Binary Relation, Basis Algebra, Approximation. Operator Form and Its Property in L-Fuzzy. Rough Sets. Zhengjiang Wu and Wenpeng Xu. School of Computer ...
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Binary Relation, Basis Algebra, Approximation Operator Form and Its Property in L-Fuzzy Rough Sets Zhengjiang Wu and Wenpeng Xu School of Computer Science and Technology, Henan Polytechnic Unviersity, Jiaozuo China Email: {wuzhengjiang,wpxu}@hpu.edu.cn
Abstract—The rough set theory usually is used in datamining. At this time, we will suppose the approximation operator to satisfy some wonderful properties. It need us to think about the basis algebra, the binary relation and the approximation operator’s form in the rough set model. Lfuzzy rough approximation operator is a general fuzzy rough approximation operator. Comparing with the others rough approximation operator, it is more clearly for the Lfuzzy rough approximation operator that not only the binary relation and the basis algebra, but also the form of approximation operator influence its properties. To inhibit the relations between binary relation on the universe, basis algebra, the approximation operators’ properties and its form in the L-fuzzy rough set, we introduce some basis algebra(residuated lattice, MTL algebra and IMTL algebra) and some binary relations(the reflexive, symmetric, ⊗-transitive, L-similar, Lindistinguishable fuzzy binary relation) into three types Lfuzzy rough approximation operators and proof some Lfuzzy rough approximation operators’ properties as its properties. As usual, we discuss the approximation operator in the constructive approach and axiomatic approach. This paper focus on the basis algebra and the form of the approximation operators. As the preliminaries, we use some spaces to discuss the properties of the non-classic logic algebra. Index Terms—L-fuzzy rough sets, approximation operators, basis algebra
I. INTRODUCTION Modeling uncertain information, including fuzziness, randomicity, incompleteness and uncomparativity, is one of the main research topics in knowledge representations. Most existing approaches are based on the extensions of classical set theory such as fuzzy set theory and rough set theory. The concept of rough set [1] was originally proposed by Pawlak as a formal tool for modeling and processing the incomplete information in information systems. In the rough set theory, the core idea of rough set is to approximate the knowledge with uncertainty by using two “certain” definitions. The two “certain” definitions are named as the lower and the upper approximation sets. The lower and the upper approximation operators are the two unary mappings in the universe. The rough set theory usually is used in data-mining. we will suppose the
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approximation operator to satisfy some wonderful properties. To avoid the supposes’ bugs and keep the supposes’ reliability, it need us to think about the basis algebra, the binary relation and the approximation operator’s form in the rough set model. The L-fuzzy rough approximation set is a general fuzzy rough approximation set. Fuzzy set is a tool for modeling and processing the fuzzy information [2]. Many attempts are tried to combine both fuzzy set and rough set. Dubois and Prade studied the fuzzification problem of rough sets [3,4]. Morsi and Yakout [5] studied a set of axioms on the fuzzy approximation operators and defined a special family of the fuzzy approximation operators by the T-norms and the residuation implicators on the [0,1]. Later, Radzikowska and Kerre [6] gave a serial of the general fuzzy rough sets with T-norms and S-norms on the [0, 1]. They named the general fuzzy rough set as the (T,I)-fuzzy rough set. In the broad family of the (T,I)fuzzy rough sets, “∧” is replaced by T-norm and “∨” is replaced by S-norm. The (T,I)-fuzzy rough set was as the bridge between the fuzzy rough set and the L-fuzzy rough set. In 2004, Radzikowska and Kerre [7] generalized the (T,I)-fuzzy rough set to the L-fuzzy rough set based on residuated latti ce and presented some basic properties of the approximation operators. In [8-11], Wu and Qin et al. discuss some kinds of L-fuzzy rough approximation operator and emphasize the importance of the basis algebra in the L-fuzzy rough approximation operator. In [12], Y.H. She discuss the L-fuzzy rough set based on residuated lattice in the axiomatic approach. In that paper, the regular residuated lattice is essentially an IMTL algebra. In this paper, L-fuzzy rough set is as an object or a carrier. The following discussion involve four elements in the approximation operator, which are the property and the form of the approximation operator, the binary relation in the universe and the basis algebra. Our focus is the basis algebra and its influences on the constructive and axiomatic approaches. In the constructive approach, the binary relation, the form of the approximation operator and the basis algebra are as the primitive notions [13, 14]. The binary relation in the universe is the formular information in the rough set. The basis algebra is the value range of set-value mapping in the approximation operator model[10]. For example, The basis algebra of a rough set is a {0, 1}Boolean algebra; The basis algebra of a fuzzy rough set is
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an interval of [0, 1]; The basis algebra of an L-fuzzy rough set is L. The properties of approximation operators depend on the basis algebra and the binary relation through the form of approximation operator. In rough set or fuzzy rough set, the effect of the basis algebra often been ignored, just because their properties (Boolean algebra and the interval of [0, 1]) are as suitable as the common sense. But in residuated lattice, ∨ and ∧ aren’t dual and transitive, ٔ is a general operator of the ∧, there is just some partial order in the lattice, etc. In the Lfuzzy rough set, the basis algebra affects the properties of approximation operators even more than the binary relation. In this paper, we select a serial of non-classic logic algebras (such as residuated lattice, MTL algebra and IMTL algebra) as the basis algebra of the L-fuzzy rough approximation operator. As the foundation of this paper, we introduce and proof some properties of those non-classic logic algebras, which can be used in this paper, in Section 3. In Section 4, the contrast of those Lfuzzy rough sets reflects the relation “the properties of Lfuzzy rough approximation operators are affected by the basis algebra and the binary relation through the it’s form” . In the axiomatic approach, we firstly find a set as the universe, and secondly set some hypotheses(, what we name as the axioms) for two operators which define in the universe, and thirdly confirm a kind binary relation by the two operators with a form as the preliminary. The hypotheses about the the operator’s form and the basis algebra are the key to explain “Why are the same conditions corresponding to different binary relations?”. The same conditions include – K(U) = U, K(A∩B) = K(A)∩K(B); – H(∅) =∅, H(A∪B) = H(A) ∪ H(B). The different binary relations include the binary relation[1], the fuzzy binary relation[13] and the fuzzy binary relation based on T-norm[5]. Actually, in the axiomatic approach, the axioms come from the properties of the approximation operators. So before we set the axioms, we should confirm the correctness of the properties of approximation operator. For the L-fuzzy rough set, we will proof it in section 4. II. PRELIMINARIES In this section, we will define the approximation space, L-fuzzy rough approximation operator and L-fuzzy set and review basic concepts in some non-classic logic algebras, e.g., residuated lattice, MTL-algebra and IMTLalgebra, which will be throughout this paper. A. Basis algebra Definition 1[15,16]. By a residuated lattice, we mean an algebra ( L, ∨ , ∧, ⊗, →, 0,1) such that (1) ( L, ∨ , ∧, 0,1) is a bound lattice with the top element 1 and the bottom element 0. (2) ⊗ : L × L → L is a binary operator and satisfies for ∀a , b, c ∈ L , a ⊗ b = b ⊗ a , a ⊗ (b ⊗ c ) = ( a ⊗ b) ⊗ c , 1⊗ a = a , a ≤ b ⇒ a ⊗ c ≤ b ⊗ c .
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(3) → : L × L → L is a residuum of ⊗ , i.e. → satisfies for all a , b, c ∈ L , a ⊗b ≤ c ⇔ a ≤ b → c . A residuated lattice ( L, ∨ , ∧, ⊗, →, 0,1) is called complete iff the underlying lattice ( L, ∨ , ∧, 0,1) is complete. Given a residuated lattice L , we define the precomplement operator ∼: L → L as follows: ∀a ∈ L , ∼ a = a →0. Definition 2[15] The residuated lattice ( L, ∨ , ∧, ⊗ , →, 0,1) is called an MTL-algebra iff it satisfies the following prelinery condition, for ∀a , b ∈ L , ( a → b) ∨ (b → a ) = 1 . Definition 3[17] The MTL-algebra LMTL is called an IMTL-algebra iff it satisfies the following condition: for ∀a ∈ LMTL , ∼∼ a = a . The operator ⊕ exists as the general ∨ , which defined by C.C. Chang in 1958 as the important operator of the MV algebra[18]. In this paper, we define it in an IMTL algebra. Definition 4. Let L be an IMTL-algebra. ⊕ : L × L → L for all a, b ∈ L , a ⊕ b = ~ a → b = ~ (~ a ⊗ ~ b)
B. L-fuzzy set and L-fuzzy binary relation The L-fuzzy set on U is a mapping FL : U → L . For every a ∈ U , FL ( x) is the degree of membership of x. The family of all L-fuzzy sets is denoted by FL (U ) . aˆ ( x) ≡ aˆ , x ∈ U is a constant L-fuzzy set. The L-fuzzy set R on U × U is called a binary L-fuzzy relation on U. An L-fuzzy relation R on U is reflexive iff R ( x, x) = 1 for every x ∈ U ; An L-fuzzy relation R on U is symmetric iff R ( x, y ) = R ( y, x) for every x, y ∈ U ; An L-fuzzy relation R on U is L-transitive iff R ( x, y ) ∧ R ( y, z ) ≤ R ( x, z ) for every x, y, z ∈ U ; An Lfuzzy relation R on U is ⊗ -transitive iff R ( x, y ) ⊗ R ( y, z ) ≤ R ( x, z ) for every x, y, z ∈ U . R is called an L-similar relation if it is reflexive and Ltransitive. R is called an L-indistinguishable relation if it is reflexive, symmetric and L-transitive. R is called an ⊗ -similar relation if it is reflexive and ⊗ -transitive. R is called an ⊗ -indistinguishable relation if it is reflexive, symmetric and ⊗ -transitive.
C. L-fuzzy rough approximation operators Definition 5[7] Suppose U is a non-empty finite universe and R is an L-fuzzy binary relation on U based on L. (U , R) is called an L-fuzzy approximation space. In this paper, If there is no special instructions, all of universes are finite. The definitions of L-fuzzy rough set have been detailedly explained in [7,9,12]. In this paper, we give the other two types L-fuzzy rough sets, which are named as “I-type L-fuzzy rough set” and “II-type L-fuzzy rough set”.
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Definition 6 Let (U , R) be an L-fuzzy approximation space. For any set A ∈ FL (U ) , x ∈ U , the lower and upper approximation RL ( A) and RL ( A) of A with respect to the approximation space (U , RL ) are L-fuzzy sets on U whose membership functions are respectively defined by RL ( A)( x) = inf ( R( x, y ) → A( y ) ) y∈U
RL ( A)( x) = sup ( R( x, y ) ⊗ A( y ) ) y∈U
The pair ( RL ( A), RL ( A) ) is referred to as a normal Lfuzzy rough set. RL , RL : FL (U ) → FL (U ) are referred to as lower and upper L-fuzzy approximation operators. RI ( A)( x) = inf ( ~ R( x, y ) ∨ A( y ) ) y∈U
RI ( A)( x) = sup ( R( x, y ) ∧ A( y ) ) y∈U
The pair ( RI ( A), RI ( A) ) is referred to as a I-type L-fuzzy rough set. RI , RI : FL (U ) → FL (U ) are referred to as the I-type lower L-fuzzy approximation operator and the Itype upper L-fuzzy approximation operator, respectively. RII ( A)( x) = ⊗ ( R( x, y ) → A( y ) ) y∈U
RII ( A)( x) = ⊕ ( R( x, y ) ⊗ A( y ) ) y∈U
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⎛ ⎞ a → ⎜ ∧bi ⎟ = ∧ ( a → bi ) , ⎝ i∈I ⎠ i∈I ⎛ ⎞ ⎜ ∨ai ⎟ → b = ∧ ( ai → b ) , i∈I ⎝ i∈I ⎠ ⎛ ⎞ a → ⎜ ∨bi ⎟ ≥ ∧ ( a → bi ) ⎝ i∈I ⎠ i∈I ⎛ ⎞ ⎜ ∧ai ⎟ → b ≥ ∨ ( ai → b ) i∈I ⎝ i∈I ⎠ (8) ~ a ∧ ~ b =~ (a ∨ b) , ~ a∨ ~ b ≤ ~ (a ∧ b) .
(9) (a ∨ b) ∧ (a ∨ c) ≥ a ∨ (b ∧ c) , (a ∧ b) ∨ (a ∧ c) ≤ a ∧ (b ∨ c) . Theorem 2. Suppose L is an MTL algebra, Then for all a, b, c ∈ L , (1) a ∨ b = ((a → b) → b) ∧ ((b → a) → a ) , (2) a ∧ b = (a ⊗ (a → b)) ∨ (b ⊗ (b → a )) , (3) a ∧ (b ∨ c ) = (a ∧ b) ∨ (a ∧ c) , a ∨ (b ∧ c ) = (a ∨ b) ∧ (a ∨ c) . Proof. (1) First we proof a ∨ b ≥ ((a → b) → b) ∧((b → a ) → a ) .
( (a → b) → b ) ∧ ( (b → a) → a ) = (( (a → b) → b ) ∧ ((b → a) → a)) ⊗ ( (a → b) ∨ (b → a) )
The pair ( RII ( A), RII ( A) ) is referred to as a II-type L-
= ( ( (a → b) → b ) ∧ ( (b → a ) → a ) ⊗ (a → b) )
fuzzy rough set. RII , RII : FL (U ) → FL (U ) are referred to as the II-type lower L-fuzzy approximation operator and the II-type upper L-fuzzy approximation operator, respectively.
= ( ( (a → b) → b ) ∧ ( (b → a ) → a ) ⊗ (a → b) ) ≤ ( ( ( a → b) → b ) ⊗ (a → b) )
III. PROPERTIES OF NON-CLASSIC LOGIC ALGEBRA
∨ ( ( (b → a ) → a ) ⊗ (b → a) )
For the L-fuzzy rough approximation operator, not only the binary relation, but also the form of approximation operator and the basis algebra influence its properties. In this section, we will introduce some properties of those non-classic logic algebras, which were used in following proofs. For the L-fuzzy rough approximation operators, the properties of non-classic logic algebras will been transmitted into its properties. Theorem 1[15,16] Suppose ( L, ∨ , ∧, ⊗, →, 0,1) is a residuated lattice, and ∼ is the precomplement operator on L . Then all a, b, c ∈ L , (1) a ⊗ b ≤ a ∧ b , a ⊗ b ≤ a → b, a → b ≥ b . (2) a ⊗ (a → b) ≤ b . (3) a → ( b → c ) = ( a ⊗ b ) → c = b → ( a → c ) . (4) If a ≤ b and c ≤ d , then d → a ≤ c → b . (5) a ≤ b ⇒ a → b = 1 . (6) a ≤ ∼∼ a , ∼∼∼ a =∼ a . (7) If L is a complete lattice, then ⎛ ⎞ ⎜ ∨ai ⎟ ⊗ b = ∨ ( ai ⊗ b ) , i∈I ⎝ i∈I ⎠
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∨ ( ( (a → b) → b ) ∧ ( (b → a) → a ) ⊗ (b → a) )
≤ b∨a On the other hand, by theorem 1-(i), a ≤ (a → b) → b , and by the same theorem (ii), it is easy to proof b ≤ (a → b) → b . That is to say a ∨ b ≤ (a → b) → b . Following the same steps, we can proof a ∨ b ≤ (b → a) → a . Thus, a ∨ b ≤ ( (b → a ) → a ) ∧ ( (a → b) → b ) .
Above all, a ∨ b = ( (b → a ) → a ) ∧ ( (a → b) → b ) . (2) First we proof a ∧ b ≤ (a ⊗ (a → b)) ∨ (b ⊗ (b → a )) . a ∧ b = (a ∧ b) ⊗ ( (a → b) ∨ (b → a ) ) = ( (a ∧ b) ⊗ (a → b) ) ∨ ( (a ∧ b) ⊗ (b → a ) )
≤ ( a ⊗ (a → b) ) ∨ ( b ⊗ (b → a) )
On the other hand, by definition 1-(ii)-(d), (a → b) ⊗ a ≤ a and by the definition 1-(iii), what is the definition of residuum, (a → b) ⊗ a ≤ b . That is to say (a → b) ⊗ a ≤ a ∧ b . Following the same steps, we can proof (b → a ) ⊗ b ≤ a ∧ b .
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Thus, ( a ⊗ (a → b) ) ∨ ( b ⊗ (b → a ) ) ≤ a ∧ b . Above all, a ∧ b = ( a ⊗ (a → b) ) ∨ ( b ⊗ (b → a ) ) . (3) For all a, b, c ∈ L a ∨ (b ∧ c) = ( (a → b ∧ c) → b ∧ c ) ∧ ( (b ∧ c → a ) → a )
= ( ( (a → b) ∧ (a → c) ) → b ∧ c ) ∧ ( ( (b → a) ∨ (c → a) ) → a )
= ( ( ( a → b) ∧ ( a → c ) ) → b ) ∧ ( ( ( a → b ) ∧ ( a → c ) ) → c ) = ( ( ( a → b) ∧ ( a → c ) ) → b ) ∧ ( ( ( a → b ) ∧ ( a → c ) ) → c )
∧ ( (c → a) → a ) ∧ ( (b → a) → a )
≥ ( ( a → b) → b ) ∧ ( ( a → c ) → c ) ∧ ( (c → a ) → a ) ∧ ( (b → a ) → a )
= (a ∨ b) ∧ (a ∨ c) ≥ a ∨ (b ∧ c ) \
Thus, a ∨ (b ∧ c) = ( a ∨ b ) ∧ ( a ∨ c ) a ∧ (b ∨ c) = ( a ⊗ (a → b ∨ c) ) ∧ ( (b ∨ c) ⊗ ((b ∨ c) → a) )
≤ ( a ⊗ (a → b) ) ∨ ( a ⊗ ( a → c ) ) ∨ ( b ⊗ ((b ∨ c ) → a ) ) ∨ ( c ⊗ ((b ∨ c) → a ) ) ≤ ( a ⊗ ( a → b) ) ∨ ( a ⊗ (a → c) ) ∨ ( b ⊗ (b → a ) ) ∨ ( c ⊗ (c → a ) )
= ( a ∧ b ) ∨ ( a ∧ c ) ≤ a ∧ (b ∨ c)
Thus, a ∧ (b ∨ c) = ( a ∧ b ) ∨ ( a ∧ c ) .
□
Compare to Theorem 1 and Theorem 2, some conclusions are established in MTL algebra but not in residuated lattice, such as - a ∨ c → b = ( a → b) ∧ ( a → c ) . - ∨ and ∧ are dual. - ∨ and ∧ content distribution law. Theorem 3. Suppose L is an IMTL algebra, and ~ is the complement operator on L. Then for all a, b, c ∈ L , (1) a ⊕ b = b ⊕ a . (2) a → b = ~ (a⊗ ~ b) =~ a ⊕ b . (3) ∼ a → b = ~ b → a . (4) a ∨ b ≤ a ⊕ b . (5) If a ≤ c , then a ⊕ b ≤ c ⊕ b .
( )
( )
(6) ∧ ai ⊕ b = ∧ ( ai ⊕ b ) , ∨ ai ⊕ b ≥ ∨ ( ai ⊕ b ) . i∈I
i∈I
i∈I
i∈I
Proof. (1)- (5) are easy to prove. (6) By Theorem 1-(vii) and Theorem 2-(iv), for ai , b ∈ L , i ∈ I ,
(
∧ ( ∼ ai ) ⊕ ∼ b =∼ ∨ ai ⊗ b
i∈I
i∈I
)
i∈ I
Thus ∧ ai ⊕ b = ∧ ( ai ⊕ b ) . i∈I
i∈I
□
In theorem 3, some conclusions is established in IMTL algebra. - There exists a direct relation between ⊗ and → in IMTL algebra. - ⊗ and ⊕ are dual. - ∨ is the smallest ⊕ .
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IV. THE L-FUZZY ROUGH APPROXIMATION OPERATOR'S PROPERTIES In this section, we will proof some properties of the three types upper and lower approximation operators. For the L-fuzzy rough set based on residuated lattice, Its properties have been discussed in detail[7,9,12]. Now, we select some useful properties and set out them in theorem 5. Theorem 5[5] Let L be a complete residuated lattice, (U , R) is L-fuzzy approximation space. Then for ∀A, B ∈ FL (U ) , (1) RL (U ) = U , RL (∅) = ∅ . (2) If A ⊆ B , then RL ( A) ⊆ RL ( B ) , RL ( A) ⊆ RL ( B ) . (3) RL ( A ∩ B ) = RL ( A) ∩ RL ( B) , RL ( A ∪ B) = RL ( A) ∪ RL ( B ) .
(4) RL ( A ∪ B ) ⊇ RL ( A) ∪ RL ( B ) , RL ( A ∩ B) ⊆ RL ( A) ∩ RL ( B) . (5) RL ( A) ⊆ ∼ RL (∼ A) , RL ( A) ⊆ ∼ RL (∼ A) . (6) ∼ RL ( A) = RL (∼ A) =∼ RL (∼∼ A) =∼∼ RL (∼ A) . (7) RL (aˆ → A) = aˆ → RL ( A) , RL (aˆ ⊗ A) = aˆ ⊗ RL ( A) .
= ∼ ∨ ( ai ⊗ b ) = ∧ ( ∼ ai ⊕ ∼ b ) i∈I
Usually, ⊗ and ⊕ are not distributive in an IMTL algebra Theorem 4. [11] In an IMTL algebra, if ⊗ are ⊕ are distribute, the IMTL algebra is a Boolean algebra. Proof. Let L be an IMTL algebra. For every a, b, c ∈ L , a → b = ~ a ⊗ b . If ⊗ is distributive about ⊕ , then a → (b ⊗ c) =∼ a ⊕ (b ⊗ c ) = (∼ a ⊕ b) ⊗ (~ a ⊕ c) = (a → b) ⊗ (a → c) Thus, (b ∧ c) → (b ⊗ c) = (b ∧ c → b) ⊗ (b ∧ c → c) = 1 ⊗ 1 = 1 That is b ∧ c ≤ b ⊗ c But we have known, in residuated lattice, b ∧ c ≥ b ⊗ c . Then in this IMTL-algebra L, b∧c = b⊗c . At the same time, = ∼ ( ∼ a ⊕ b) ⊕ ( a ⊗ c ) = ( a → b) → (a ⊗ c ) . Because a ⊗ (b → c) = (a → a ) → (a ⊗ b) = 1 → (a ⊗ b) = a ∧ b , L is a BL algebra. So L is an MV algebra. For every a ∈ L , a ∧ ~ a = a ⊗ (a → 0) = (a → a) → (a ⊗ 0) = 1 → 0 = 0 . Thus L is a Boolean algebra.
The property “ ∼ RL ( A) = RL (∼ A) ” is named as “the lower approximation operator is the semi-dual L-fuzzy operator of the upper approximation operator”. A. I-Type L-fuzzy rough set based on residuated lattice Theorem 6 Let L be a complete residuated lattice, (U , R) is L-fuzzy approximation space. Then for all A, B ∈ FL (U ) , (1) RI (U ) = U , RI (∅) = ∅ . (2) If A ⊆ B , then RI ( A) ⊆ RI ( B ) , RI ( A) ⊆ RI ( B) .
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(3) RI ( A) ⊆ ∼ RI (∼ A) , RI ( A) ⊆ ∼ RI (∼ A) . (4) ∼ RI ( A) ⊇ RI (∼ A) . (5) RI ( A ∩ B) ⊆ RI ( A) ∩ RI ( B) , RI ( A ∪ B) ⊇ RI ( A) ∪ RI ( B) . (6) RI ( A ∪ B) ⊇ RI ( A) ∪ RI ( B) , RI ( A ∩ B) ⊆ RI ( A) ∩ RI ( B) Proof. (i) and (ii) are easy to proof. (3) For any x ∈ U , A ∈ FL (U ) ,
∼ RI (∼ A)( x) =∼ sup ( R ( x, y )∧ ∼ A( y ) ) y∈U
= inf ( ∼ ( R ( x, y )∧ ∼ A( y ) ) ) y∈U
≥ inf ( ∼ R ( x, y )∨ ∼∼ A( y ) ) y∈U
≥ inf ( ∼ R ( x, y ) ∨ A( y ) ) = RI ( A)( x) y∈U
∼ RI (∼ A)( x) =∼ inf ( ∼ R( x, y )∨ ∼ A( y ) ) y∈U
≥ sup ( ∼ ( ∼ R( x, y )∨ ∼ A( y ) ) ) y∈U
= sup ( ∼∼ R( x, y )∨ ∼∼ A( y ) ) y∈U
≥ sup ( R( x, y ) ∨ A( y ) ) = RI ( A)( x) y∈U
Thus RI ( A) ⊆∼ RI (∼ A) , RI ( A) ⊆∼ RI (∼ A) . (4) For all x ∈ U , A ∈ FL (U ) , ∼ RI ( A)( x) =∼ sup ( R( x, y ) ∧ A( y ) ) y∈U
= inf ( ∼ ( R ( x, y ) ∧ A( y ) ) ) y∈U
≥ inf ( ∼ R ( x, y )∨ ∼ A( y ) ) y∈U
= RI (∼ A)( x) (5) It is easy to proof by (2). □ The difference between the conditions of theorem 5 and theorem 6 is the form of the approximation operator. One is RL ( A)( x) = inf ( RL ( x, y ) → A( y ) ) y∈U
RL ( A)( x) = sup ( RL ( x, y ) ⊗ A( y ) ) y∈U
The other one is RI ( A)( x) = inf ( ~ RL ( x, y ) ∨ A( y ) ) y∈U
RI ( A)( x) = sup ( RL ( x, y ) ∧ A( y ) ) y∈U
The main differences between the conclusions of theorem 5 and theorem 6 are following: - ∼ RL ( A) = RL (∼ A) . - RI ( A) ⊆ ∼ RI (∼ A) , RI ( A) ⊆ ∼ RI (∼ A) . - RL ( A ∩ B) = RL ( A) ∩ RL ( B ) , RL ( A ∪ B) = RL ( A) ∪ RL ( B) - RI ( A ∩ B) ⊆ RI ( A) ∩ RI ( B) , RI ( A ∪ B ) ⊇ RI ( A) ∪ RI ( B) .
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- RL (aˆ → A) = aˆ → RL ( A) , RL (aˆ ⊗ A) = aˆ ⊗ RL ( A) . Those differences between the forms of the approximation operators cause the differences between the conclusions. In nature, the difference of the conclusions come from the basis algebra, so we prefer the parlance "The properties of the approximation operator depend on the basis algebra and binary relation" and ignore the function of the approximation operators' forms. Theorem 7. Let L be a complete residuated lattice, and (U , R) be an L-fuzzy approximation space based on L. If R is a serial L-fuzzy binary relation, then for all A ∈ FL (U ) , (1) RI ( A) ⊆ RI ( A) . (2) RI (U ) = U , RI (∅) = ∅ . Theorem 8. Let L be a complete residuated lattice, and (U , R) be an L-fuzzy approximation space based on L. If R is a reflexive L-fuzzy binary relation, then for all A ∈ FL (U ) , RI ( A) ⊆ A ⊆ RI ( A) . Theorem 9. Let L be a complete residuated lattice, and (U;R) be an L-fuzzy approximation space based on L. If R is a symmetric L-fuzzy binary relation, then for all A ∈ FL (U ) , (1) RI (1x )( y ) = RI (1y )( x) . (2) RI (1U /{ x} )( y ) = RI (1U /{ y} )( x) . B. I-Type L-fuzzy rough set based on MTL algebra The basis algebra of I-Type L-fuzzy rough set based on MTL algebra is an MTL algebra. By theorem 2, ∨ and ∧ are dual and distributive. Compared with Theorem 6, we can add some properties for the I-Type L-fuzzy rough approximation operator based on MTL algebra and those properties is set out in following theorem. Theorem 10. Let L be a complete MTL algebra, and (U , R) be an L-fuzzy approximation space based on L. for all A, B ∈ FL (U ) , (1) RI (aˆ ∪ A) = aˆ ∪ RI ( A) , RI (aˆ ∩ A) = aˆ ∩ RI ( A) . (2) RI ( A ∩ B ) = RI ( A) ∩ RI ( B ) , RI ( A ∪ B ) = RI ( A) ∪ RI ( B ) . Proof. By theorem 2, ∨ and ∧ are dual and distributive. So for every A, B ∈ FL (U ) , x ∈ U , the following equations can be confirmed. RI (aˆ ∨ A)( x) = inf ( ~ RL ( x, y ) ∨ α ∨ A( y ) ) y∈U
= α ∨ inf ( ~ RL ( x, y ) ∨ A( y ) ) y∈U
= α ∨ inf ( ~ RL ( x, y ) ∨ A( y ) ) y∈U
RI ( A ∩ B)( x) = inf ( ~ RL ( x, y ) ∨ ( A( y ) ∧ B ( y )) ) y∈U
= inf ( (~ RL ( x, y ) ∨ A( y )) ∧ (~ RL ( x, y ) ∨ B ( y )) ) y∈U
= inf (~ RL ( x, y ) ∨ A( y )) ∧ inf (~ RL ( x, y ) ∨ B( y )) y∈U
= RI ( A)( x) ∧ RI ( B )( x)
y∈U
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Similar, we can proof RI (aˆ ∧ A) = aˆ ∧ RI ( A) and RI ( A ∪ B ) = RI ( A) ∪ RI ( B ) . □ Theorem 11. Let L be a complete MTL algebra, and (U , R) be an L-fuzzy approximation space based on L. If R is a serial L-fuzzy binary relation, then for all a ∈ L , RI (aˆ ) = aˆ = RI (aˆ ) . Theorem 12. Let L be a complete MTL algebra, and (U , R) be an L-fuzzy approximation space based on L. If R is a L-transitive L-fuzzy binary relation, then for all A ∈ FL (U ) ,
(1) RI ( RI ( A)) ⊆ RI ( A) . (2) RI ( RI ( A)) ⊇ RI ( A) . Theorem 13. Let L be a complete MTL algebra, and (U , R) be an L-fuzzy approximation space based on L. If R is a L-similar L-fuzzy binary relation, then for all A ∈ FL (U ) , (1) RI ( RI ( A)) = RI ( A) . (2) RI ( RI ( A)) = RI ( A) . Theorem 14. Let L be a complete MTL algebra, and (U , R) be an L-fuzzy approximation space based on L. If R is a L-undistinguishing L-fuzzy binary relation, then for all A ∈ FL (U ) , (1) RI ( RI ( A)) = RI ( A) . (2) RI ( RI ( A)) = RI ( A) . C. II-Type L-fuzzy rough set based on IMTL algebra Observe the form the II-type approximation operator. The form only include three operators: ⊗ , ⊕ and ∼ . In theorem 3, we have confirmed that ⊗ and ⊕ are dual, so it is easy to proof the duality of RII and RII . In theorem 4, we have to draw a conclusion “If ⊗ and ⊕ are distributive, the IMTL algebra must be a Boolean algebra.” with some sorrows. The distributivity of ⊗ and ⊕ embarrass us to confirm some properties, which are very useful for II-Type L-fuzzy rough set’s axiomatic approach. Those important properties, which are very useful for II-Type L-fuzzy rough set’s axiomatic approach, include: - RII (aˆ → A) = aˆ → RII ( A) , - RII (aˆ ⊗ A) = aˆ ⊗ RII ( A) . - RII ( A ∩ B ) = RII ( A) ∩ RII ( B) , - RII ( A ∪ B) = RII ( A) ∪ RII ( B) . In the theorem 15, we will proof those conclusions in detail. Theorem 15. Suppose L is a complete IMTL algebra, and (U , R) is an L-fuzzy approximation space based on L. It holds for all A, B ∈ FL (U ) , a ∈ L , (1) RII (U ) = U , RII (∅) = ∅ . (2) If A ⊆ B , then RII ( A) ⊆ RII ( B ) , RII ( A) ⊆ RII ( B ) . (3) RII ( A) = ∼ RII (∼ A) , RII ( A) ⊆ ∼ RII (∼ A) . (4) RII ( A ∩ B ) ⊆ RII ( A) ∩ RII ( B ) ,
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RII ( A ∪ B) ⊇ RII ( A) ∪ RII ( B) . (4) RII ( A ∪ B ) ⊇ RII ( A) ∪ RII ( B ) , RL ( A ∩ B ) ⊆ RL ( A) ∩ RL ( B ) . Proof. (1) For every x ∈ U , RII (∅)( x) = ⊕ ( R ( x, y ) ⊗ 0 ) = 0 , y∈U
RII (U )( x) = ⊗ ( ∼ R ( x, y ) ⊕ 1) ≥ ⊗ ( ∼ R ( x, y ) ∨ 1) = 1 ; y∈U
y∈U
Thus, RII (∅) = ∅ , RII (U ) = U . (2) is easy to proof. (3) For every x ∈ U , A ∈ FL (U ) ,
∼ RII (∼ A)( x) = ∼ ⊕ ( R ( x, y )⊗ ∼ A( y ) ) y∈U
= ⊗ ( ∼ ( R ( x, y )⊗ ∼ A( y ) ) ) y∈U
= ⊗ ( ∼ R ( x, y )⊕ ∼∼ A( y ) ) y∈U
= ⊗ ( R ( x, y ) → A( y ) ) = RII ( A)( x) y∈U
∼ RII (∼ A)( x) = ∼ ⊗ ( R ( x, y ) →∼ A( y ) ) y∈U
= ⊕ ( ∼ ( ∼ R( x, y )⊕ ∼ A( y ) ) ) y∈U
= ⊕ ( ∼∼ R ( x, y )⊗ ∼∼ A( y ) ) y∈U
= ⊕ ( R ( x, y ) ⊗ A( y ) ) = RII ( A)( x) y∈U
Thus ∼ RII (∼ A) = RII ( A) , ∼ RII (∼ A) = RII ( A) . (4) and (5) are easy to proof by (2). □ Theorem 16. Let L be a complete IMTL algebra, and (U , R) be an L-fuzzy approximation space based on L. If R is a serial L-fuzzy binary relation, then for all A ∈ FL (U ) , RII ( A) ⊆ RII ( A) . Proof. By theorem 1, for every a, b ∈ L , a ∧ b ≥ a ⊗ b , then a ∨ b ≤ a ⊕ b . For every x ∈ U , A ∈ FL (U ) , there exist y0 ∈ U , such that R ( x, y0 ) = 1 . RII ( A)( x) = ⊗ ( R ( x, y ) → A( y ) ) y∈U
≤ inf ( R( x, y ) → A( y ) ) y∈U
≤ sup ( R ( x, y ) ⊗ A( y ) ) y∈U
:
≤ ⊕ ( R ( x, y ) ⊗ A( y ) ) = RII ( A)( x) y∈U
Thus, RII ( A) ⊆ RII ( A) . □ Theorem 17. Let L be a complete IMTL algebra, and (U , R) be an L-fuzzy approximation space based on L. If R is a reflexive L-fuzzy binary relation, then for all A ∈ FL (U ) , (1) RII ( A) ⊆ A ⊆ RII ( A) . (2) RII ( A → B) ⊆ A → B ⊆ RII ( A) → B . (3) RII ( A → B) ⊇ A → B ⊇ RII ( A) → B . Proof. (i) For every x ∈ U , A ∈ FL (U ) . RII ( A)( x) = ⊕ ( R( x, y ) ⊗ A( y ) ) y∈U
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≥ sup ( R ( x, y ) ⊗ A( y ) ) ≥ R ( x, x ) ⊗ A( x)
TABLE I. DEFINITION OF “ ⊗ ”
y∈U
= A( x) . RII ( A)( x) = ⊗ ( R ( x, y ) → A( y ) ) y∈U
≤ inf ( R ( x, y ) → A( y ) ) ≤ R ( x, x) → A( x) y∈U
= A( x) .
Thus RII ( A) ⊆ A ⊆ RII ( A) . (ii) and (iii) are easy to proof by (i).
□
Theorem 18. Let L be a complete IMTL algebra, and (U , R) be an L-fuzzy approximation space based on L. If R is a symmetric L-fuzzy binary relation, then for all A ∈ FL (U ) , (1) RII (1x )( y ) = RII (1y )( x) .
(2) RII (1U /{ y} )( x) = RII (1U /{ x} )( y ) . Proof. For every x, y ∈ U , RII (1x )( y ) = ⊕ ( R ( y, z ) ⊗ 1x ( z ) ) = R( y, x ) ; z∈U
RII (1y )( x) = ⊕ ( R ( x, z ) ⊗ 1y ( z ) ) = R ( x, y ) : z∈U
RII (1U /{ y} )( x) = ⊗ ( R( x, z ) → 1U /{ y} ( z ) ) = ∼ R ( x, y ) ; z∈U
RII (1U /{ x} )( y ) = ⊗ ( R( y, z ) → 1U /{ x} ( z ) ) = ∼ R( y, x) . z∈U
Because of the symmetric of R, RII (1x )( y ) = RII (1y )( x) , RII (1U /{ y} )( x) = RII (1U /{ x} )( y ) .
□
Generally, RII (aˆ → A) = aˆ → RII ( A) ,
RII (aˆ ⊗ A) = aˆ ⊗ RII ( A) . Such as the following example Example 1. Let L6 = ( L, ∨ , ∧, ⊗, →, 0,1) be is a lattice implication algebra [19]. We define ⊗ and ⊕ on Lattice implication lattice by Table I and II In the L6 , (b ⊗ a) ⊕ (b ⊗ a) = d ⊕ d = a , b ⊗ (a ⊕ a) = b ⊗ a = d Let U = {a, d } , A = {a / x,1 / y} , R ( x, y ) = a , then
R ( x, x ) = 1 ,
RII (bˆ ⊗ A)( x ) = ( R ( x, x ) ⊗ b ⊗ A( x ) ) ⊕ ( R ( x, y ) ⊗ b ⊗ A( y ) )
= (1 ⊗ b ⊗ a ) ⊕ ( a ⊗ b ⊗ 1) = a . bˆ ⊗ RII ( A)( x) = b ⊗ ( ( R ( x, x) ⊗ A( x) ) ⊕ ( R( x, y ) ⊗ A( y ) ) )
= b ⊗ ( (1 ⊗ a ) ⊕ ( a ⊗ 1) ) = d .
At this time, RI (bˆ ⊗ A) ≠ bˆ ⊗ RI ( A) .
⊗
0
a
b
c
d
1
0 a b c d 1
0 0 0 0 0 0
0 a d 0 d a
0 d c c 0 b
0 0 c c 0 c
0 d 0 0 0 d
0 a b c d 1
TABLE II. DEFINITION OF “ ⊗ ”
⊕
0
a
b
c
d
1
0 a b c d 1
0 a b c d 1
a a 1 1 a 1
b 1 1 b 1 1
c 1 b c b 1
d a 1 b a 1
1 1 1 1 1 1
About the fuzzy rough approximation operator, two ways have been discussed in [5] and [14,20]. The difference of two ways exists in the beginning--the axioms, which one chooses R( A) =∼ R (∼ A) , R (∅) = ∅ , R ( A ∪ B) = R ( A) ∪ R ( B) , but the other one chooses R (∅) = ∅ , R ( A ∪ B) = R ( A) ∪ R ( B) , R (U ) = U , R ( A ∩ B ) = R ( A) ∩ R ( B ) . In [14,20], the upper approximation operator and the lower approximation operator have some relations, such as “their are dual”. At this time, the axiomatic space equals to a topology space and the upper and lower approximation operators are corresponding to the interior and closure operators respectively. In the other way[5], the upper approximation operator and the lower approximation operator are defined independently. Essentially, the two ways are similar. Firstly, choose some properties of the approximation operators as the axioms in an universe. Secondly, proof the existence of the binary relation in the universe with a certain form of approximation operator. The form has been set before the second step. Theorem 19. Suppose L is a complete residuated lattice, {∼ a | a ∈ L} = L , K,H:FL (U ) → FL (U ) , and K is
□
the semi-dual L-fuzzy operator of H(for all A ∈ FL (U ) , K (~ A) =~ H ( A) . There exists an L-fuzzy binary
V. AXIOMATIC APPROACH OF THE L-FUZZY ROUGH SET
relation R to satisfy K ( A) = RL ( A) and H ( A) = RL ( A) iff the K and H satisfy: for all A, B ∈ FL (U ) , a ∈ L , (1) αˆ ⊗ H ( A) = H (αˆ ⊗ A) . (2) H ( A ∪ B ) = H ( A) ∪ H ( B) .
In the axiomatic approach, we usually choose two unary operators in the universe. Through adding the axiom set to the two unary operators, they induce the relation and the approximation space. The axioms are the properties of which the approximation operators satisfy. © 2011 ACADEMY PUBLISHER
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(3) K ( A ∩ B ) = K ( A) ∩ K ( B) Proof. “ ⇒ ” It is directly by Theorems 5. “ ⇐ ” Let R ( x, y ) = H (1y )( x) . For all A ∈ FL (U ) , x ∈U . RL ( A)( x) = sup( R( x, y ) ⊗ A( y )) y∈U
= sup( H (1y )( x) ⊗ A( y )) y∈U
By Theorem 1-(vii), RL ( A) = ∪ ( H (1y ) ⊗ A( y )) y∈U
= ∪ ( H (1y ⊗ A( y ))) y∈U
semi-dual L-fuzzy operator of H(for all A ∈ FL (U ) , K (~ A) =~ H ( A) . There exists an L-fuzzy binary relation R to satisfy K ( A) = RL ( A) and H ( A) = RL ( A) iff the K and H satisfy: for all A, B ∈ FL (U ) , a ∈ L , (1) αˆ ⊗ H ( A) = H (αˆ ⊗ A) . (2) H ( A ∪ B ) = H ( A) ∪ H ( B) . With the duality of the ∨ and ∧ (, which have been proved in Theorem 2), the conditions in theorem 19 can be reduced to – L is a complete residuated lattice and {∼ a | a ∈ L} = L . - RL ( A) = ∪ ( H (1y ) ⊗ A( y )) .
= H ⎛⎜ ∪ (1y ⊗ A( y )) ⎞⎟ = H ( A ) . ⎝ y∈U ⎠ Thus for all A ∈ FL (U ) , RL ( A) = H ( A ) .
y∈U
- K (~ A) = ~ H ( A) - αˆ ⊗ H ( A) = H (αˆ ⊗ A) , - H ( A ∪ B ) = H ( A) ∪ H ( B ) . The reason is the following equations can be proposed. K (~ A∩ ~ B) = K (~ ( A ∪ B)) =~ H ( A ∪ B)
Because the K is the semi-dual operator of H, the following equations can be proposed. ∼ R ( x, y ) =∼ H (1y )( x) = K (0U /{ y} )( x)
αˆ → K ( A) = αˆ →~ H ( A) =~ (αˆ ⊗ H ( A) )
=~ ( H ( A) ∪ H ( B ) ) =~ H ( A)∩ ~ H ( B )
=~ H (αˆ ⊗ A ) = K ( ~ (αˆ ⊗ A ) ) = K (αˆ →~ A ) .
= K (~ A) ∩ K (~ B ) .
By Theorem 1-(vii), we have RL (∼ A) = ∩ ( A( y ) → K (0U /{ y} )) y∈U
= ∩ ( K ( A( y ) → 0U /{ y} )) y∈U
= K ⎛⎜ ∩ ( A( y ) → 0U /{ y} ) ⎞⎟ ⎝ y∈U ⎠ = K ( ∼ A)
By the assumption
{∼ a | a ∈ L} = L
, for every
B ∈ FL (U ) , there exists ~ A such that B =~ A . Thus for
all A ∈ FL (U ) , K ( A) = RL ( A) . There exists six conditions in Theorem 19. - L is a complete residuated lattice {∼ a | a ∈ L} = L .
□ and
Theorem 21. Suppose L is a complete IMTL algebra, K,H:FL (U ) → FL (U ) , and K and H are the dual L-fuzzy operators. There exists an L-fuzzy binary relation R to satisfy K ( A) = RL ( A) and H ( A) = RL ( A) iff the K and H satisfy: for all A, B ∈ FL (U ) , a ∈ L , (1) αˆ ⊕ K ( A) = K (αˆ ⊕ A) . (2) K ( A ∩ B ) = K ( A) ∩ K ( B) . In Theorem 21, “ αˆ ⊕ K ( A) = K (αˆ ⊕ A) ” replaces “ αˆ → K ( A) = K (αˆ → A) ” and “ K and H are dual” replaces “K is the semi-dual L-fuzzy operator of H”. With the duality of the upper and lower approximation operators, the conditions in theorem 20 can be reduced to - K ( A) =~ H (~ A) .
- RL ( A) = ∪ ( H (1y ) ⊗ A( y )) .
- RL ( A) = ∪ ( H (1y ) ⊗ A( y ))
y∈U
- K (~ A) =~ H ( A) - K ( A ∩ B ) = K ( A) ∩ K ( B) . - αˆ ⊗ H ( A) = H (αˆ ⊗ A) , □ - H ( A ∪ B ) = H ( A) ∪ H ( B ) . The first condition is limited by the basis algebra. The second condition is the form of the approximation operator. The third condition is the relation of the upper and lower approximation operators. Without the duality of ∨ and ∧ , the equation K ( A ∩ B ) = K ( A) ∩ K ( B) can’t be proved by H ( A ∪ B ) = H ( A) ∪ H ( B ) . The fifth and sixth conditions are independent. All of them are necessary in Theorem 19. Theorem 20. Suppose L is a complete MTL algebra, ∼ { a | a ∈ L} = L , K,H:FL (U ) → FL (U ) , and K is the
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y∈U
- αˆ ⊗ H ( A) = H (αˆ ⊗ A) - H ( A ∪ B ) = H ( A) ∪ H ( B ) . or - H ( A) =~ K (~ A) , - RL (∼ A) = ∩ ( A( y ) → K (0U /{ y} )) , y∈U
- αˆ → K ( A) = K (αˆ → A) , - K ( A ∩ B ) = K ( A) ∩ K ( B) . Theorem 19-Theorem 21 are the axiomatic representative theorem of the normal L-fuzzy rough approximation operators. If we use the condition αˆ ∩ H ( A) = H (αˆ ∩ A)
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to replace
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αˆ ⊗ H ( A) = H (αˆ ⊗ A) ,
then Theorem 20 will change to the axiomatic representative theorem of the I-type L-fuzzy rough approximation operators. Unfortunately, by Theorem 15, the II-type approximation operators can’t satisfy the or properties “ RII ( A ∪ B ) = RII ( A) ∪ RII ( B) ” “ RII ( A ∩ B ) ⊆ RII ( A) ∩ RII ( B ) ”. They are necessary condition for the proof of the axiomatic representative theorem of the II-type L-fuzzy rough approximation operators Above all, if we want to find a certain binary relation such that K ( A) = R ( A) and H ( A) = R ( A) , the following conditions are necessary. - αˆ ⊗ H ( A) = H (αˆ ⊗ A) - H ( A ∪ B ) = H ( A) ∪ H ( B ) . - αˆ → K ( A) = K (αˆ → A) , - K ( A ∩ B ) = K ( A) ∩ K ( B) We name above those as “the necessary conditions for the axiomatic approach”. The function of the condition “K are H are dual” or “K is the semi-dual operator of H” is just to reduce the necessary conditions for the axiomatic approach. It is not essential for the axiomatic approach. Without those necessary conditions, we can also define the approximation operators with some axioms, but we can not find the certain binary relation and prove R(A) = H(A) and R(A) = K(A). So in this section, we have not discussed the I-type L-fuzzy rough approximation operator based on residuated lattice and II-type L-fuzzy rough approximation operator. VI. CONCLUSION In this paper, we discuss four elements of the approximation operators. They are the property and form of approximation operator, the basis algebra and binary relation. The four element of the approximation operator are been organized in constructive and axiomatic approaches as the following Figure 1. The Figure 1 shows some meanings about the relation of those four elements. If we will suppose the approximation operator to satisfy some properties such as the necessary conditions for the axiomatic approach, we should consider the limit of the basis algebra, the form of the approximation operator. For example, it is unable to hypothesize that the L-fuzzy rough approximation operator satisfies RL ( A) = ∼ RL (∼ A) with a residuated lattice, but we can set the same properties for a L-fuzzy rough approximation operator based on IMTL-algebra or a Itype L-fuzzy rough approximation operator based on MTLalgebra. As for the reason–why not can we hypothesize RL ( A) =∼ RL (∼ A) for the approximation operator based on residuated lattice? it should be discussed in the other approach, in which our discussion will start with the property and form of approximation operator, binary relation.
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Figure 1. The relations Between Property of Approxiomation Operator, Binary relation, the Form of Approximation Operator and Basis Algebra
ACKNOWLEDGMENT This work was supported by the National Natural Science Foundation of China (Grant No. 60873108), Henan Province Key Scientific and Technological Project (082102210079). REFERENCES [1] Z. Pawlak: Rough sets, International Journal of Computer and Information Science, 1982, 341-356. [2] L.A. Zadeh, Fuzzy sets, Information Control. 8, 1965, 338353. [3] D. Dubois, H. Prade: Rough fuzzy sets and fuzzy rough sets, Internat. J. General Systems 17(2-3), 1990, 191-209. [4] D. Dubois, H. Prade: Putting fuzzy sets and rough sets together, In Intelligent Decision Support, (Edited by R. Slowinski), Kluwer Academic, Dordrecht, 1992, 203-232. [5] N. N. Morsi, M. M. Yakout: Axiomatics for fuzzy rough sets, Fuzzy Sets and Systems 100, 1998, 327-342. [6] A. M. Radzikowska, E. E. Kerre: A comparative study of fuzzy rough sets, Fuzzy Sets and System, 126, 2002, 137155. [7] A. M. Radzikowska, E. E. Kerre: Transactions on Rough Sets II J.F. Peters et al. Eds. LNCS 3135. 2004, 278C296. [8] Zh. J. Wu, K.Y. Qin: L-fuzzy Close-Topological and Lfuzzy Approximation Space. Intelligent Decision Making Systems(Proceedings of the 4th International ISKE Conference), (Koen Vanhoof, Da Ruan, Tianrui Li, Geert Wets editors) Hasselt, Belgium, 27-28 November 2009. 619-624. [9] Zh. J. Wu, W. F. Du, K. Y. Qin: The properties of L-fuzzy rough set based on complete residuated lattice. 2008 International Symposium on Information Science and Engieering, Shanghai, China, 2008, 617-621. [10] Zh. J. Wu, L. X. Yang, T. R. Li, K. Y. Qin: The basis algebra in L-fuzzy rough sets. 2009 International Conference on Rough Set and Knowledge Technology, The Gold Coast, Austrlia, 2009, 320-325. [11] Zh. J. Wu, X.X. He: A general L-fuzzy rough set based on IMTL-algebra. Computational Intelligence, the 9th FLINS Conference, E’mei, China, 2010, 111-115. [12] Y.H.She, G.J. Wang: An axiomatic approach of fuzzy rough sets based on residuated lattices. Computers and Mathematics with Applications 58 2009,189-201. [13] W. Z.Wu, J. S. Mi,W. X. Zhang: Generalized fuzzy rough sets, Information Sciences 151, 2003, 263-282. [14] W. Z. Wu, W. X. Zhang: Constructive and axiomatic approaches of fuzzy approximation operators, Information Sciences, 159, 2004, 233-254. [15] J. Pavelka: On fuzzy logic I: Many-valued rules of inference, II: Enriched residuated lattices and semantics of propositional calculi, III: Semantical completeness of some many-valued propositional calculi. Zeitschr. F. Math. Logik und Grundlagend. Math., 25, 1979, 45-52, 119-134, 447-464.
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[16] F. Esteva, L. Godo: Monoidal t-norm-based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems 124, 2001, 271-288. [17] D. Pei: On equivalent forms of fuzzy logic systems NM and IMTL, Fuzzy Sets and Systems, 138, 2003, 187-195. [18] C.C. Chang : Algebraic analysis of many logic. Trans Amer Math Soc, 88 1958, 467-490. [19] Y. Xu, D. Ruan, K. Qin, J. Liu: Lattice-valued logic. Springs-Verlag (2003) 28-57. [20] Y. Y. Yao: Constructive and algebraic methods of the theory of rough sets, Information Sciences, 109 1998, 2147.
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Zhengjiang Wu was born in 1981 in Linfen, Shanxi Provence, China. He received the M.S. degree and the Ph.D. degree from Southwest Jiaotong University, Chengdu, China, in 2005 and 2009, respectively. He is working as a part-time Postdoctoral Fellow with Southwest Jiaotong University from 2009 to now. Since 2009, he has been a teacher at Henan Polytechnic University, Jiaozuo, Henan, China. His research interests include Fuzzy Rough Set Theory, Granular Computer, Data Mining. Wenpeng Xu, graduated from Zhejiang University and received the Master degree in Mechanic Engineer in 2004, Hangzhou, China. Currently he is an associate professor in College of Computer and Technology, Henan Polytechnic University, Jiaozuo, Henan, China. His research interests include CAD, Computer Graphics.
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