A (y ), 1y ,

(40)

y∈U

388

then for any x ∈ U we have

     T A(y ), 1y (x )  y∈U     = T A ( y ), H 1 y (x ) y∈U     = T A ( y ), H 1 y ( x ) y∈U     = T A ( y ), H 1 x ( y ) y∈U   

H (A )(x ) = H

= A, H 1x

= HT (A )(x ) −1

389 390

T

by Eq. (40 ) by (ATU ) by the definition by (1 ) by Eq. (5 ) by Eq. (19 ).

H −1 (A ).

Thus we conclude H (A ) = T “(2)⇒(1)” For any (x, y) ∈ U × U, since

H (A )(y ) = HT−1 (A )(y ), ∀A ∈ F (U ), 391

(41)

by taking A = 1x in Eq. (41) and in terms of Eq. (5), we have

 

 

H 1x (y ) = HT−1 1x (y )    = 1x , H 1y T     = T 1 x ( z ), H 1 y ( z ) z∈U          = T 0, H 1y (z ) ∨ T 1, H 1y (x ) {z∈U  |z=x}

= H 1 y ( x ). 392 393 394 395 396

 Now, we are to examine that we only need one axiom to characterize the symmetric S-lower fuzzy rough approximation operator. Theorem 16. Let L : F (U ) → F (U ) be a fuzzy operator. Then there exists a symmetric fuzzy relation R on U such that Eq. (30) holds iff L satisfies following (ASLS): Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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397

(ASLS) ∀A, A j ∈ F (U ), ∀a, a j ∈ [0, 1], j ∈ J, where J is any index set,





S  (A ) ∩ L a, L−1 S

 

S aj , A j





  

=S  a, L(A ) ∩

j∈J

398 399 Q3 400 401 402

 

S aj , L A j

.

(42)

j∈J

Proof. “⇒” If there exists a symmetric fuzzy relation R on U such that Eq. (30) holds, then, by Theorem 5, we know that L satisfies (ASL). Since R is a symmetric fuzzy relation on U, by property (FL6) in Theorem 1, Theorems 3 and 7, and Lemma 1, we have L(B ) = L−1 (B ) for all B ∈ F (U ). Therefore, by Theorem 5, we conclude that L satisfies (ASLS) . S “⇐” Assume that L satisfies (ASLS). On one hand, let J = {1}, by taking a = a1 = 1 and A = A1 = 1U = U =  1 in (ASLS), we have 1, L−1 ( 1 )) ∩ L(S( 1,  1 )) = S( 1, L( 1 )), then, by (S6) in Proposition 2, we conclude  1) =  S( 1 ∩ L( 1, which follows that S

 

 

L  1 = 1U = U. 1 = L 1U = L(U ) =  403

(43)

On the other hand, for any A ∈ F (U ), let J = {1}, by taking a1 = 1, a = 0, and A1 = A in (ASLS), then we have

















S  (A ) ∩ L S( 0, L−1 1, A ) = S  0, L(A ) ∩ S  1, L(A ) . S 404

405 407 408 409 410 411 412

A ∈ F (U ).

(45)

Finally, for all A j ∈ F (U ) and all aj ∈ [0, 1], j ∈ J, where J is any index set, by setting a = 1 in (ASLS), then (ASLS) is degenerated to (ASL). Consequently, by Theorem 5, there exists a fuzzy relation R on U such that Eq. (30) holds. Furthermore, by Lemma 1, Eq. (45), and Theorems 1 and 3, we see that R is symmetric.  Analogous to Theorem 16, the following theorem shows that the set of axioms {(AFU1), (AFU2), (AFUS)} for characterizing the symmetric T-upper fuzzy rough approximation operator can be replaced by a single axiom. Theorem 17. Let H : F (U ) → F (U ) be a fuzzy operator. Then there exists a symmetric fuzzy relation R on U such that Eq. (31) holds iff H satisfies following (ATUS): (ATUS) ∀A, A j ∈ F (U ), ∀a, a j ∈ [0, 1], j ∈ J, where J is any index set,





T  a, HT−1 (A ) ∪ H

 

T aj , A j





  

=T  a, H (A ) ∪

j∈J

413 414 Q4 415 416

(44)

Therefore, by (S6) and (S7) in Proposition 2, we conclude

L−1 ( A ) = L ( A ), S 406

418 420 421 422 423 424 425

426 427 428 429

430

.

(46)

j∈J

Proof. “⇒” If there exists a symmetric fuzzy relation R on U such that Eq. (31) holds, then, by Theorem 6, we know that H satisfies (ATU). Since R is a symmetric fuzzy relation on U, by property (FU6) in Theorem 1, Theorems 3 and 8, and Lemma 2, we have H (B ) = HT−1 (B ) for all B ∈ F (U ). Therefore, by Theorem 6, we conclude that H satisfies (ATUS). “⇐” Assume that H satisfies (ATUS). For any A ∈ F (U ), let J = {1}, by taking a1 = 0, a = 1, and A1 = A in (ATUS), then

















(47)

Therefore, by (T6) and (T7) in Proposition 1, we have

HT−1 (A ) = H (A ), 419

 

T aj , H A j

1, HT−1 (A ) ∪ H T ( 0, A ) = T  1, H (A ) ∪ T  0, H (A ) . T  417

15

A ∈ F (U ).

(48)

On the other hand, for all A j ∈ F (U ) and all aj ∈ [0, 1], j ∈ J, where J is any index set, by setting a = 0 in (ATUS), then (ATUS) is degenerated to (ATU). Hence, by Theorem 6, there exists a fuzzy relation R on U such that Eq. (31) holds. Furthermore, by Lemma 2, Eq. (48), and Theorems 1 and 3, we see that R is symmetric.  By employing the inf S-product and sup T-product in Section 2, we can obtain another pair of single axioms for characterizing the symmetric S-lower and T-upper fuzzy rough approximation operators. Theorem 18. Let L : F (U ) → F (U ) be a fuzzy operator. Then there exists a symmetric fuzzy relation R on U such that Eq. (30) holds iff L satisfies following (ASLS) : (ASLS) ∀A, B ∈ F (U ),



A, L(B )



S





= L ( A ), B .

(49)

S

Proof. By Theorem 7, we know that L satisfies axiom (ASL) iff [A, L(B )]S = [L−1 (A ), B]S for all A, B ∈ F (U ). Thus we only need to S

prove that L satisfies (ASLS) iff [A, L(B )]S = [L−1 (A ), B]S and L(A ) = L−1 (A ) for all A, B ∈ F (U ). S S In fact, the sufficiency is obvious. For the necessity, if L satisfies (ASLS) , for any A ∈ F (U ) and x ∈ U, by taking B = 1U−{x} in (ASLS) , then, by Definition 5, we see that



A, L(B )



S





= A, L 1U−{x}



S

= L−1 (A )(x ). S

(50)

On the other hand, Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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L ( A ), B





= L(A ), 1U−{x}

S

=









S

=





S L(A )(y ), 1U−{x} (y )

y∈U

S L(A )(y ), 1







∧ S L(A )(x ), 0



{y∈U |y=x}

= L(A )(x ). 431

Hence L−1 (A )(x ) = L(A )(x ) for all x ∈ U. It follows that S

∀A ∈ F (U ).

L−1 ( A ) = L ( A ), S 432 433 434 435 436 437 438

439 440 441 442

443

(51)



Combining Eq. (51) with (ASLS) we conclude that A, L(B )





S



= L−1 ( A ), B S

S

for all A, B ∈ F (U ). 

Theorem 18 shows that the axiom (ASLS) is equivalent to the axiom (ASLS). Analogously, we examine that the axiom (ATUS) for characterizing the symmetric T-upper fuzzy rough approximation operator can be equivalently replaced by condition (ATUS) in the following theorem. Theorem 19. Let H : F (U ) → F (U ) be a fuzzy operator. Then there exists a symmetric fuzzy relation R on U such that Eq. (31) holds iff H satisfies following (ATUS) : (ATUS) ∀A, B ∈ F (U ),



A, H (B )





T



= H ( A ), B .

(52)

T

Proof. By Lemma 2, we only need to prove that H satisfies (ATUS) iff (A, H (B ))T = (HT−1 (A ), B )T and H (A ) = HT−1 (A ) for all A, B ∈ F (U ). In fact, the sufficiency is obvious. For the necessity, assume that H satisfies (ATUS) , for any A ∈ F (U ) and x ∈ U, by taking B = 1x in (ATUS) , by Definition 5, we see that



A, H (B )





T

On the other hand,



H ( A ), B

 

= A, H 1x



T

T



= H ( A ), 1 x =





= HT−1 (A )(x ).

 T

=





(53)



T H (A )(y ), 1x (y )

y∈U

T H (A )(y ), 0







∨ T H (A )(x ), 1



{y∈U |y=x}

= H (A )(x ). 444

Hence HT−1 (A )(x ) = H (A )(x ) for all x ∈ U. It follows that

HT−1 (A ) = H (A ), 445

∀A ∈ F (U ).

(54)

Combining Eq. (54) with (ATUS) , we conclude that (A, H (B ))T = (HT−1 (A ), B )T for all A, B ∈ F (U ). 

446

4.5. Axioms for T-transitive (S, T)-fuzzy rough approximation operators

447

450

In this subsection we will explore single axioms to characterize the (S, T)-fuzzy rough approximation operators generated by a T-transitive fuzzy relation. According to [23], axiomatic characterizations of T-transitive (S, T)-fuzzy rough approximation operators are summarized as the following theorem.

451

Theorem 20 ([23]). Let L, H : F (U ) −→ F (U ) be two fuzzy operators. If T and S are dual with each other, then:

448 449

452 453 454 455 456 457 458 459 460 461

(1) there exists a T-transitive   fuzzy relation R on U such that Eq. (30) holds iff L satisfies properties {(AFL1), (AFL2), (AFLT)}, where (AFLT ) L(A ) ⊆ L L(A ) , ∀A ∈ F (U ). (2) there exists fuzzy relation R on U such that Eq. (31) holds iff H satisfies properties {(AFU1), (AFU2), (AFUT)}, where  a T-transitive  (AFUT ) H H (A ) ⊆ H (A ), ∀A ∈ F (U ). Remark 5. (AFL1), (AFL2) and (AFLT) (resp. (AFU1), (AFU2), and (AFUT)) in Theorem 20 are independent, i.e., any two of them cannot deduce the third one [11,23]. Thus {(AFL1), (AFL2), (AFLT)} (resp. {(AFU1), (AFU2), (AFUT)}) is the basic set of axioms to characterize T-transitive lower (resp. upper) fuzzy rough approximation operator. The next theorem shows that the set of axioms {(AFL1), (AFL2), (AFLT)} can be replaced by a single axiom. Theorem 21. Let L : F (U ) → F (U ) be a fuzzy operator. If T and S are dual with each other, then there exists a T-transitive fuzzy relation R on U such that Eq. (30) holds iff L satisfies following (ASLT): Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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462

 

S aj , A j



=

 

465 466 467

  

S aj , L L A j

∩

j∈J

.

(55)

j∈J

Proof. “⇒” If there exists a T-transitive fuzzy relation R on U such that Eq. (30) holds, then, by Theorem 3, we conclude that L(B)⊆L(L(B)) for all B ∈ F (U ) and hence S( a, L(B )) ⊆ S( a, L(L(B ))) for all a ∈ [0, 1] and all B ∈ F (U ). Therefore, by Theorem 5, it is easy to deduce that L satisfies (ASLT). “⇐” Assume that L satisfies (ASLT). For any B ∈ F (U ), let J = {1}, by taking a1 = 0 and A1 = B in (ASLT), we have L(S( 0, B )) = 0, L(B )) ∩ S( 0, L(L(B ))). By (S7) in Proposition 2, we see that L(B ) = L(B ) ∩ L(L(B )). It follows that S(





∀B ∈ F (U ).

L (B ) ⊆ L L (B ) , 468

   

S aj , L A j

j∈J

463

17

(ASLT) ∀A j ∈ F (U ), ∀a j ∈ [0, 1], j ∈ J, where J is any index set,

L

464

(56)

That is, L satisfies (FLT) in Theorem 3. Moreover, inclusion relation (56) implies Eq. (57) below,















b, L(B ) = S  b, L(B ) ∩ S  b, L L(B ) , ∀B ∈ F (U ), ∀b ∈ [0, 1]. S 469 470 471 472 473 474 475

Analogous to Theorem 21, the following theorem shows that the set of axioms {(AFU1), (AFU2), (AFUT)} for characterizing the T-transitive upper fuzzy rough approximation operator can be replaced by one axiom. Theorem 22. Let H : F (U ) → F (U ) be a fuzzy operator. Then there exists a T-transitive fuzzy relation R on U such that Eq. (31) holds iff H satisfies following (ATUT): (ATUT) ∀A j ∈ F (U ), ∀a j ∈ [0, 1], j ∈ J, where J is any index set,

 

T aj , A j



=

 

478 479 480

483

.

(58)

j∈J

Proof. “⇒” If there exists a T-transitive fuzzy relation R on U such that Eq. (31) holds, then, by Theorem 3, we conclude that a, H (H (B ))) ⊆ T ( a, H (B )) for all a ∈ [0, 1] and all B ∈ F (U ). Then, by Theorem 6, it is H(H(B))⊆H(B) for all B ∈ F (U ) and hence T ( easy to verify that H satisfies (ATUT). “⇐” Assume that H satisfies (ATUT). For any B ∈ F (U ), let J = {1}, by taking a1 = 1, and A1 = B in (ATUT), we can conclude H (B ) = H (H (B )) ∪ H (B ). Then





∀B ∈ F (U ).

(59)

That is, H satisfies (FUT) in Theorem 3. Moreover, inclusion relation (59) implies Eq. (60) below,





T  b, H H (B ) 482

  

T aj , H H A j

∪

j∈J

H H ( B ) ⊆ H ( B ), 481

    

T aj , H A j

j∈J

476

(57)

Hence, it is easy to see that L satisfies (ASL). Thus, by Theorem 5, there exists a fuzzy relation R on U such that Eq. (30) holds. Furthermore, by inclusion relation (56) and Theorem 3, we conclude that R is a T-transitive fuzzy relation. 

H

477

[m3Gsc;December 4, 2015;19:32]











∪T  b, H (B ) = T  b, H (B ) , ∀B ∈ F (U ), ∀b ∈ [0, 1].

(60)

Hence, by (ATUT), we observe that H satisfies (ATU). Therefore, by Theorem 6, there exists a fuzzy relation R on U such that Eq. (31) holds. Furthermore, by inclusion relation (59) and Theorem 3, we conclude that R is a T-transitive fuzzy relation. 

485

In what follows, we will show that (S, T)-fuzzy rough approximation operators generated by various compositive special fuzzy relations can also be characterized by single axioms.

486

4.6. Axioms for serial and symmetric (S, T)-fuzzy rough approximation operators

487

In this subsection we will explore single axioms to characterize the (S, T)-fuzzy rough approximation operators generated by a serial and symmetric fuzzy relation. According to Theorems 9 and 15, we observe that {(AFL1), (AFL2), (AFL0), (AFLS)} (resp. {(AFU1), (AFU2), (AFU0), (AFUS)}) is the basic set of axioms for characterizing the S-lower (resp. T-upper) fuzzy rough approximation operator generated by a serial and symmetric fuzzy relation. In the next theorem, we will examine that {(AFL1), (AFL2), (AFL0), (AFLS)} can be replaced by a single axiom.

484

488 489 490 491 492 493 494 495

Theorem 23. Let L : F (U ) → F (U ) be a fuzzy operator. Then there exists a serial and symmetric fuzzy relation R on U such that Eq. (30) holds iff L satisfies following (ASLS0): (ASLS0) ∀A, A j ∈ F (U ), ∀a, a j ∈ [0, 1], j ∈ J, where J is any index set,



 

U − L ∅U





∩S  (A ) ∩ L a, L−1 S

 

S aj , A j





  

=S  a, L(A ) ∩

j∈J

496 497 498 499

 

S aj , L A j

.

(61)

j∈J

Proof. “⇒” If there exists a serial and symmetric fuzzy relation R on U such that Eq. (30) holds, then, by Theorem 2 and Lemma 1, we see that L(∅U ) = ∅U and L−1 (A ) = L(A ) for all A ∈ F (U ), which follows from Theorem 5 that L satisfies (ASLS0). S “⇐” Assume that L satisfies (ASLS0). By taking J = {1}, a = a = 1 and A = A = U =  1 in (ASLS0), we then conclude that 1

(U − L(∅U )) ∩ L(U ) = U, hence U − L(∅U ) = U, which follows that

 

L ∅U = ∅U .

1

(62)

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Thus (ASLS0) is degenerated to (ASLS). Therefore, by Theorem 16, there exists a symmetric fuzzy relation R on U such that Eq. (30) holds. Moreover, by Eq. (62) and Theorem 2, we conclude that R is serial.  Analogous to Theorem 23, the following theorem shows that the set of axioms {(AFU1), (AFU2), (AFU0), (AFUS)} for characterizing the serial and symmetric T-upper fuzzy rough approximation operator can be replaced by one axiom. Theorem 24. Let H : F (U ) → F (U ) be a fuzzy operator. Then there exists a serial and symmetric fuzzy relation R on U such that Eq. (31) holds iff H satisfies following (ATUS0): (ATUS0) ∀A, A j ∈ F (U ), ∀a, a j ∈ [0, 1], j ∈ J, where J is any index set,









U − H (U ) ∪ T  a, HT−1 (A ) ∪ H

 

T aj , A j





  

=T  a, H (A ) ∪

j∈J

507 508 509 510

[m3Gsc;December 4, 2015;19:32]

 

T aj , H A j

.

(63)

j∈J

Proof. “⇒” If there exists a serial and symmetric fuzzy relation R on U such that Eq. (31) holds, by Theorem 2 and Lemma 2, we see that H (U ) = U and HT−1 (A ) = H (A ) for all A ∈ F (U ), which follows from Theorem 6 that H satisfies (ATUS0). “⇐” Assume that H satisfies (ATUS0). By taking J = {1}, a = a = 0 and A = A = U =  1 in (ATUS0), we then conclude that 1

(U − H (U )) ∪ H (∅U ) = ∅U , hence U − H (U ) = ∅U , which follows that

1

H (U ) = U

(64)

512

Thus (ATUS0) is degenerated to (ATUS). Therefore, by Theorem 17, there exists a symmetric fuzzy relation R on U such that Eq. (31) holds. Moreover, by Eq. (64) and Theorem 2, we conclude that R is serial. 

513

4.7. Axioms for serial and T-transitive (S, T)-fuzzy rough approximation operators

514

In this subsection we will explore single axioms to characterize the (S, T)-fuzzy rough approximation operators generated by a serial and T-transitive fuzzy relation. According to Theorems 9 and 20, we observe that {(AFL1), (AFL2), (AFL0), (AFLT)} (resp. {(AFU1), (AFU2), (AFU0), (AFUT)}) is the basic set of axioms for characterizing the S-lower (resp. T-upper) fuzzy rough approximation operator generated by a serial and T-transitive fuzzy relation. In the next theorem, we will examine that the set of axioms {(AFL1), (AFL2), (AFL0), (AFLT)} can be replaced by a single axiom.

511

515 516 517 518 519 520 521 522

Theorem 25. Let L : F (U ) → F (U ) be a fuzzy operator. If T and S are dual with each other, then there exists a serial and T-transitive fuzzy relation R on U such that Eq. (30) holds iff L satisfies following (ASLT0): (ASLT0)∀A j ∈ F (U ), ∀a j ∈ [0, 1], j ∈ J, where J is any index set,



 

U − L ∅U

∩L

 

S aj , A j



=

 

j∈J

523 524 525 526

   

S aj , L A j

  

S aj , L L A j

∩

j∈J

.

(65)

j∈J

Proof. “⇒” If there exists a serial and T-transitive fuzzy relation R on U such that Eq. (30) holds, then, by Theorem 2, we see that L(∅U ) = ∅U , which implies that U − L(∅U ) = U. Consequently, by Theorem 21, we conclude that L satisfies (ASLT0). “⇐” Assume that L satisfies (ASLT0). By taking J = {1}, a1 = 1 and A1 = U =  1 in (ASLT0), we then conclude that (U − L(∅U )) ∩ L(U ) = U, hence U − L(∅U ) = U, which follows that

 

L ∅U = ∅U . 527 528 529 530 531 532 533

(66)

Thus (ASLT0) is degenerated to (ASLT). Therefore, by Theorem 21, there exists a T-transitive fuzzy relation R on U such that Eq. (30) holds. Moreover, by Eq. (66) and Theorem 2, we conclude that R is serial.  Analogous to Theorem 25, the following theorem shows that the set of axioms {(AFU1), (AFU2), (AFU0), (AFUT)} for characterizing the serial and T-transitive upper fuzzy rough approximation operator can be replaced by one axiom. Theorem 26. Let H : F (U ) → F (U ) be a fuzzy operator. Then there exists a serial and T-transitive fuzzy relation R on U such that Eq. (31) holds iff H satisfies following (ATUT0): (ATUT0) ∀A j ∈ F (U ), ∀a j ∈ [0, 1], j ∈ J, where J is any index set,





U − H (U ) ∪ H

 

T aj , A j

j∈J

534 535 536 537

539

=

 

    

T aj , H A j

j∈J

  

T aj , H H A j

∪

.

(67)

j∈J

Proof. “⇒” If there exists a serial and T-transitive fuzzy relation R on U such that Eq. (31) holds, then, by Theorem 2, we see that H (U ) = U, which implies that U − H (U ) = ∅U . Consequently, by Theorem 22, we conclude that H satisfies (ATUT0). “⇐” Assume that H satisfies (ATUT0). By taking J = {1}, a1 = 0 and A1 = U =  1 in (ATUT0), we then conclude that (U − H (U )) ∪ H (∅U ) = ∅U , hence U − H (U ) = ∅U , which follows that

H (U ) = U. 538



(68)

Thus (ATUT0) is degenerated to (ATUT). Therefore, by Theorem 22, there exists a T-transitive fuzzy relation R on U such that Eq. (31) holds. Moreover, by Eq. (68) and Theorem 2, we see that R is serial.  Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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19

540

4.8. Axioms for reflexive and symmetric (S, T)-fuzzy rough approximation operators

541

In this subsection we will explore single axioms to characterize the (S, T)-fuzzy rough approximation operators generated by a reflexive and symmetric fuzzy relation. According to Theorems 12 and 15, we observe that {(AFL1), (AFL2), (AFLR), (AFLS)} (resp. {(AFU1), (AFU2), (AFUR), (AFUS)}) is the basic set of axioms for characterizing the S-lower (resp. T-upper) fuzzy rough approximation operator generated by a reflexive and symmetric fuzzy relation. In the next theorem we will examine that {(AFL1), (AFL2), (AFLR), (AFLS)} can be replaced by a single axiom.

542 543 544 545 546 547 548 549

Theorem 27. Let L : F (U ) → F (U ) be a fuzzy operator. Then there exists a reflexive and symmetric fuzzy relation R on U such that Eq. (30) holds iff L satisfies following (ASLRS): (ASLRS) ∀A, A j ∈ F (U ), ∀a, a j ∈ [0, 1], j ∈ J, where J is any index set,





S  (A ) ∩ L a, L−1 S

 

S aj , A j





  

=S  a, L(A ) ∩

j∈J

550 551



 

S aj , L A j

⊆

j∈J

553 554

556

j∈J

559 560 561 562 563





(70)

∀B ∈ F (U ).

(71)

Hence, for all A j ∈ F (U ) and all aj ∈ [0, 1], j ∈ J, where J is any index set, it can be seen that inclusion relation (70) holds, which implies that

 

   

S aj , A j

∩



=

j∈J



 

S aj , L A j

Theorem 28. Let H : F (U ) → F (U ) be a fuzzy operator. Then there exists a reflexive and symmetric fuzzy relation R on U such that Eq. (31) holds iff H satisfies following (ATURS): (ATURS) ∀A, A j ∈ F (U ), ∀a, a j ∈ [0, 1], j ∈ J, where J is any index set,





 

T aj , A j





  

=T  a, H (A ) ∪





T aj , A j ⊆

j∈J

567 568

570

573 574

.

(73)

j∈J





 

T aj , H A j

.

(74)

∀B ∈ F (U ).

(75)

Hence, for all A j ∈ F (U ) and all aj ∈ [0, 1], j ∈ J, where J is any index set, it can be seen that inclusion relation (74) holds, which implies that

  j∈J

571



Thus, by Theorem 17, we see that H satisfies (ATURS). “⇐” Assume that H satisfies (ATURS). For any B ∈ F (U ), by taking J = {1}, a1 = 1, a = 0, and A1 = A = B in (ATURS), then we conclude that H (B ) = H (B ) ∪ B. Consequently,

    

T aj , H A j

572

j∈J

T aj , A j

∪

j∈J

B ⊆ H ( B ), 569

    

T aj , H A j

Proof. “⇒” If there exists a reflexive and symmetric fuzzy relation R on U such that Eq. (31) holds, then, by Theorem 3, we have B⊆H(B) for all B ∈ F (U ). Hence, for all A j ∈ F (U ) and all aj ∈ [0, 1], j ∈ J, where J is any index set, we have



566

(72)

Analogous to Theorem 27, the following theorem shows that the set of axioms {(AFU1), (AFU2), (AFUR), (AFUS)} for characterizing the reflexive and symmetric T-upper fuzzy rough approximation operator can be replaced by one axiom.

j∈J

565

.

j∈J

Therefore, (ASLRS) is degenerated to (ASLS). Thus, by Theorem 16, there exists a symmetric fuzzy relation R on U such that Eq. (30) holds. Moreover, by inclusion relation (71) and Theorem 3, we conclude that R is reflexive. 

T  a, HT−1 (A ) ∪ H 564

(69)

S aj , A j .

j∈J

557

.

j∈J

Thus, by Theorem 16, it is easy to verify that Eq. (69) holds, which means that L satisfies (ASLRS). “⇐” Assume that L satisfies (ASLRS). For any B ∈ F (U ), by taking J = {1}, a1 = 0, a = 1, and A1 = A = B in (ASLRS), we conclude that L(B ) = L(B ) ∩ B. Consequently,

S aj , L A j

558



j∈J

L(B ) ⊆ B, 555

S aj , A j

∩

Proof. “⇒” If there exists a reflexive and symmetric fuzzy relation R on U such that Eq. (30) holds, then, by Theorem 3, we have L(B)⊆B for all B ∈ F (U ). Hence, for all A j ∈ F (U ) and all aj ∈ [0, 1], j ∈ J, where J is any index set, we have

552

   

S aj , L A j

T aj , A j

∪

j∈J



=





 

T aj , H A j

.

(76)

j∈J

Therefore, (ATURS) is degenerated to (ATUS). Thus, by Theorem 17, there exists a symmetric fuzzy relation R on U such that Eq. (31) holds. Moreover, by inclusion relation (75) and Theorem 3, we see that R is reflexive.  By employing the inf S-product and sup T-product in Section 2, we can obtain another pair of single axioms for characterizing the reflexive and symmetric S-lower and T-upper fuzzy rough approximation operators. Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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575 576 577

578 579

580 581 582

583

584

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Theorem 29. Let L : F (U ) → F (U ) be a fuzzy operator. Then there exists a reflexive and symmetric fuzzy relation R on U such that Eq. (30) holds iff L satisfies following (ASLRS) : (ASLRS) ∀A, B ∈ F (U ),



A, B ∩ L(B )







= L ( A ), B .

S

(77)

S

Proof. “⇒” Assume that there exists a reflexive and symmetric fuzzy relation R on U such that Eq. (30) holds. On one hand, since R is symmetric, by Theorem 18, we have



A, L(B )







= L(A ), B , ∀A, B ∈ F (U ).

S

(78)

S

On the other hand, since R is reflexive, by Theorem 3, we see that inclusion relation (71) holds. Combining inclusion relation (71) with Eq. (78), we conclude Eq. (77), which means that L satisfies (ASLRS) . “⇐” Assume that L satisfies (ASLRS) . For any A, B ∈ F (U ), by (O1) and (O7) in Proposition 4, we have



A, B ∩ L(B )

Hence



L ( A ), B







= A, B

S







∧ A, L(B )

S



S





= L(A ), B , ∀A, B ∈ F (U ).



≤ A, B , ∀B ∈ F (U ).

S

(80)

S

Then, by (O4) in Proposition 4, we have

L(A ) ⊆ A, ∀A ∈ F (U ). 585 586 587 588 589 590 591 592

593 594

595 596 597

598

(81)

Thus (ASLRS) is degenerated to (ASLS) . Therefore, by Theorem 18, there exists a symmetric fuzzy relation R on U such that Eq. (30) holds. Moreover, by inclusion relation (81) and Theorem 3, we conclude that R is reflexive.  Theorem 29 shows that the axiom (ASLRS) is equivalent to the axiom (ASLRS). Analogously, we examine that the axiom (ATURS) for characterizing the reflexive and symmetric T-upper fuzzy rough approximation operator can be equivalently replaced by condition (ATURS) in the following theorem. Theorem 30. Let H : F (U ) → F (U ) be a fuzzy operator. Then there exists a reflexive and symmetric fuzzy relation R on U such that Eq. (31) holds iff H satisfies following (ATURS) : (ATURS) ∀A, B ∈ F (U ),



A, B ∪ H (B )





T



= H ( A ), B .

(82)

T

Proof. “⇒” Assume that there exists a reflexive and symmetric fuzzy relation R on U such that Eq. (31) holds. On one hand, since R is symmetric, by Theorem 19, we have



A, H (B )





T



= H (A ), B , ∀A, B ∈ F (U ).

(83)

T

On the other hand, since R is reflexive, by Theorem 3, we see that inclusion relation (75) holds. Combining Eq. (83) with inclusion relation (75), we conclude Eq. (82), which means that H satisfies (ATURS) . “⇐” Assume that H satisfies (ATURS) . For any A, B ∈ F (U ), by (I7) in Proposition 3, we have



A, B ∪ H (B )

Hence



A, B





T

599

(79)

S



T



= A, B





T

∨ A, H (B )



T





= H (A ), B , ∀A, B ∈ F (U ).

(84)

T



≤ H (A ), B , ∀B ∈ F (U ).

(85)

T

Then, by (I4) in Proposition 3, we have

A ⊆ H (A ), ∀A ∈ F (U ).

(86)

601

Thus (ATURS) is degenerated to (ATUS) . Therefore, by Theorem 19, there exists a symmetric fuzzy relation R on U such that Eq. (31) holds. Moreover, by inclusion relation (86) and Theorem 3, we conclude that R is reflexive. 

602

4.9. Axioms for reflexive and T-transitive (S, T)-fuzzy rough approximation operators

603

In this subsection we will explore single axioms to characterize the (S, T)-fuzzy rough approximation operators generated by a reflexive and T-transitive fuzzy relation. According to Theorems 12 and 20, we observe that {(AFL1), (AFL2), (AFLR), (AFLT)} (resp. {(AFU1), (AFU2), (AFUR), (AFUT)}) is the basic set of axioms for characterizing the S-lower (resp. T-upper) fuzzy rough approximation operator generated by a reflexive and T-transitive fuzzy relation. In the next theorem, we will examine that {(AFL1), (AFL2), (AFLR), (AFLT)} can be replaced by a single axiom.

600

604 605 606 607 608 609 610

Theorem 31. Let L : F (U ) → F (U ) be a fuzzy operator. If T and S are dual with each other, then there exists a reflexive and T-transitive fuzzy relation R on U such that Eq. (30) holds iff L satisfies following (ASLRT): Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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611

(ASLRT) ∀A j ∈ F (U ), ∀a j ∈ [0, 1], j ∈ J, where J is any index set,

L

 

S aj , A j



=

 

S aj , A j

j∈J

612 613 614 615

  

   

S aj , L A j

∩

j∈J

  

S aj , L L A j

∩

j∈J

21

.

(87)

j∈J

Proof. “⇒” Assume that there exists a reflexive and T-transitive fuzzy relation R on U such that Eq. (30) holds. Since R is reflexive, by Theorem 13, L satisfies (ASLR). On the other hand, since R is T-transitive, by Theorem 21, L obeys (ASLT). Combining (ASLR) with (ASLT), we conclude Eq. (87), which means that L satisfies (ASLRT). “⇐” Assume that L satisfies (ASLRT). For any B ∈ F (U ), by taking J = {1}, a1 = 0 and A1 = B in (ASLRT), we obtain





L (B ) = B ∩ L (B ) ∩ L L (B ) . 616

Hence we have

∀B ∈ F (U ),

L(B ) ⊆ B, 617

and





(89)

∀B ∈ F (U ).

L (B ) ⊆ L L (B ) , 618

(88)

(90)

That is, L satisfies properties (FLR) and (FLT) in Theorem 3. Moreover, inclusion relations (89) and (90) imply Eq. (91) below:



















S b, L(B ) = S  b, B ∩ S  b, L(B ) ∩ S  b, L L(B ) , 619 620 621 622 623 624 625 626

Analogous to Theorem 31, the following theorem shows that the set of axioms {(AFU1), (AFU2), (AFUR), (AFUT)} for characterizing the reflexive and T-transitive upper fuzzy rough approximation operator can be replaced by one axiom. Theorem 32. Let H : F (U ) → F (U ) be a fuzzy operator. Then there exists a reflexive and T-transitive fuzzy relation R on U such that Eq. (31) holds iff H satisfies following (ATURT): (ATURT) ∀A j ∈ F (U ), ∀a j ∈ [0, 1], j ∈ J, where J is any index set,

 

T aj , A j



=

 

T aj , A j

j∈J

627 629 630

   

j∈J

    

T aj , H A j

∪

j∈J

  

T aj , H H A j

∪





(93)

Hence we have

and



∀B ∈ F (U ),



H H ( B ) ⊆ H ( B ), 633

(94)

∀B ∈ F (U ).

(95)

Moreover, inclusion relations (94) and (95) imply Eq. (96) below:



















T  b, H (B ) = T  b, B ∪ T  b, H (B ) ∪ T  b, H H (B ) , ∀b ∈ [0, 1], ∀B ∈ F (U ). 634 635 636 637 638 639 640 641 642 643 644 645

(92)

Proof. “⇒” Assume that there exists a reflexive and T-transitive fuzzy relation R on U such that Eq. (31) holds. Since R is reflexive, by Theorem 14, H satisfies (ATUR). On the other hand, since R is T-transitive, by Theorem 22, H obeys (ATUT). Combining (ATUR) with (ATUT), we conclude Eq. (92), which means that H satisfies (ATURT). “⇐” Assume that H satisfies (ATURT). For any B ∈ F (U ), by taking J = {1}, a1 = 1, and A1 = B in (ATURT), we obtain

B ⊆ H ( B ), 632

.

j∈J

H (B ) = B ∪ H (B ) ∪ H H (B ) . 631

(91)

Consequently, by (ALSRT), it is easy to see that L satisfies (ALS). Thus, by Theorem 5, there exists a fuzzy relation R on U such that Eq. (30) holds. Furthermore, by inclusion relations (89) and (90), and Theorem 3, we conclude that R is a reflexive and T-transitive fuzzy relation. 

H

628

∀b ∈ [0, 1], ∀B ∈ F (U ).

(96)

Consequently, by (ATURT), it is easy to see that H satisfies (AUT). Thus, by Theorem 6, there exists a fuzzy relation R on U such that Eq. (31) holds. Furthermore, by inclusion relations (94) and (95), and Theorem 3, we conclude that R is a reflexive and T-transitive fuzzy relation.  Remark 6. According to [23], a relation-based T-upper fuzzy rough approximation operator is a fuzzy closure operator (which satisfies fuzzy closure axioms) if and only if the fuzzy relation in the approximation space is reflexive and T-transitive. Thus, our results show that, if a fuzzy operator H : F (U ) → F (U ) satisfies the condition (ATURT), then it must obey the following four fuzzy closure axioms: (Cl1) (Cl2) (Cl3) (Cl4)

A ⊆ H (A ), ∀A ∈ F (U ), H (A ∪ B) = H (A ) ∪ H (B ), ∀A, B ∈ F (U ), H H (A ) = H (A ), ∀A ∈ F (U ), , ∀α ∈ [0, 1]. ) = α H (α

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646

4.10. Axioms for symmetric and T-transitive (S, T)-fuzzy rough approximation operators

647

In this subsection we will explore single axioms to characterize the (S, T)-fuzzy rough approximation operators generated by a symmetric and T-transitive fuzzy relation. According to Theorems 15 and 20, we observe that {(AFL1), (AFL2), (AFLS), (AFLT)} (resp. {(AFU1), (AFU2), (AFUS), (AFUT)}) is the basic set of axioms for characterizing the S-lower (resp. T-upper) fuzzy rough approximation operator generated by a symmetric and T-transitive fuzzy relation. In the next theorem, we will examine that {(AFL1), (AFL2), (AFLS), (AFLT)} can be replaced by a single axiom.

648 649 650 651 652 653 654 655

Theorem 33. Let L : F (U ) → F (U ) be a fuzzy operator. If T and S are dual with each other, then there exists a symmetric and Ttransitive fuzzy relation R on U such that Eq. (30) holds iff L satisfies following (ASLST): (ASLST) ∀A, A j ∈ F (U ), ∀a, a j ∈ [0, 1], j ∈ J, where J is any index set,





S  (A ) ∩ L a, L−1 S

 

S aj , A j





  

=S  a, L(A ) ∩

j∈J

656 657

j∈J





661 662 663 664 665 666 667 668 669 670





It follows that







(98)









∀a ∈ [0, 1], ∀B ∈ F (U ). 



∀a ∈ [0, 1], ∀B ∈ F (U ).

=S  a, L(B ) ,

Theorem 34. Let H : F (U ) → F (U ) be a fuzzy operator. Then there exists a symmetric and T-transitive fuzzy relation R on U such that Eq. (31) holds iff H satisfies following (ATUST): (ATUST) ∀A, A j ∈ F (U ), ∀a, a j ∈ [0, 1], j ∈ J, where J is any index set,





 

T aj , A j





  

=T  a, H ( A ) ∪





∀B ∈ F (U ).





It follows that







676 677 678 679 680 681 682 683 684

.

(101)

j∈J

(102)









⊆T  a, H (B ) ,

∀a ∈ [0, 1], ∀B ∈ F (U ).





T  a, H (B ) ∪ T  a, H H ( B ) 675

j∈J

  

T aj , H H A j

∪

Then, by (T2) in Proposition 1, we conclude that

T  a, H H (B ) 674

    

T aj , H A j

Proof. “⇒” Assume that there exists a symmetric and T-transitive fuzzy relation R on U such that Eq. (31) holds. Since R is T-transitive, by Theorem 3, we have

H H ( B ) ⊆ H ( B ), 673

(100)

Analogous to Theorem 33, the following theorem shows that the set of axioms {(AFU1), (AFU2), (AFUS), (AFUT)} for characterizing the symmetric and T-transitive upper fuzzy rough approximation operator can be replaced by one axiom.

j∈J

671

(99)

On the other hand, since R is symmetric, by Theorem 16, L satisfies (ASLS). Thus, by Eq. (100), it is easy to verify that L obeys (ASLST). “⇐” Assume that L satisfies (ASLST). For any B ∈ F (U ), by taking J = {1}, a1 = 0, a = 1, and A1 = A = B in (ASLST), we have L(B ) = L(B ) ∩ L(L(B )), which implies that inclusion relation (98) holds. Then we see that (ASLST) is degenerated to (ASLS). Thus, by Theorem 16, there exists a symmetric fuzzy relation such that Eq. (30) holds. Moreover, by inclusion relation (98) and Theorem 3, we see that R is T-transitive. 

T  a, HT−1 (A ) ∪ H

672

(97)

Then, by (S2) in Proposition 2, we conclude that

S  a, L(B ) ∩ S  a, L L ( B ) 660

.

j∈J

∀B ∈ F (U ).

S  a, L(B ) ⊆ S( a, L L(B ) , 659

  

S aj , L L A j

∩

Proof. “⇒” Assume that there exists a symmetric and T-transitive fuzzy relation R on U such that Eq. (30) holds. Since R is T-transitive, by Theorem 3, we have

L (B ) ⊆ L L (B ) , 658

   

S aj , L A j



=T  a, H ( B ) ,

∀a ∈ [0, 1], ∀B ∈ F (U ).

(103)

(104)

On the other hand, since R is symmetric, by Theorem 17, H satisfies (ATUS). Thus, by Eq. (104), it is easy to verify that H obeys (ATUST). “⇐” Assume that H satisfies (ATUST). For any B ∈ F (U ), by taking J = {1}, a1 = 1, a = 0, and A1 = A = B in (ATUST), we have H (B ) = H (B ) ∪ H (H (B )), which implies that inclusion relation (102) holds. Hence we see that (ATUST) is degenerated to (ATUS). Therefore, by Theorem 17, there exists a symmetric fuzzy relation such that Eq. (31) holds. Moreover, by inclusion relation (102) and Theorem 3, we see that R is T-transitive.  By employing the inf S-product and sup T-product in Section 2, we can obtain another pair of single axioms for characterizing the symmetric and T-transitive S-lower and T-upper fuzzy rough approximation operators. Theorem 35. Let L : F (U ) → F (U ) be a fuzzy operator. If T and S are dual with each other, then there exists a symmetric and Ttransitive fuzzy relation R on U such that Eq. (30) holds iff L satisfies following (ASLST) : Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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685

686 687

688 689 690

691

692

693

694 695

696

697

(ASLST) ∀A, B ∈ F (U ),





A, L(B ) ∩ L L(B )

699 700 701 702 703 704

705 706

707 708 709

710

711

712



S



= L ( A ), B .

23

(105)

S

Proof. “⇒” Assume that there exists a symmetric and T-transitive fuzzy relation R on U such that Eq. (30) holds. On one hand, since R is symmetric, by Theorem 18, we have



A, L(B )







= L(A ), B , ∀A, B ∈ F (U ).

S

(106)

S

On the other hand, since R is T-transitive, by Theorem 3, we see that inclusion relation (98) holds. Combining inclusion relation (98) with Eq. (106), we conclude Eq. (105), which means that L satisfies (ASLST) . “⇐” Assume that L satisfies (ASLST) . For any A, B ∈ F (U ), by (O1) and (O7) in Proposition 4, we have





A, L(B ) ∩ L L(B )

Then



L ( A ), B







S



= A, L(B )



S





∧ A, L L(B )





S



= L ( A ), B .

(107)

S



≤ A, L(B ) , ∀A, B ∈ F (U ).

S

(108)

S

By exchanging A and B in inequality (108) and in terms of (O1) in Proposition 4, we then obtain



A, L(B )





S

A, L(B )

(109)

S

Consequently





≤ L(A ), B , ∀A, B ∈ F (U ).







= L(A ), B , ∀A, B ∈ F (U ).

S

(110)

S

Therefore, by Theorem 18, there exists a symmetric fuzzy relation R on U such that Eq. (30) holds. On the other hand, Eq. (107) implies



L ( A ), B







≤ A, L L(B )

S



S

, ∀A, B ∈ F (U ).

(111)

Hence, by Eq. (110), we conclude



A, L(B )







≤ A, L L(B )

S



S

, ∀A, B ∈ F (U ).

(112)

It follows from (O1) and (O4) in Proposition 4 that





L (B ) ⊆ L L (B ) , 698



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∀B ∈ F (U )

(113)

Thus, by Theorem 3, we see that R is T-transitive.  Theorem 35 shows that the axiom (ASLST) is equivalent to the axiom (ASLST). Analogously, we examine that the axiom (ATUST) for characterizing the symmetric and T-transitive upper fuzzy rough approximation operator can be equivalently replaced by condition (ATUST) in the following theorem. Theorem 36. Let H : F (U ) → F (U ) be a fuzzy operator. Then there exists a symmetric and T-transitive fuzzy relation R on U such that Eq. (31) holds iff H satisfies following (ATUST) : (ATUST) ∀A, B ∈ F (U ),





A, H (B ) ∪ H H (B )





T



= H ( A ), B .

(114)

T

Proof. “⇒” Assume that there exists a symmetric and T-transitive fuzzy relation R on U such that Eq. (31) holds. On one hand, since R is symmetric, by Theorem 19, we have



A, H (B )



T





= H (A ), B , ∀A, B ∈ F (U ).

(115)

T

On the other hand, since R is T-transitive, by Theorem 3, we see that inclusion relation (102) holds. Combining inclusion relation (102) with Eq. (115), we conclude Eq. (114), which means that H satisfies (ATUST) . “⇐” Assume that H satisfies (ATUST) . For any A, B ∈ F (U ), by (I1) and (I7) in Proposition 3, we have





A, H (B ) ∪ H H (B )

Hence



A, H (B )

 T







T

= A, H (B )



T





∨ A, H H (B )





T



= H ( A ), B .

(116)

T



≤ H (A ), B , ∀A, B ∈ F (U ).

(117)

T

By exchanging A and B in inequality (117) and in terms of (I1) in Proposition 3, we then obtain



H ( A ), B





T

Consequently



H ( A ), B



(118)

T



T



≤ A, H (B ) , ∀A, B ∈ F (U ).



= A, H (B ) , ∀A, B ∈ F (U ).

(119)

T

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Therefore, by Theorem 19, there exists a symmetric fuzzy relation R on U such that Eq. (31) holds. On the other hand, Eq. (116) implies





A, H H (B )





T



≤ H (A ), B , ∀A, B ∈ F (U ).

Then, by Eq. (119), we conclude





A, H H (B )





T

(120)

T



≤ A, H (B ) , ∀A, B ∈ F (U ).

(121)

T

716

It follows from (I1) and (I4) in Proposition 3 that

717

Thus, by Theorem 3, we observe that R is T-transitive. 

∀B ∈ F (U ).

H (H (B )) ⊆ H (B ),

(122)

718

4.11. Axioms for T-similarity (S, T)-fuzzy rough approximation operators

719

In this subsection we will explore single axioms to characterize the (S, T)-fuzzy rough approximation operators generated by a fuzzy T-similarity relation. According to Theorems 12, 15 and 20, we observe that {(AFL1), (AFL2), (AFLR), (AFLS), (AFLT)} (resp. {(AFU1), (AFU2), (AFUR), (AFUS), (AFUT)}) is the basic set of axioms for characterizing the S-lower (resp. T-upper) fuzzy rough approximation operator generated by a T-similarity fuzzy relation. In the next theorem, we will examine that {(AFL1), (AFL2), (AFLR), (AFLS), (AFLT)} can be replaced by a single axiom.

720 721 722 723 724 725 726 727

Theorem 37. Let L : F (U ) → F (U ) be a fuzzy operator. If T and S are dual with each other, then there exists a T-similarity fuzzy relation R on U such that Eq. (30) holds iff L satisfies following (ASLE): (ASLE) ∀A, A j ∈ F (U ), ∀a, a j ∈ [0, 1], j ∈ J, where J is any index set,





S  (A ) ∩ L a, L−1 S

 

S aj , A j





  

=S  a, L(A ) ∩

j∈J

728 729 730 731 732 733

S aj , A j

  

j∈J

   

S aj , L A j

∩

j∈J

  

S aj , L L A j

∩

.

(123)

j∈J

Proof. “⇒” Assume that there exists a T-similarity fuzzy relation R on U such that Eq. (30) holds. For all A, A j ∈ F (U ) and all a, aj ∈ [0, 1], j ∈ J, where J is any index set, since R is symmetric and T-transitive, by Theorem 33, L satisfies (ASLST). On the other hand, since R is reflexive, by Theorem 13, L obeys (ASLR). Combining (ASLST) with (ASLR), we conclude Eq. (123), which means that L satisfies (ASLE). “⇐” Assume that L satisfies (ASLE). For any B ∈ F (U ), by taking J = {1}, a1 = 0, a = 1, and A1 = A = B in (ASLE), we can conclude





L (B ) = B ∩ L (B ) ∩ L L (B ) . 734

Hence we have

L(B ) ⊆ B, 735

and

∀B ∈ F (U ),





L (B ) ⊆ L L (B ) , 736

Consequently

and

(125)

∀B ∈ F (U ).

















S b, L(B ) ⊆ S  b, B , 737

739 740 741 742 743 744



∀b ∈ [0, 1], ∀B ∈ F (U ).

(128)

Analogous to Theorem 37, the following theorem shows that the set of axioms {(AFU1), (AFU2), (AFUR), (AFUS), (AFUT)} for characterizing the T-similarity upper fuzzy rough approximation operator can be replaced by one axiom. Theorem 38. Let H : F (U ) → F (U ) be a fuzzy operator. Then there exists a T-similarity fuzzy relation R on U such that Eq. (31) holds iff H satisfies following (ATUE): (ATUE) ∀A, A j ∈ F (U ), ∀a, a j ∈ [0, 1], j ∈ J, where J is any index set,





 

T aj , A j

j∈J

746

(127)

Thus we can observe that (ASLE) is degenerated to (ASLS). Then, by Theorem 16, there exists a symmetric fuzzy relation such that Eq. (30) holds. Moreover, by inclusion relations (125) and (126), and Theorem 3, we see that R is reflexive and T-transitive. 

T  a, HT−1 (A ) ∪ H 745

(126)

∀b ∈ [0, 1], ∀B ∈ F (U ),

S b, L(B ) ⊆ S  b, L L(B ) , 738

(124)





  

=T  a, H ( A ) ∪

T aj , A j

j∈J

   

    

T aj , H A j

∪

j∈J

  

T aj , H H A j

∪

.

(129)

j∈J

Proof. “⇒” Assume that there exists a T-similarity fuzzy relation R on U such that Eq. (31) holds. For all A, A j ∈ F (U ) and all a, aj ∈ [0, 1], j ∈ J, where J is any index set, since R is symmetric and T-transitive, by Theorem 34, H satisfies (ATUST). On the other Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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hand, since R is reflexive, by Theorem 14, H obeys (ATUR). Combining (ATUST) with (ATUR), we conclude Eq. (129), which means that H satisfies (ATUE). “⇐” Assume that H satisfies (ATUE). For any B ∈ F (U ), by taking J = {1}, a1 = 1, a = 0, and A1 = A = B in (ATUE), we have





H (B ) = B ∪ H (B ) ∪ H H (B ) . 750

and

∀B ∈ F (U ),





Consequently













T  b, B ⊆ T  b, H (B ) , 753

and

T  b, H H (B ) 754 755 756 757 758 759 760 761

762 763

764

765

767

768

769

770 771

772

773 774 775 776 777

(132)

∀b ∈ [0, 1], ∀B ∈ F (U ),





⊆T  b, H (B ) ,

(133)

∀b ∈ [0, 1], ∀B ∈ F (U ).

(134)

Thus we can observe that (ATUE) is degenerated to (ATUS). Then, by Theorem 17, there exists a symmetric fuzzy relation on U such that Eq. (31) holds. Moreover, by inclusion relations (131) and (132), and Theorem 3, we see that R is reflexive and T-transitive.  By employing the inf S-product and sup T-product in Section 2, we can obtain another pair of single axioms for characterizing the T-similarity (S, T)-fuzzy rough approximation operators. Theorem 39. Let L : F (U ) → F (U ) be a fuzzy operator. If T and S are dual with each other, then there exists a T-similarity fuzzy relation R on U such that Eq. (30) holds iff L satisfies following (ASLE) : (ASLE) ∀A, B ∈ F (U ),





A, B ∩ L(B ) ∩ L L(B )





S



= L ( A ), B .

(135)

S

Proof. “⇒” Assume that there exists a T-similarity fuzzy relation R on U such that Eq. (30) holds. On one hand, since R is symmetric and T-transitive, by Theorem 35, we have





A, L(B ) ∩ L L(B )





S



= L(A ), B , ∀A, B ∈ F (U ).

(136)

S

On the other hand, since R is reflexive, by Theorem 3, we conclude

L(B ) ⊆ B, 766



(131)

∀B ∈ F (U ).

H H ( B ) ⊆ H ( B ), 752

(130)

Hence we conclude

B ⊆ H ( B ), 751

25

∀B ∈ F (U ).

(137)

Combining Eq. (136) with inclusion relation (137), we obtain Eq. (135), which means that L satisfies (ASLE) . “⇐” Assume that L satisfies (ASLE) . For any A, B ∈ F (U ), by (O1) and (O7) in Proposition 4, we have





A, B ∩ L(B ) ∩ L L(B )

Hence



L ( A ), B





S





S

= A, B



S



∧ A, L(B )



S





∧ A, L L(B )





S



= L ( A ), B .

(138)

S



≤ A, L(B ) , ∀A, B ∈ F (U ).

(139)

S

By exchanging A and B in inequality (139) and in terms of (O1) in Proposition 4, we then obtain



A, L(B )



S

Consequently



A, L(B )

S





≤ L(A ), B , ∀A, B ∈ F (U ).

(140)

S





= L(A ), B , ∀A, B ∈ F (U ).

(141)

S

Therefore, by Theorem 18, there exists a symmetric fuzzy relation R on U such that Eq. (30) holds. On the other hand, Eq. (138) implies



and



L ( A ), B L ( A ), B





S



(142)

S



S



≤ A, B , ∀A, B ∈ F (U ),



≤ A, L L(B )

 S

, ∀A, B ∈ F (U ).

(143)

Similar to the proofs in Theorems 29 and 35, inequalities (142) and (143) imply inclusion relations (125) and (126), respectively. Then, by Theorem 3, we see that R is reflexive and T-transitive. Thus we have proved that R is a T-similarity fuzzy relation.  Theorem 39 shows that the axiom (ASLE) is equivalent to the axiom (ASLE). Analogously, we examine that the axiom (ATUE) for characterizing the T-similarity upper fuzzy rough approximation operator can be equivalently replaced by condition (ATUE) in the following theorem. Please cite this article as: W.-Z. Wu et al., Axiomatic characterizations of (S, T )-fuzzy rough approximation operators, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.028

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Theorem 40. Let H : F (U ) → F (U ) be a fuzzy operator. Then there exists a T-similarity fuzzy relation R on U such that Eq. (31) holds iff H satisfies following (ATUE) : (ATUE) ∀A, B ∈ F (U ),





A, B ∪ H (B ) ∪ H H (B )

785

786

787

788

789 790





A, H (B ) ∪ H H (B )





= H ( A ), B .

T

(144)

T





T



= H (A ), B , ∀A, B ∈ F (U ).

(145)

T

On the other hand, since R is reflexive, by Theorem 3, we conclude

∀B ∈ F (U ).

(146)

Combining Eq. (145) with inclusion relation (146), we obtain Eq. (144), which means that H satisfies (ATUE) . “⇐” Assume that H satisfies (ATUE) . For any A, B ∈ F (U ), by (I1) and (I7) in Proposition 3, we have





A, B ∪ H (B ) ∪ H H (B )

Hence



A, H (B )





T





T

= A, B



T



∨ A, H (B )



T



∨ A, H (H (B ))



T

= ( H ( A ), B )T .



≤ H (A ), B , ∀A, B ∈ F (U ).

(147)

(148)

T

By exchanging A and B in inequality (148) and in terms of (I1) in Proposition 3, we then obtain



H ( A ), B





T



A, H (B )





≤ A, H (B ) , ∀A, B ∈ F (U ).



T

(149)

T

Consequently



= H (A ), B , ∀A, B ∈ F (U ).

(150)

T

Therefore, by Theorem 19, there exists a symmetric fuzzy relation R on U such that Eq. (31) holds. On the other hand, Eq. (147) implies



A, B





T

791



Proof. “⇒” Assume that there exists a T-similarity fuzzy relation R on U such that Eq. (31) holds. On one hand, since R is symmetric and T-transitive, by Theorem 36, we have

B ⊆ H ( B ), 784

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and





≤ A, H (B ) , ∀A, B ∈ F (U ),



A, H H (B )

(151)

T





T



≤ A, H (B ) , ∀A, B ∈ F (U ).

(152)

T

793

Similar to the proofs in Theorems 30 and 36, inequalities (151) and (152) imply inclusion relations (146) and (132), respectively. Then, by Theorem 3, we see that R is reflexive and T-transitive. Thus we have proved that R is a T-similarity fuzzy relation. 

794

5. Conclusion

795

805

A basic problem in the axiomatic approach in rough set theory is to seek for a minimal set of axioms for an abstract lower/upper approximation operator, which is necessary and sufficient for the existence of an approximation space producing the same abstract approximation operator. In this paper, we have developed a general framework for the study of axiomatic approach to relation-based (S, T)-fuzzy rough approximation operators determined by a triangular conorm S and a triangular norm T on the unit interval [0, 1]. We have shown that the S-lower (and, respectively, T-upper) fuzzy rough approximation operator generated by a generalized fuzzy relation can be described by only one axiom. We have also proved that each of (S, T)-fuzzy rough approximation operators corresponding to some special types of fuzzy relations such as serial, reflexive, symmetric, and T-transitive fuzzy relations as well as any of their compositions can be characterized by single axioms. This work may be viewed as the extension of Liu [8], and it may also be taken as the completeness of Mi et al. [11] and Wu [23]. We believe that the axiomatic approach we offer here will help us to gain much more insights into the mathematical structures of (S, T)-fuzzy rough approximation operators.

806

Acknowledgment

807

The authors would like to thank the anonymous referees and the Editor for their valuable comments and suggestions. This work was supported by grants from the National Natural Science Foundation of China(Nos. 61573321, 61272021, 61202206, 61363056, and 61173181), the Zhejiang Provincial Natural Science Foundation of China (Nos. LZ12F03002 and LY14F030001) and Chongqing Key Laboratory of Computational Intelligence (No. CQ-LCI-2013-01).

792

796 797 798 799 800 801 802 803 804

808 809 810

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