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Intuitionistic L-fuzzy β-covering Rough Set

Intuitionistic L-fuzzy β-covering Rough Set FUZZY INFORMATION AND ENGINEERING 2020, VOL. 12, NO. 1, 19–37 https://doi.org/10.1080/16168658.2020.1778829 Peng Yu, Xiao-gang An and Xiao-hong Zhang School of arts and Sciences, Shaanxi University of Science and Technology, Xi’an, People’s Republic of China ABSTRACT ARTICLE HISTORY Received 7 April 2020 By introducing the concepts of intuitionistic L-fuzzy β-covering and Revised 15 May 2020 intuitionistic L-fuzzy β-neighborhood, we define three kinds of intu- Accepted 1 June 2020 itionistic L-fuzzy β-covering rough set models. The basic properties of those intuitionistic L-fuzzy β-covering rough set models are inves- KEYWORDS tigated. Moreover, we define the other three kinds of intuitionistic L- Intuitionistic L-fuzzy set; fuzzy β-covering rough set models by using the former three models. covering rough set; rough Finally, we present the matrix representations of the newly defined set; Alexandrov topolgy lower and upper approximation operators so that the calculation of lower and upper approximations of subsets can be converted into operations on matrices. 1. Introduction Rough set (RS) theory was firstly introduced by Pawlak [1]. It may be seen as an extension of set theory and has been found to be a new mathematical tool to deal with insufficient and incomplete information systems. In Pawlak’s rough set model, the equivalence relation is an important concept that is used to construct the lower and upper approximations of an arbitrary subset of the universe of discourse. However, the condition of the equivalence relation is highly restrictive and may limit the application of rough sets in many practical problems. Hence, numerous extensions of Pawlak’s rough set were proposed by replacing the equivalence relation with a few mathematical concepts that are more general in nature, for example, arbitrary binary relations [2–4], neighborhood systems, covering-based and Boolean algebras [5–8]. Covering rough set theory is an important generalization of the classical rough set theory. It was firstly introduced by Zakowski [9]. After then, many kinds of different cov- ering rough sets are introduced [10–13]. Meanwhile, fuzzy covering rough set and fuzzy β-covering rough set are important generalizations of the classical covering rough set theory [14]. The concept of an intuitionistic fuzzy (IF for short) set, initiated by Atanassov [15–17], is another important tool for dealing with imperfect and imprecise information. Compared with Zadeh’s fuzzy set, an IF set is more objective than a fuzzy set to describe the vagueness CONTACT Peng Yu yupeng@sust.edu.cn © 2020 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society of China & Operations Research Society of Guangdong Province. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 20 P. YU ET AL. of data, because IF set gives both a membership and a non-membership degree of which an element belongs to a set [18–21]. As an important generalization of IF set, Atanassov and Stoeva defined intuitionistic L-fuzzy (ILF for short ) set in 1984, which actually is an IF set based on residuated lattice L. ILF set and covering rough set are the important generalizations of the classical fuzzy set and rough set, but related work is rarely done to combine the ILF set theory with the covering rough set theory. Motivated by this, in this paper, by introducing the concepts of intuitionistic L-fuzzy β-covering and intuitionistic L-fuzzy β-neighborhood, we define sev- eral differentiable rough set models, obtain some basic properties of those models, and present the matrix representations of the newly defined lower and upper approximation operators. 2. Preliminaries In this section, we introduce some basic concepts of residuated lattice and intuitionistic L-fuzzy set used in this paper. We refer to [22–26] for residuated lattice theory and to [18] for intuitionistic L-fuzzy set. 2.1. Complete Regular Residuated Lattice Definition 2.1 ([22]): A residuated lattice is an algebraic structure L = (L; ∧, ∨, ∗, →,0,1) of type (2, 2, 2, 2, 0, 0) such that (i) (L; ∧, ∨,0,1) is a bounded lattice with the least element 0 and the greatest element 1; (ii) (L; ∗,1) is a commutative monoid with the identity 1; (iii) L satisfies the adjointness property, i.e. ∀ x, y, z ∈ L, x ∗ y ≤ z iff x ≤ y → z. A residuated lattice is said to be complete if the underlying lattice is complete. Definition 2.2 ([26]): The negator on L is the mapping ¬ : L → L defined by ¬a = a → 0 for every a ∈ L.If ¬¬a = a for all a ∈ L, then L is called a regular residuated lattice. In this paper, if there is no further statement, L always denotes a complete regular residuated lattice. Theorem 2.1 ([18]): Let L be a complete regular residuated lattice. Then (R1) a ∗ b ≤ a ∧ b; (R2) a = 1 → a; (R3) a ≤ b ⇔ a → b = 1; (R4) a → (b → c) = (a ∗ b) → c; (R5) a → b ≥ b; (R6) a ∗ ( b ) = (a ∗ b ); i i i i (R7) a → ( b ) = (a → b ); i i i i FUZZY INFORMATION AND ENGINEERING 21 (R8)( b ) → a = (b → a); i i i i (R9) ¬( a ) = ¬a , ¬( a ) = ¬a . i i i i i∈ i∈ i∈ i∈ 2.2. Intuitionistic L-fuzzy Set In this subsection, we introduce the notion of ILF set that can be regarded as the general- ization of IF set induced by Atanssov [1, 2]. At first, we introduce a special lattice L.Thiscan be seen as a generalization of the set {(x, y) ∈ [0, 1] × [0, 1]|x + y ≤ 1}. Definition 2.3 ([18]): L ={(x , x )|x , x ∈ L, x ≤¬x }. The partial ordering ≤ is defined 1 2 1 2 1 2 ˜ as follows: x ≤ y ⇔ x ≤ y , x ≥ y , ∀x = (x , x ), y = (y , y ) ∈ L. 1 1 2 2 1 2 1 2 The pair (L, ≤ ) is a complete lattice, with the smallest element 0 = (0, 1) and the greatest ˜ ˜ L L element 1 = (1, 0). The meet operator ∧ and the join operaton ∨ on L which are connected to the partial ordering ≤ are defined as follows: x ∧ y = (x ∧ y , x ∨ y ), x ∨ y = (x ∨ y , x ∧ y ). ˜ 1 1 2 2 ˜ 1 1 2 2 L L Definition 2.4 ([18]): Let U be a non-empty universe of discourse. An ILF set A in U is an object having the form A ={< x, μ (x), ν (x)> |x ∈ U}, A A where μ : U → L and ν : U → L satisfy μ (x) ≤¬ν (x) for all x ∈ U. A A A A They are called the degree of L membership and the degree of L non-membership of the element x to A, respectively. The family of all ILF subsets in U is denoted by ILF(U). We denote A(x) = (μ (x), ν (x)), then A ∈ ILF(U) if and only if A ∈ L for all x ∈ U. (α, β) A A denotes the constant ILF sets for all (α, β) ∈ L.If U is a non-empty finite set, then ILF set A can be denoted by a matrix A = ((μ (x ), ν (x )), ... , (μ (x ), ν (x ))). A 1 A 1 A n A n For any A, B, A (i ∈ ) ∈ ILF(U), some operations are introduced as follows: (1) A ⊆ B iff A(x) ≤ B(x). (2) A = B iff A ⊆ B and B ⊆ A. (3) A B ={< x, μ (x) ∧ μ (x), ν (x) ∨ ν (x)> |x ∈ U} means (A B)(x) = A(x) ∧ A B A B ˜ B(x). (4) A B ={< x, μ (x) ∨ μ (x), ν (x) ∧ ν (x)> |x ∈ U} means (A B)(x) = A(x) ∨ A B A B ˜ B(x). (5) A ={< x, μ (x), ν (x)> |x ∈ U} means ( A )(x) = ( ) i A A i ˜ i∈ i∈ i i∈ i i∈ i∈ μ (x). (6) A ={< x, μ (x), ν (x)> |x ∈ U} means ( A )(x) = ( ) i A A i ˜ i∈ i∈ i i∈ i i∈ i∈ μ (x). (7) ∼ A ={< x, ν (x), μ (x)> |x ∈ U}. A A 22 P. YU ET AL. 3. Intuitionistic L-fuzzy Rough Set 3.1. Intuitionistic L-fuzzy Covering In this subsection, we introduce the concepts of intuitionistic L-fuzzy β-covering and intu- itionistic L-fuzzy β-neighborhood of a non-empty universe of discourse U, and investigate the basic properties of it. Definition 3.1: Let U be an arbitrary universal set, and L be the intuitionistic L-fuzzy power set of U.For each β ∈ L,wecall C ={C , C , ... , C }, with C ∈ ILF(U)(i = 1, 2, ... , m),an 1 2 m i intuitionistic L-fuzzy β-covering of U,iffor each x ∈ U, C ∈ C exists, such that C (x) ≥ β.We i i also call (U, C) an intuitionistic L-fuzzy β-covering approximation space. Definition 3.2: Let C ={C , C , ... , C } be an intuitionistic L-fuzzy β-covering of U for 1 2 m some β ∈ L.For each x ∈ U, we define the intuitionistic L-fuzzy β-neighborhood N of x as N =∩{C ∈ C|C (x) ≥ β}. x i i Proposition 3.1: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. C ={C , C , ... , C } being a fuzzy β-covering of U for some β ∈ L, N ={y ∈ U|N (y) ≥ β}, 1 2 m then (i) x ∈ N ; β β (ii) N = N . x y y∈N Proof: (i) For all x ∈ U, N (x) =∩{C ∈ C|C (x) ≥ β}(x) = C (x) ≥ β, hence x ∈ N . x i i i ˜ C (x)≥ β x i L β β β β β (ii) On the one hand, since x ∈ N ,wehave N ⊆ N ( β N ) = β N . x x x y y y=x,y∈N y∈N x x β β β On the other hand, let y ∈ N , then N (y) ≥ β,wehave N ⊆{C |C (y) ≥ β}. Therefore, x x i i β β β β β β N ⊇ N .Let z ∈ N , then N (z) ≥ β,wehave N (z) ≥ β and z ∈ N .Furthermore,wehave x y y x y x β β β β N ⊆ N , N ⊆ N . y x y x y∈N Proposition 3.2: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. If β β 2 1 β ≤ β , then N ≥ N . 1 ˜ 2 x x Proof: Since β ≤ β , then for every c ∈{c |c (x) ≥ β }, c (x) ≥ β ≥ β , this means c ∈ 1 2 j i i 2 j 2 1 j {c |c (x) ≥ β }. Hence, {c |c (x) ≥ β }⊆{c |c (x) ≥ β }; furthermore, N =∩{c |c (x) ≥ i i 1 i i 2 i i 1 x i i β }⊆∩{c |c (x) ≥ β }= N . 2 i i 1 x Proposition 3.3: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. If y ∈ β β β N and z ∈ N , then z ∈ N . x y x FUZZY INFORMATION AND ENGINEERING 23 β β β β β Proof: Assume that y ∈ N and z ∈ N , then N (y) ≥ β and N (z) ≥ β. This means N ∈ x y x x y β β β β {c |c (y) ≥ β} and N ≥ N =∩{c |c (y) ≥ β}. Furthermore, we have N (z) ≥ N (z) ≥ β i i x y i i x y and z ∈ N . Example 3.1: Let L ={0, a, b, c, d, e,1}, → and ⊗ be defined as follows: → 0 ab c d e 1 ⊗ 0 ab c d e 1 0 1 1 11 111 0 0000 00 0 a e 1 1 1 111 a 0000 00 a b be 1e e 1 1 b 000 aa ab c ce e 1 e 11 c 00 a 0 aa c d de e e 111 d 00 aa 0 ad e aee e e 11 e 00 aa aa e 1 0 ab c d e 1 1 0 ab c d e 1 The partial order ≤ on L is defined by Figure 1 Figure 1. Partial order set L. 24 P. YU ET AL. It is easy to check that L is a complete regular residuated lattice, and L ={(0, 0), (0, a), (0, b), (0, c), (0, d), (0, e), (0, 1), (a,0), (a, a), (a, b), (a, c), (a, d), (a, e), (b,0), (b, a), (b, b), (c,0), (c, a), (c, c), (d,0), (d, a), (d, d), (e,0), (e, a), (1, 0)}. Let U ={x , x , x , x }, β = (a, e) ∈ L, 1 2 3 4 (a, b) (0, e) (d, a) (0, b) C = + + + , x x x x 1 2 3 4 (b,0) (b, a) (a, b) (0, e) C = + + + , x x x x 1 2 3 4 (0, c) (a, c) (a, d) (b, b) C = + + + , x x x x 1 2 3 4 then {C , C , C } is a fuzzy β-covering of U. 1 2 3 (a, b) (0, e) (a, b) (0, e) N = C ∩ C = + + + , 1 2 x x x x 1 2 3 4 (0, c) (a, c) (a, e) (0, e) N = C ∩ C = + + + , x 2 3 x x x x 1 2 3 4 (0, e) (0, e) (a, e) (0, e) N = C ∩ C ∩ C = + + + , 1 2 3 x x x x 1 2 3 4 (0, c) (a, c) (a, d) (b, b) N = C = + + + . x 3 x x x x 1 2 3 4 β β β β Furthermore, we have N ={x , x }, N ={x , x }, N ={x }, N ={x , x , x }. 1 3 2 3 3 2 3 4 x x x x 1 2 3 4 3.2. Intuitionistic L-fuzzy Covering Rough Set In this subsection, we will define three kinds of basic intuitionistic L-fuzzy β-covering rough set models and investigate the properties of it. At first, we define the I-type L-fuzzy β-covering rough set model by using the L-fuzzy β-covering of U. Definition 3.3: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space with C ={C , C , ... , C } being a fuzzy β-covering of U for some β ∈ L.Foreach A ∈ ILF(U),we 1 2 m β β define the I-type lower approximation R (A) and the I-type upper approximation R (A) I I of x as R (A)(x) =∧ A(y), y∈N R (A)(x) =∨ β A(y). y∈N In fact, the I-type L-fuzzy β-covering model above is a generalization of the classical fuzzy rough set model, which simply uses N induced by intuitionistic L-fuzzy β-neighborhood N to replace the equivalence class in the fuzzy rough set model. Therefore, it is one of the most basic covering rough set models. FUZZY INFORMATION AND ENGINEERING 25 Definition 3.4: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space with C ={C , C , ... , C } being a fuzzy β-covering of U for some β ∈ L.Foreach A ∈ ILF(U),we 1 2 m β β define the II-type lower approximation R (A) and the II-type upper approximation R (A) II II of x as β β R (A)(x) =∧ β (¬N (y) ∨ A(y)), II y∈N β β R (A)(x) =∨ β (N (y) ∧ A(y)). II y∈N Definition 3.5: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space with C ={C , C , ... , C } being a fuzzy β-covering of U for some β ∈ L.Foreach A ∈ ILF(U),we 1 2 m β β define the III-type lower approximation R (A) and the III-type upper approximation R (A) III III of x as β β R (A)(x) =∧ (¬N (y) ∨ A(y)), y∈U x III β β R (A)(x) =∨ (N (y) ∧ A(y)). y∈U x III In the definition of III-type lower (upper) approximation operator, if we consider N (y) as an intuitionistic L-fuzzy binary relation on U, ¬N (y) ∨ A(y) is understood as a classical fuzzy implication operator I(a, b) = (1 − a) ∨ b, then III-type lower (upper) approximation opera- tor is a generalization of lower (upper) approximation operator of definition 4.4 in literature [23]. In fact, if we take arbitrary fuzzy implication I based on ILF sets instead of classic fuzzy implication I, then we can define some other types of intuitionistic L-fuzzy covering rough set. Example 3.2: In Example 3.1, let A = (0, d)/x + (a, c)/x + (b, a)/x + (e, a)/x , 1 2 3 4 β = (a, b), then C ={C , C , C } being a fuzzy β-covering of U,and 1 2 3 (a, b) (0, e) (a, b) (0, e) β β N = C ∩ C = + + + , N ={x , x }. x 1 2 1 3 1 1 x x x x 1 2 3 4 (b,0) (b, a) (a, b) (0, e) N = C = + + + , N ={x , x , x }. x 2 1 2 3 2 x x x x x 1 2 3 4 (a, b) (0, e) (a, b) (0, e) N = C ∩ C = + + + , N ={x , x }. x 1 2 1 3 3 x x x x x 1 2 3 4 (0, c) (a, c) (a, d) (b, b) N = C = + + + , N ={x }. 3 4 4 x x x x x 1 2 3 4 Furthermore, we have R (A)(x ) = A(x ) ∧ A(x ) = (0, d) ∧ (b, a) = (0, d), 1 1 3 R (A)(x ) = A(x ) ∧ A(x ) ∧ A(x ) = (0, d) ∧ (a, c) ∧ (b, a) = (0, e), 2 1 2 3 R (A)(x ) = A(x ) ∧ A(x ) = (0, d) ∧ (b, a) = (0, d), 3 1 3 R (A)(x ) = A(x ) = (e, a). 4 4 I 26 P. YU ET AL. Hence, (0, d) (0, e) (0, d) (e, a) R (A) = + + + , x x x x 1 2 3 4 β β β R (A)(x ) = (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) 1 1 1 3 3 x x II 1 1 = (¬(a, b) ∨ (0, d)) ∧ (¬(a, b) ∨ (b, a)) = (b, a), β β β β R (A)(x ) = (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) 2 x 1 1 x 2 2 x 3 3 II 2 2 2 = (¬(b,0) ∨ (0, d)) ∧ (¬(b, a) ∨ (a, c)) ∧ (¬(a, b) ∨ (a, b)) = (0, a), β β β R (A)(x ) = (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) 3 1 1 3 3 x x II 3 3 = (¬(a, b) ∨ (0, d)) ∧ (¬(a, b) ∨ (b, a)) = (b, a), β β R (A)(x ) = (¬N (x ) ∨ A(x )) 4 x 4 4 II = (¬(b, b) ∨ (e, a)) = (b, b), (b, a) (0, a) (b, a) (e, a) R (A) = + + + . II x x x x 1 2 3 4 β β β β R (A)(x ) = (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) 1 x 1 1 x 2 2 x 3 3 III 1 1 1 ∧ (¬N (x ) ∨ A(x )) x 4 4 = (¬(a, b) ∨ (0, d)) ∧ (¬(0, e) ∨ (a, c)) ∧ (¬(a, b) ∨ (b, a)) ∧ (¬(0, e) ∨ (e, a)) = (b, a), β β β β R (A)(x ) = (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) 2 x 1 1 x 2 2 x 3 3 III 2 2 2 ∧ (¬N (x ) ∨ A(x )) 4 4 = (¬(b,0) ∨ (0, d)) ∧ (¬(b, a) ∨ (a, c)) ∧ (¬(a, b) ∨ (b, a)) ∧ (¬(0, e) ∨ (e, a)) = (0, a), β β R (A)(x ) = R (A)(x ) = (b, a), 3 1 III III β β β β R (A)(x ) = (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) 4 x 1 1 x 2 2 x 3 3 4 4 4 III ∧ (¬N (x ) ∨ A(x )) x 4 4 = (¬(0, c) ∨ (0, d)) ∧ (¬(a, c) ∨ (a, c)) ∧ (¬(a, d) ∨ (b, a)) ∧ (¬(b, b) ∨ (e, a)) = (c, a), (b, a) (0, a) (b, a) (c, a) R (A) = + + + . III x x x x 1 2 3 4 FUZZY INFORMATION AND ENGINEERING 27 Similarly, we have (b, a) (b, a) (b, a) (b, c) R (A) = + + + , x x x x 1 2 3 4 (a, b) (a, a) (a, b) (b, b) R (A) = + + + , II x x x x 1 2 3 4 (a, a) (a, a) (a, a) (b, a) R (A) = + + + . III x x x x 1 2 3 4 Remark 3.1: In Example 3.2, we can see that β β R (A)(x ) = (0, d) = (b, a) = R (A)(x ), 1 1 I II β β R (A)(x ) = (b, b) = (c, a) = R (A)(x ), 4 4 II III β β R (A)(x ) = (0, d) = (b, a) = R (A)(x ). 1 1 I III β β R (A)(x ) = (b, a) = (a, b) = R (A)(x ), 1 1 I II β β R (A)(x ) = (a, b) = (a, a) = R (A)(x ), 1 1 II III β β R (A)(x ) = (b, a) = (a, a) = R (A)(x ), 1 1 I III β β β we have R (A), R (A), R (A) as three kinds of different lower approximation operators and I II III β β β R (A), R (A), R (A) as three kinds of different lower approximation operators. But they have I II III the following relationships: Theorem 3.1: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space, then β β β β (i) R (A) ≤ R (A), R (A) ≤ R (A); I II II I β β β β (ii) R (A) ≤ R (A), R (A) ≤ R (A). III II II III Proof: (i) Since A(y) ≤∼ N (y) ∨ A(y), β β β R (A)(x) =∧ β A(y) ≤∧ β (∼ N (y) ∨ A(y)) = R (A)(x). I II y∈N y∈N x x β β Similarly, we have R (A) ≤ R (A). II I (ii) Since N ⊆ U,wehave β β β β R (A)(x) =∧ (∼ N (y) ∨ A(y)) ≤∧ (∼ N (y) ∨ A(y)) = R (A)(x). y∈U x x III II y∈N β β Similarly, we have R (A) ≤ R (A). II III 28 P. YU ET AL. Theorem 3.2: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. β β (i) if β = 1 , then R (A) = R (A); I II β β (ii) if β = 0 , then R (A) = R (A). II III Proof: β β (i) If β = 1 , then ∀y ∈ N ;wehave N (y) = 1 . Hence R (A)(x) = L L x x II β β ∧ (¬N (y) ∨ A(y)) =∧ (0 ∨ A(y)) =∧ A(y) = R (A)(x). β β β x L y∈N y∈N y∈N x x x β β β (ii) If β = 0 , then N = U;wehave R (A)(x) =∧ β (¬N (y) ∨ A(y)) =∧ (¬N (y) ∨ L x y∈U x x II y∈N A(y)) = R (A)(x). III Example 3.3: In Example 3.1, let (1, 0) (0, e) (d, a) (0, b) C = + + + , x x x x 1 2 3 4 (b,0) (1, 0) (1, 0) (0, e) C = + + + , x x x x 1 2 3 4 (0, c) (1, 0) (a, d) (1, 0) C = + + + , x x x x 1 2 3 4 (0, c) (1, 0) (a, e) (1, 0) C = + + + . x x x x 1 2 3 4 β = (1, 0) ∈ L, then C , C , C , C being an L-fuzzy β-covering of U. Suppose that A = 1 2 3 4 β β (e, a)/x + (b, a)/x + (a, c)/x + (0, e)/x ,wehave R (A)(x ) = (e, a) = (a, a) = R (A) 1 2 3 4 1 I III β β (x ).If β = (0, 1) ∈ L,wehave R (A)(x ) = (0, e) = (e, a) = R (A)(x ). 1 1 1 I III β β Theorem 3.3: For the operators R (A) and R (A), the following properties hold: I I β β (i) R (0 ) = 0 , R (0 ) = 0 ; ¯ ¯ ¯ ¯ L L L L I I β β (ii) R (1 ) = 1 , R (1 ) = 1 ; ¯ ¯ ¯ ¯ L L L L I I β β β β (iii) R (A) =∼ R (∼ A) and R (A) =∼ R (∼ A); I I I I β β β β β β (iv) R (A ∩ B) = R (A) ∩ R (B), R (A ∪ B) = R (A) ∪ R (B); I I I I I I β β β β β β (v) R (A ∪ B) ⊇ R (A) ∪ R (B), R (A ∩ B) ⊆ R (A) ∩ R (B); I I I I I I β β (vi) ∀A ∈ ILF(U), R (A) ⊆ A ⊆ R (A); I I β β β β (vii) If A ⊆ B, then R (A) ⊆ R (B), R (A) ⊆ R (B); I I I I β β β β β β (viii) R (R (A)) = R (A), R (R (A)) = R (A). I I I I I I Proof: Straightforward.  FUZZY INFORMATION AND ENGINEERING 29 β β Theorem 3.4: For the operators R (A) and R (A), the following properties hold: II II β β (i) R (1 ) = 1 , R (0 ) = 0 ; ¯ ¯ ¯ ¯ L L L L II II β β β β (ii) R (A) =∼ R (∼ A) and R (A) =∼ R (∼ A); II II II II β β β β β β (iii) R (A ∩ B) = R (A) ∩ R (B), R (A ∪ B) = R (A) ∪ R (B); II II II II II II β β β β β β (iv) R (A ∪ B) ⊇ R (A) ∪ R (B), R (A ∩ B) ⊆ R (A) ∩ R (B); II II II II II II β β β β (v) If A ⊆ B, then R (A) ⊆ R (B), R (A) ⊆ R (B); II II II II β β β (vi) R (A) ≤∼ β ∨ Aand R (A) ≥ β ∧ A. Particularly, when β = 1 , we have that R (A) ⊆ II II II A ⊆ R (A). II Proof: β β β β (i) R (1 ) =∧ (∼ N (y) ∨ (1, 0)) = 1 , R (0 ) =∨ (N (y) ∧ (0, 1)) = 0 . β β ¯ x ¯ ¯ x ¯ II L L II L L y∈N y∈N x x (ii) β β R (∼ A)(x) =∧ (∼ N (y)∨∼ A(y)) II y∈N =∧ (∼ (N (y) ∧ A(y)) y∈N =∼ ∨ (N (y) ∧ A(y)) y∈N =∼ R (A)(x). II β β Similarly, we have R (A) =∼ R (∼ A). II II (iii) For each x ∈ U, β β R (A ∩ B)(x) =∧ (∼ N (y) ∨ (A ∩ B)(y)) II y∈N β β =∧ ((∼ N (y) ∨ A(y)) ∧ (∼ N (y) ∨ B(y))) x x y∈N β β =∧ β (∼ N (y) ∨ A(y)) ∧ (∧ β (∼ N (y) ∨ B(y))) x x y∈N y∈N x x β β = R (A)(x) ∩ R (B)(x). II II β β β Similarly, we have R (A ∪ B) = R (A) ∪ R (B). II II II (iv) For each x ∈ U, β β R (A ∪ B)(x) =∧ (N (y) ∧ (A ∪ B)(y)) II y∈N β β =∧ ((N (y) ∧ A(y)) ∨ (N (y) ∧ B(y))) x x y∈N β β ≥∧ (N (y) ∧ A(y)) ∨ (∧ (N (y) ∧ B(y))) β β x x y∈N y∈N x x β β = R (A)(x) ∪ R (B)(x). II II 30 P. YU ET AL. β β β Similarly, we have R (A ∩ B) ⊆ R (A) ∩ R (B). II II II (v) Since A ⊆ B,wehave A(x) ≤ B(x) for every x ∈ U. Furthermore, β β R (A)(x) =∧ β (∼ N (y) ∨ A(y)) II y∈N ≤∧ β (∼ N (y) ∨ B(y)) y∈N = R (B)(x). II β β β (vi) For each x ∈ U, R (A)(x) =∧ β (∼ N (y) ∨ A(y)) ≤∼ N (x) ∨ A(x) ≤∼ β ∨ A(x), x x II y∈N β β then R (A) ≤∼ β ∨ A(x). Similarly, we have R (A) ≥ β ∧ A. Particularly, when β = 1 , II II L β β we have R (A) ⊆ A ⊆ R (A). II II β β β β β Remark 3.2: In Example 3.2, R (R (A))(x ) = (a, a) = (0, a) = R (A)(x ), R (R (A))(x ) = 2 2 2 II II II II II β β β β β β β (a, b) = (a, a) = R (A)(x ).Wehave R (R (A)) = R (A), R (R (A)) = R (A),thatistosay II II II II II II II β β the idempotent of lower and upper approximate operators R ,R does not hold. II II β β Theorem 3.5: For the III-type approximate operators R and R , the following properties III III hold: β β (i) R (0 ) = 0 , R (1 ) = 1 ; ¯ ¯ ¯ ¯ III L L III L L β β β β (ii) R (A) =∼ R (∼ A) and R (A) =∼ R (∼ A); III I III III β β β β β β (iii) R (A ∩ B) = R (A) ∩ R (B), R (A ∪ B) = R (A) ∪ R (B); III III III III III III β β β β β β (iv) R (A ∪ B) ⊇ R (A) ∪ R (B), R (A ∩ B) ⊆ R (A) ∩ R (B); III III III III III III β β β β (v) If A ⊆ B, then R (A) ⊆ R (B), R (A) ⊆ R (B). III III III III Definition 3.6: A subset τ ⊆ ILF(U) is called an ILF topology if it satisfies: (i) 1 ,0 ∈ τ; ¯ ¯ L L (ii) A, B ∈ τ implies A B ∈ τ ; (iii) {A |i ∈ }∈ τ implies A ∈ τ. i i i∈ Theorem 3.6: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. Then τ ={A ∈ ILF(U)|R (A) = A} is an ALexandrov topology. Proof: FUZZY INFORMATION AND ENGINEERING 31 (i) By (ii) of Theorem 3.3, we have R (1 ) = 1 ,1 ∈ τ . By (i) of Theorem 3.3, we have ¯ ¯ ¯ C I L L L β β R (0 ) ⊆ 0 , R (0 ) = 0 ,0 ∈ τ . ¯ ¯ ¯ ¯ ¯ I L L I L L L β β β (ii) Let A, B ∈ τ , by (iii) of Theorem 3.3, we have R (A ∩ B) = R (A) ∩ R (B) = A ∩ B, A ∩ I I I B ∈ τ . β β (iii) Let {A |i ∈ }∈ τ, by (iv) of Theorem 3.3, we have A = R (A ) ⊆ R ( A ). Hence, i i i i i∈ I I β β A ⊆ R ( A ).At the same time, by (vi) of Theorem 3.3, we have R ( A ) ⊆ i i i i∈ I i∈ I i∈ A . Hence, A = R ( A ). τ is an ALexandrov topology. i i i i∈ i∈ I i∈ Corollary 3.1: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. If β = 1 , then τ ={A ∈ ILF(U)|R (A) = A} is an ALexandrov topology. ¯ C II β β Proof: By Theorem 3.2(i), when β = 1 ,wehave R = R , hence τ is an ALexandrov L I II topology. 3.3. Some Other Rough Approximate Operators Based on I,II,III-type Rough Approximate Operators The Pawlak approximation operators satisfy many properties [[4],[5]]. When generaliz- ing Pawlak approximations, one task is to specify a subset of these properties that new approximation operators are required to preserve. Another task is to search for possible generalizations of the various notions used in the three definitions. Furthermore, one must make sure that the generalized approximation operators satisfy the required properties. In this paper, we consider the following properties suggested by PomykaIa [10]: for all A ∈ ILF (i) R (∼ A) =∼ R (A), (ii) R (A) ⊆ A ⊆ R (A). Definition 3.7: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space with C ={C , C , ... , C } being a fuzzy β-covering of U for some β ∈ L.For each A ∈ ILFS,we 1 2 m define other three kinds of new lower approximation operators and upper approximation operators of A as β β β β β β R (A)(x) = R (A)(x) ∧ R (A)(x), R (A)(x) = R (A)(x) ∨ R (A)(x), IV I II IV I II β β β β β β R (A)(x) = R (A)(x) ∧ R (A)(x), R (A)(x) = R (A)(x) ∨ R (A)(x), V I III V I III β β β β β β R (A)(x) = R (A)(x) ∧ R (A)(x), R (A)(x) = R (A)(x) ∨ R (A)(x). VI IV V VI IV V β β β β β β Then (R , R (A)), (R , R (A)),(R , R (A)) are called IV,V,VI-type rough operators of A, IV IV IV IV IV IV respectively. Theorem 3.7: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. Then 32 P. YU ET AL. β β (i) R (∼ A) =∼ R (A); IV IV β β (ii) R (∼ A) =∼ R (A); V V β β (iii) R (∼ A) =∼ R (A). VI VI Proof: We only prove (i), (ii) and (iii) can be proved similarly. β β β R (∼ A) = R (∼ A) ∧ R (∼ A) IV I II β β =∼ R (A)∧∼ R (A) I II β β =∼ (R (A) ∨ R (A)) I II =∼ R (A). IV Theorem 3.8: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. Then β β (i) R (A) ⊆ A ⊆ R (A); IV IV β β (ii) R (A) ⊆ A ⊆ R (A); V V β β (iii) R (A) ⊆ A ⊆ R (A). VI VI β β Proof: (i) By (vi) of Theorem 3.3, for every x ∈ U,wehave R (A)(x) ≤ A(x) ≤ R (A)(x). Then I I β β β β R (A)(x) = R (A)(x) ∧ R (A)(x) ≤ A(x) ∧ R (A)(x) ≤ A(x) IV I II II and β β β β R (A)(x) = R (A)(x) ∨ R (A)(x) ≥ A(x) ∨ R (A)(x) ≥ A(x). IV I II II β β Hence, R (A) ⊆ A ⊆ R (A). IV IV β β β β β β The relationships between R , R , R , R , R , R can be described as follows (Figure 2): IV IV V V VI VI Theorem 3.9: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. Then β β β β (i) R ≤ R , R ≤ R ; IV I V I β β β β (ii) R ≤ R , R ≤ R ; VI V VI IV β β β β (iii) R ≤ R , R ≤ R ; IV VI V VI β β β β (iv) R ≤ R , R ≤ R . I IV I V β β Remark 3.3: In Example 3.2, R (A) = (0, d)/x + (0, e)/x + (0, d)/x + (c, a)/x , R (A) = 1 2 3 4 V IV β β (0, e)/x + (0, e)/x + (0, e)/x + (b, b)/x . R (A)(x ) and R (A)(x ) are not comparable, 1 2 3 4 4 4 V Iv FUZZY INFORMATION AND ENGINEERING 33 Figure 2. The relationships between three new lower and upper approximation operators based on I,II,III-type Rough Approximate Operators. β β β hence R (A) and R (A) are incomparable. At the same time, in Example 3.2, R (A)(x ) = V Iv IV β β β β β (b, a) ≥ (b, c) = R (A)(x ) and R (A)(x ) = (b, e) ≤ (b, a) = R (A)(x ). R (A) and R (A) 3 4 4 V IV V V IV are also incomparable. 3.4. Matrix Representations of Lower and Upper Approximation Operators In this subsection, we will present matrix representations of the lower and upper approxi- mation operators defined in Definitions 3.3–3.5. The matrix representations of the approx- imation operators make it possible to calculate the lower and upper approximations of subsets through the operations on matrices, which is algorithmic, and can easily be implemented through the computer. 34 P. YU ET AL. Definition 3.8: Let A = (a ) and B = (b ) be two matrices. We define C = A B = ij n×m kj m×l (c ) and D = A B = (d ) as follows: ij n×l ij n×l c = (∼ a ∨ b ), i = 1, 2, ... , n, j = 1, 2, ... , l, ij ik kj k=1 d = (a ∧ b ), i = 1, 2, ... , n, j = 1, 2, ... , l. ij ik kj k=1 Obviously, for arbitrary A, B, C,if B ≤ C, then A B ≤ A C, B A ≥ C A, A B ≤ A C and B A ≤ C A. Meanwhile, A B = B A and A B = B A. Definition 3.9: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. We β β β define matrix M , M , M as follows: I II III 1 , y ∈ N , β L M (x, y) = 0 , otherwise, β β N (y), y ∈ N , β x M (x, y) = II 0 , otherwise, β β M (x, y) = N (y). III Here N is defined as in Definition 3.2. β β β β From Definition 3.9, we can see that M  M and M  M . II I II III Theorem 3.10: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space, A ∈ β β β β T T ILF(U). Then R (A) = M A , R (A) = M A (i ∈{I, II, II}). i i i i Example 3.4: In Example 3.2, take β = (a, b), then ⎛ ⎞ ⎛ ⎞ 1 0 1 0 (a, b) 0 (a, b) 0 L L L L L L ⎜ ⎟ ⎜ ⎟ 1 1 1 0 (b,0)(b, a)(a, b) 0 β L L L L β L ⎜ ⎟ ⎜ ⎟ M = , M = , I II ⎝ ⎠ ⎝ ⎠ 1 0 1 0 (a, b) 0 (a, b) 0 L L L L L L 0 0 0 1 0 0 0 (b, b) L L L L L L L ⎛ ⎞ (a, b)(0, e)(a, b)(0, e) ⎜ ⎟ (b,0)(b, a)(a, b)(0, e) ⎜ ⎟ M = . III ⎝ ⎠ (a, b)(0, e)(a, b)(0, e) (0, c)(a, c)(a, d)(b, b) (0, d) (a, c) (b, a) (e, a) If A = + + + ,we have x x x x 1 2 3 4 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 1 0 (0, d) (0, d) L L L L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1 1 1 0 (a, c) (0, e) β β L L L L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ R (A) = M A = = , I I ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 1 0 1 0 (b, a) (0, d) L L L L 0 0 0 1 (e, a) (e, a) L L L L FUZZY INFORMATION AND ENGINEERING 35 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ (a, b) 0 (a, b) 0 (0, d) (b, a) L L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (b,0)(b, a)(a, b) 0 (a, c) (0, a) β β T L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = , R (A) = M A = II II ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (a, b) 0 (a, b) 0 (b, a) (b, a) L L 0 0 0 (b, b) (e, a) (e, a) L L L ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ (a, b)(0, e)(a, b)(0, e) (0, d) (b, a) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (b,0)(b, a)(a, b)(0, e) (a, c) (0, a) β β ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ R (A) = M A = = , III II ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (a, b)(0, e)(a, b)(0, e) (b, a) (b, a) (0, c)(a, c)(a, d)(b, b) (e, a) (c, a) ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 1 0 (0, d) (b, a) L L L L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1 1 1 0 (a, c) (b, a) β β L L L L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ R (A) = M A = = , I I ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 1 0 1 0 (b, a) (b, a) L L L L 0 0 0 1 (e, a) (e, a) L L L L ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ (a, b) (a, b) 0 (a, b) 0 (0, d) L L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (b,0)(b, a)(a, b) 0 (a, c) (a, a) β β L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ R (A) = M A = = , II II ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (a, b) 0 (a, b) 0 (b, a) (a, b) L L 0 0 0 (b, b) (e, a) (b, b) L L L ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ (a, b)(0, e)(a, b)(0, e) (0, d) (a, b) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (b,0)(b, a)(a, b)(0, e) (a, c) (a, a) β β ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ R (A) = M A = = . III III ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (a, b)(0, e)(a, b)(0, e) (b, a) (a, b) (0, c)(a, c)(a, d)(b, b) (e, a) (b, a) 4. Conclusions The theories of covering rough sets, intuitionistic L-fuzzy sets, are important for dealing with uncertainty and inaccuracy problems. In order to handle these uncertainty and inaccu- racy problems more effectively, the combination of the covering and intuitionistic L-fuzzy sets is further researched in this paper. The concepts of intuitionistic L-fuzzy β-covering and intuitionistic L-fuzzy β-neighborhood are proposed. Several intuitionistic L-fuzzy β- covering rough set models are discussed. Some properties of these models are proved and demonstrated by some examples. The research results fill in the blanks of the study of the intuitionistic L-fuzzy rough set with the method of covering. We will investigate the applications of the presented models and construct multi-granulation intuitionistic L-fuzzy covering rough sets, which will be part of the future research directions considered by our group. Disclosure statement No potential conflict of interest was reported by the author(s). Notes on contributors Peng YU was born in 1981 in Ningxia, China. He received the Ph.D. degree in Mathematic from Shaanxi Normal University, xian, china in 2019. He is currently an associate professor of Shaanxi University of Science and Technology. He has published 10 research papers on fuzzy reasoning and rough set in 36 P. YU ET AL. international journals. His current research interests include Non-classic logic, fuzzy reasoning and rough set. Xiao-gang An was born in 1975 in Shaanxi, China. He received the master’s degree in computer science from Southwest Jiaotong University, Chongqing, china in 2005. He is currently a lecturer of Shaanxi University of Science and Technology. He has published three research papers on Logic algebra in international journals. His current research interests include Logic algebra and rough set. Xiao-hong Zhang was born in 1965 in Shaanxi, China. He received the Ph.D. degree in computer sci- ence from Northwestern Polytechnical University, xian, china in 2005. He is currently an fuzz professor of Shaanxi University of Science and Technology. He has published 60 research papers on logic alge- bra and rough set in international journals. His current research interests include The logical basis of artificial intelligence, approximate reasoning, Logic algebra and rough set. Funding Scientific Research Program Funded by Shaanxi Provincial Education Department [No. 18JK0099] and the Scientific Research Start-up Fund of Shaanxi University of Science and Technology [2019BJ-41]. References [1] Pawlak Z. Rough set. Int J Comput Inf Sci II. 1982;11(5):341–356. [2] Greco S, Matarazzo B, Slowinski R. Rough approximation by dominance relations. Int J Intell Syst. 2002;17:153–171. [3] Liu G, Zhu W. The algebraic structures of generalized rough set theory. Inf Sci. 2008;178: 4105–4113. [4] Yao YY. Constructive and algebraic methods of theory of rough sets. Inf Sci. 1998;109:21–47. [5] Yao YY. Relational interpretations of neighborhood operators and rough set approximation operators. Inf Sci. 1998;111:239–259. [6] Wu WZ, Zhang WX. Neighborhood operator systems and appoximations. Inf Sci. 2002;144: 201–217. [7] Pomykala JA. Approximation operations in approximation space. Bull Polish Acad Sci. 1987;35:653–662. [8] Pomykala JA. On definability in the nondeterministic information system. Bull Polish Acad Sci. 1998;36:193–210. [9] Zakowski W. Approximations in the space (μ, ). Demonstratio Math. 1983;16:761–769. [10] Ma LW. Classification of coverings in the finite approximation spaces. Inf Sci. 2014;276:31–41. [11] Couso I, Dubois D. Rough sets, coverings and incomplete information. Fund Inform. 2011;108:223–247. [12] Yao YY, Yao B. Covering based rough set approximations. Inf Sci. 2012;200:91–107. [13] Xue ZA, Si XM, Xue TY et al. Multi-granulation covering rough intuitionistic fuzzy sets. J Intell Fuzzy Syst. 2017;32:899–911. [14] Yang B, Hu BQ. On some types of fuzzy covering-based rough sets. Fuzzy Sets Syst. 2017;312:36–65. [15] Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986;20:87–96. [16] Atanassov K. Intuitionistic fuzzy sets: theory and applications. Physica-Verlag; 1999. [17] Atanassov K. Index matrices: towards an augmented matrix calculus. Springer; 2014. [18] Zhong Y, Yan CH. Intuitionistic fuzzy rough sets. Intuitionistic fuzzy preorders and intuitionistic fuzzy topologies. Fuzzy Inf Eng. 2016;82:255–279. [19] Zhan ZM. Generalized intuitionistic fuzzy rough sets based on intuitionistic fuzzy coverings. Inf Sci. 2012;198:186–206. [20] Chen TY. A comparative analysis of score functions for multiple criteria decision making in intuitionistic fuzzy settings. Inf Sci. 2011;181:3652–3676. [21] Zhang X, Zheng Y. Linguistic quantifiers modeled by interval-valued intuitionistic Sugeno inte- grals. J Intell Fuzzy Syst. 2015;29(2):583–592. FUZZY INFORMATION AND ENGINEERING 37 [22] Galatos N, Jipsen P, Kowalski T et al. Residuated lattices: an algebraic glimpse at substructural logics. Amsterdam: Elsevier; 2007. (Studies in Logic and the Foundations of Mathematics; vol. 151). [23] Ono H. Substructural logics and residuated lattices – an introduction; 2003. p. 193–228. (Trends in Logic: 50 years of Studia Logica; vol. 21). [24] Blount K, Tsinakis C. The structure of residuated lattice. Int J Algebra Comput. 2003;13:437–461. [25] Yang J, Zhang XG. Some weaker versions of topological residuated lattices. Fuzzy Sets Syst. 2019;373:62–77. [26] Pei DW. The characterization of residuated and regular residuated lattices. Acta Math Sin. 2002;45:271–278. Chinese. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Intuitionistic L-fuzzy β-covering Rough Set

Intuitionistic L-fuzzy β-covering Rough Set

Abstract

By introducing the concepts of intuitionistic L-fuzzy β-covering and intuitionistic L-fuzzy β-neighborhood, we define three kinds of intuitionistic L-fuzzy β-covering rough set models. The basic properties of those intuitionistic L-fuzzy β-covering rough set models are investigated. Moreover, we define the other three kinds of intuitionistic L-fuzzy β-covering rough set models by using the former three models. Finally, we present the matrix representations of the...
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© 2020 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society of China & Operations Research Society of Guangdong Province.
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1616-8666
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1616-8658
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10.1080/16168658.2020.1778829
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FUZZY INFORMATION AND ENGINEERING 2020, VOL. 12, NO. 1, 19–37 https://doi.org/10.1080/16168658.2020.1778829 Peng Yu, Xiao-gang An and Xiao-hong Zhang School of arts and Sciences, Shaanxi University of Science and Technology, Xi’an, People’s Republic of China ABSTRACT ARTICLE HISTORY Received 7 April 2020 By introducing the concepts of intuitionistic L-fuzzy β-covering and Revised 15 May 2020 intuitionistic L-fuzzy β-neighborhood, we define three kinds of intu- Accepted 1 June 2020 itionistic L-fuzzy β-covering rough set models. The basic properties of those intuitionistic L-fuzzy β-covering rough set models are inves- KEYWORDS tigated. Moreover, we define the other three kinds of intuitionistic L- Intuitionistic L-fuzzy set; fuzzy β-covering rough set models by using the former three models. covering rough set; rough Finally, we present the matrix representations of the newly defined set; Alexandrov topolgy lower and upper approximation operators so that the calculation of lower and upper approximations of subsets can be converted into operations on matrices. 1. Introduction Rough set (RS) theory was firstly introduced by Pawlak [1]. It may be seen as an extension of set theory and has been found to be a new mathematical tool to deal with insufficient and incomplete information systems. In Pawlak’s rough set model, the equivalence relation is an important concept that is used to construct the lower and upper approximations of an arbitrary subset of the universe of discourse. However, the condition of the equivalence relation is highly restrictive and may limit the application of rough sets in many practical problems. Hence, numerous extensions of Pawlak’s rough set were proposed by replacing the equivalence relation with a few mathematical concepts that are more general in nature, for example, arbitrary binary relations [2–4], neighborhood systems, covering-based and Boolean algebras [5–8]. Covering rough set theory is an important generalization of the classical rough set theory. It was firstly introduced by Zakowski [9]. After then, many kinds of different cov- ering rough sets are introduced [10–13]. Meanwhile, fuzzy covering rough set and fuzzy β-covering rough set are important generalizations of the classical covering rough set theory [14]. The concept of an intuitionistic fuzzy (IF for short) set, initiated by Atanassov [15–17], is another important tool for dealing with imperfect and imprecise information. Compared with Zadeh’s fuzzy set, an IF set is more objective than a fuzzy set to describe the vagueness CONTACT Peng Yu yupeng@sust.edu.cn © 2020 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society of China & Operations Research Society of Guangdong Province. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 20 P. YU ET AL. of data, because IF set gives both a membership and a non-membership degree of which an element belongs to a set [18–21]. As an important generalization of IF set, Atanassov and Stoeva defined intuitionistic L-fuzzy (ILF for short ) set in 1984, which actually is an IF set based on residuated lattice L. ILF set and covering rough set are the important generalizations of the classical fuzzy set and rough set, but related work is rarely done to combine the ILF set theory with the covering rough set theory. Motivated by this, in this paper, by introducing the concepts of intuitionistic L-fuzzy β-covering and intuitionistic L-fuzzy β-neighborhood, we define sev- eral differentiable rough set models, obtain some basic properties of those models, and present the matrix representations of the newly defined lower and upper approximation operators. 2. Preliminaries In this section, we introduce some basic concepts of residuated lattice and intuitionistic L-fuzzy set used in this paper. We refer to [22–26] for residuated lattice theory and to [18] for intuitionistic L-fuzzy set. 2.1. Complete Regular Residuated Lattice Definition 2.1 ([22]): A residuated lattice is an algebraic structure L = (L; ∧, ∨, ∗, →,0,1) of type (2, 2, 2, 2, 0, 0) such that (i) (L; ∧, ∨,0,1) is a bounded lattice with the least element 0 and the greatest element 1; (ii) (L; ∗,1) is a commutative monoid with the identity 1; (iii) L satisfies the adjointness property, i.e. ∀ x, y, z ∈ L, x ∗ y ≤ z iff x ≤ y → z. A residuated lattice is said to be complete if the underlying lattice is complete. Definition 2.2 ([26]): The negator on L is the mapping ¬ : L → L defined by ¬a = a → 0 for every a ∈ L.If ¬¬a = a for all a ∈ L, then L is called a regular residuated lattice. In this paper, if there is no further statement, L always denotes a complete regular residuated lattice. Theorem 2.1 ([18]): Let L be a complete regular residuated lattice. Then (R1) a ∗ b ≤ a ∧ b; (R2) a = 1 → a; (R3) a ≤ b ⇔ a → b = 1; (R4) a → (b → c) = (a ∗ b) → c; (R5) a → b ≥ b; (R6) a ∗ ( b ) = (a ∗ b ); i i i i (R7) a → ( b ) = (a → b ); i i i i FUZZY INFORMATION AND ENGINEERING 21 (R8)( b ) → a = (b → a); i i i i (R9) ¬( a ) = ¬a , ¬( a ) = ¬a . i i i i i∈ i∈ i∈ i∈ 2.2. Intuitionistic L-fuzzy Set In this subsection, we introduce the notion of ILF set that can be regarded as the general- ization of IF set induced by Atanssov [1, 2]. At first, we introduce a special lattice L.Thiscan be seen as a generalization of the set {(x, y) ∈ [0, 1] × [0, 1]|x + y ≤ 1}. Definition 2.3 ([18]): L ={(x , x )|x , x ∈ L, x ≤¬x }. The partial ordering ≤ is defined 1 2 1 2 1 2 ˜ as follows: x ≤ y ⇔ x ≤ y , x ≥ y , ∀x = (x , x ), y = (y , y ) ∈ L. 1 1 2 2 1 2 1 2 The pair (L, ≤ ) is a complete lattice, with the smallest element 0 = (0, 1) and the greatest ˜ ˜ L L element 1 = (1, 0). The meet operator ∧ and the join operaton ∨ on L which are connected to the partial ordering ≤ are defined as follows: x ∧ y = (x ∧ y , x ∨ y ), x ∨ y = (x ∨ y , x ∧ y ). ˜ 1 1 2 2 ˜ 1 1 2 2 L L Definition 2.4 ([18]): Let U be a non-empty universe of discourse. An ILF set A in U is an object having the form A ={< x, μ (x), ν (x)> |x ∈ U}, A A where μ : U → L and ν : U → L satisfy μ (x) ≤¬ν (x) for all x ∈ U. A A A A They are called the degree of L membership and the degree of L non-membership of the element x to A, respectively. The family of all ILF subsets in U is denoted by ILF(U). We denote A(x) = (μ (x), ν (x)), then A ∈ ILF(U) if and only if A ∈ L for all x ∈ U. (α, β) A A denotes the constant ILF sets for all (α, β) ∈ L.If U is a non-empty finite set, then ILF set A can be denoted by a matrix A = ((μ (x ), ν (x )), ... , (μ (x ), ν (x ))). A 1 A 1 A n A n For any A, B, A (i ∈ ) ∈ ILF(U), some operations are introduced as follows: (1) A ⊆ B iff A(x) ≤ B(x). (2) A = B iff A ⊆ B and B ⊆ A. (3) A B ={< x, μ (x) ∧ μ (x), ν (x) ∨ ν (x)> |x ∈ U} means (A B)(x) = A(x) ∧ A B A B ˜ B(x). (4) A B ={< x, μ (x) ∨ μ (x), ν (x) ∧ ν (x)> |x ∈ U} means (A B)(x) = A(x) ∨ A B A B ˜ B(x). (5) A ={< x, μ (x), ν (x)> |x ∈ U} means ( A )(x) = ( ) i A A i ˜ i∈ i∈ i i∈ i i∈ i∈ μ (x). (6) A ={< x, μ (x), ν (x)> |x ∈ U} means ( A )(x) = ( ) i A A i ˜ i∈ i∈ i i∈ i i∈ i∈ μ (x). (7) ∼ A ={< x, ν (x), μ (x)> |x ∈ U}. A A 22 P. YU ET AL. 3. Intuitionistic L-fuzzy Rough Set 3.1. Intuitionistic L-fuzzy Covering In this subsection, we introduce the concepts of intuitionistic L-fuzzy β-covering and intu- itionistic L-fuzzy β-neighborhood of a non-empty universe of discourse U, and investigate the basic properties of it. Definition 3.1: Let U be an arbitrary universal set, and L be the intuitionistic L-fuzzy power set of U.For each β ∈ L,wecall C ={C , C , ... , C }, with C ∈ ILF(U)(i = 1, 2, ... , m),an 1 2 m i intuitionistic L-fuzzy β-covering of U,iffor each x ∈ U, C ∈ C exists, such that C (x) ≥ β.We i i also call (U, C) an intuitionistic L-fuzzy β-covering approximation space. Definition 3.2: Let C ={C , C , ... , C } be an intuitionistic L-fuzzy β-covering of U for 1 2 m some β ∈ L.For each x ∈ U, we define the intuitionistic L-fuzzy β-neighborhood N of x as N =∩{C ∈ C|C (x) ≥ β}. x i i Proposition 3.1: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. C ={C , C , ... , C } being a fuzzy β-covering of U for some β ∈ L, N ={y ∈ U|N (y) ≥ β}, 1 2 m then (i) x ∈ N ; β β (ii) N = N . x y y∈N Proof: (i) For all x ∈ U, N (x) =∩{C ∈ C|C (x) ≥ β}(x) = C (x) ≥ β, hence x ∈ N . x i i i ˜ C (x)≥ β x i L β β β β β (ii) On the one hand, since x ∈ N ,wehave N ⊆ N ( β N ) = β N . x x x y y y=x,y∈N y∈N x x β β β On the other hand, let y ∈ N , then N (y) ≥ β,wehave N ⊆{C |C (y) ≥ β}. Therefore, x x i i β β β β β β N ⊇ N .Let z ∈ N , then N (z) ≥ β,wehave N (z) ≥ β and z ∈ N .Furthermore,wehave x y y x y x β β β β N ⊆ N , N ⊆ N . y x y x y∈N Proposition 3.2: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. If β β 2 1 β ≤ β , then N ≥ N . 1 ˜ 2 x x Proof: Since β ≤ β , then for every c ∈{c |c (x) ≥ β }, c (x) ≥ β ≥ β , this means c ∈ 1 2 j i i 2 j 2 1 j {c |c (x) ≥ β }. Hence, {c |c (x) ≥ β }⊆{c |c (x) ≥ β }; furthermore, N =∩{c |c (x) ≥ i i 1 i i 2 i i 1 x i i β }⊆∩{c |c (x) ≥ β }= N . 2 i i 1 x Proposition 3.3: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. If y ∈ β β β N and z ∈ N , then z ∈ N . x y x FUZZY INFORMATION AND ENGINEERING 23 β β β β β Proof: Assume that y ∈ N and z ∈ N , then N (y) ≥ β and N (z) ≥ β. This means N ∈ x y x x y β β β β {c |c (y) ≥ β} and N ≥ N =∩{c |c (y) ≥ β}. Furthermore, we have N (z) ≥ N (z) ≥ β i i x y i i x y and z ∈ N . Example 3.1: Let L ={0, a, b, c, d, e,1}, → and ⊗ be defined as follows: → 0 ab c d e 1 ⊗ 0 ab c d e 1 0 1 1 11 111 0 0000 00 0 a e 1 1 1 111 a 0000 00 a b be 1e e 1 1 b 000 aa ab c ce e 1 e 11 c 00 a 0 aa c d de e e 111 d 00 aa 0 ad e aee e e 11 e 00 aa aa e 1 0 ab c d e 1 1 0 ab c d e 1 The partial order ≤ on L is defined by Figure 1 Figure 1. Partial order set L. 24 P. YU ET AL. It is easy to check that L is a complete regular residuated lattice, and L ={(0, 0), (0, a), (0, b), (0, c), (0, d), (0, e), (0, 1), (a,0), (a, a), (a, b), (a, c), (a, d), (a, e), (b,0), (b, a), (b, b), (c,0), (c, a), (c, c), (d,0), (d, a), (d, d), (e,0), (e, a), (1, 0)}. Let U ={x , x , x , x }, β = (a, e) ∈ L, 1 2 3 4 (a, b) (0, e) (d, a) (0, b) C = + + + , x x x x 1 2 3 4 (b,0) (b, a) (a, b) (0, e) C = + + + , x x x x 1 2 3 4 (0, c) (a, c) (a, d) (b, b) C = + + + , x x x x 1 2 3 4 then {C , C , C } is a fuzzy β-covering of U. 1 2 3 (a, b) (0, e) (a, b) (0, e) N = C ∩ C = + + + , 1 2 x x x x 1 2 3 4 (0, c) (a, c) (a, e) (0, e) N = C ∩ C = + + + , x 2 3 x x x x 1 2 3 4 (0, e) (0, e) (a, e) (0, e) N = C ∩ C ∩ C = + + + , 1 2 3 x x x x 1 2 3 4 (0, c) (a, c) (a, d) (b, b) N = C = + + + . x 3 x x x x 1 2 3 4 β β β β Furthermore, we have N ={x , x }, N ={x , x }, N ={x }, N ={x , x , x }. 1 3 2 3 3 2 3 4 x x x x 1 2 3 4 3.2. Intuitionistic L-fuzzy Covering Rough Set In this subsection, we will define three kinds of basic intuitionistic L-fuzzy β-covering rough set models and investigate the properties of it. At first, we define the I-type L-fuzzy β-covering rough set model by using the L-fuzzy β-covering of U. Definition 3.3: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space with C ={C , C , ... , C } being a fuzzy β-covering of U for some β ∈ L.Foreach A ∈ ILF(U),we 1 2 m β β define the I-type lower approximation R (A) and the I-type upper approximation R (A) I I of x as R (A)(x) =∧ A(y), y∈N R (A)(x) =∨ β A(y). y∈N In fact, the I-type L-fuzzy β-covering model above is a generalization of the classical fuzzy rough set model, which simply uses N induced by intuitionistic L-fuzzy β-neighborhood N to replace the equivalence class in the fuzzy rough set model. Therefore, it is one of the most basic covering rough set models. FUZZY INFORMATION AND ENGINEERING 25 Definition 3.4: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space with C ={C , C , ... , C } being a fuzzy β-covering of U for some β ∈ L.Foreach A ∈ ILF(U),we 1 2 m β β define the II-type lower approximation R (A) and the II-type upper approximation R (A) II II of x as β β R (A)(x) =∧ β (¬N (y) ∨ A(y)), II y∈N β β R (A)(x) =∨ β (N (y) ∧ A(y)). II y∈N Definition 3.5: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space with C ={C , C , ... , C } being a fuzzy β-covering of U for some β ∈ L.Foreach A ∈ ILF(U),we 1 2 m β β define the III-type lower approximation R (A) and the III-type upper approximation R (A) III III of x as β β R (A)(x) =∧ (¬N (y) ∨ A(y)), y∈U x III β β R (A)(x) =∨ (N (y) ∧ A(y)). y∈U x III In the definition of III-type lower (upper) approximation operator, if we consider N (y) as an intuitionistic L-fuzzy binary relation on U, ¬N (y) ∨ A(y) is understood as a classical fuzzy implication operator I(a, b) = (1 − a) ∨ b, then III-type lower (upper) approximation opera- tor is a generalization of lower (upper) approximation operator of definition 4.4 in literature [23]. In fact, if we take arbitrary fuzzy implication I based on ILF sets instead of classic fuzzy implication I, then we can define some other types of intuitionistic L-fuzzy covering rough set. Example 3.2: In Example 3.1, let A = (0, d)/x + (a, c)/x + (b, a)/x + (e, a)/x , 1 2 3 4 β = (a, b), then C ={C , C , C } being a fuzzy β-covering of U,and 1 2 3 (a, b) (0, e) (a, b) (0, e) β β N = C ∩ C = + + + , N ={x , x }. x 1 2 1 3 1 1 x x x x 1 2 3 4 (b,0) (b, a) (a, b) (0, e) N = C = + + + , N ={x , x , x }. x 2 1 2 3 2 x x x x x 1 2 3 4 (a, b) (0, e) (a, b) (0, e) N = C ∩ C = + + + , N ={x , x }. x 1 2 1 3 3 x x x x x 1 2 3 4 (0, c) (a, c) (a, d) (b, b) N = C = + + + , N ={x }. 3 4 4 x x x x x 1 2 3 4 Furthermore, we have R (A)(x ) = A(x ) ∧ A(x ) = (0, d) ∧ (b, a) = (0, d), 1 1 3 R (A)(x ) = A(x ) ∧ A(x ) ∧ A(x ) = (0, d) ∧ (a, c) ∧ (b, a) = (0, e), 2 1 2 3 R (A)(x ) = A(x ) ∧ A(x ) = (0, d) ∧ (b, a) = (0, d), 3 1 3 R (A)(x ) = A(x ) = (e, a). 4 4 I 26 P. YU ET AL. Hence, (0, d) (0, e) (0, d) (e, a) R (A) = + + + , x x x x 1 2 3 4 β β β R (A)(x ) = (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) 1 1 1 3 3 x x II 1 1 = (¬(a, b) ∨ (0, d)) ∧ (¬(a, b) ∨ (b, a)) = (b, a), β β β β R (A)(x ) = (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) 2 x 1 1 x 2 2 x 3 3 II 2 2 2 = (¬(b,0) ∨ (0, d)) ∧ (¬(b, a) ∨ (a, c)) ∧ (¬(a, b) ∨ (a, b)) = (0, a), β β β R (A)(x ) = (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) 3 1 1 3 3 x x II 3 3 = (¬(a, b) ∨ (0, d)) ∧ (¬(a, b) ∨ (b, a)) = (b, a), β β R (A)(x ) = (¬N (x ) ∨ A(x )) 4 x 4 4 II = (¬(b, b) ∨ (e, a)) = (b, b), (b, a) (0, a) (b, a) (e, a) R (A) = + + + . II x x x x 1 2 3 4 β β β β R (A)(x ) = (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) 1 x 1 1 x 2 2 x 3 3 III 1 1 1 ∧ (¬N (x ) ∨ A(x )) x 4 4 = (¬(a, b) ∨ (0, d)) ∧ (¬(0, e) ∨ (a, c)) ∧ (¬(a, b) ∨ (b, a)) ∧ (¬(0, e) ∨ (e, a)) = (b, a), β β β β R (A)(x ) = (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) 2 x 1 1 x 2 2 x 3 3 III 2 2 2 ∧ (¬N (x ) ∨ A(x )) 4 4 = (¬(b,0) ∨ (0, d)) ∧ (¬(b, a) ∨ (a, c)) ∧ (¬(a, b) ∨ (b, a)) ∧ (¬(0, e) ∨ (e, a)) = (0, a), β β R (A)(x ) = R (A)(x ) = (b, a), 3 1 III III β β β β R (A)(x ) = (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) ∧ (¬N (x ) ∨ A(x )) 4 x 1 1 x 2 2 x 3 3 4 4 4 III ∧ (¬N (x ) ∨ A(x )) x 4 4 = (¬(0, c) ∨ (0, d)) ∧ (¬(a, c) ∨ (a, c)) ∧ (¬(a, d) ∨ (b, a)) ∧ (¬(b, b) ∨ (e, a)) = (c, a), (b, a) (0, a) (b, a) (c, a) R (A) = + + + . III x x x x 1 2 3 4 FUZZY INFORMATION AND ENGINEERING 27 Similarly, we have (b, a) (b, a) (b, a) (b, c) R (A) = + + + , x x x x 1 2 3 4 (a, b) (a, a) (a, b) (b, b) R (A) = + + + , II x x x x 1 2 3 4 (a, a) (a, a) (a, a) (b, a) R (A) = + + + . III x x x x 1 2 3 4 Remark 3.1: In Example 3.2, we can see that β β R (A)(x ) = (0, d) = (b, a) = R (A)(x ), 1 1 I II β β R (A)(x ) = (b, b) = (c, a) = R (A)(x ), 4 4 II III β β R (A)(x ) = (0, d) = (b, a) = R (A)(x ). 1 1 I III β β R (A)(x ) = (b, a) = (a, b) = R (A)(x ), 1 1 I II β β R (A)(x ) = (a, b) = (a, a) = R (A)(x ), 1 1 II III β β R (A)(x ) = (b, a) = (a, a) = R (A)(x ), 1 1 I III β β β we have R (A), R (A), R (A) as three kinds of different lower approximation operators and I II III β β β R (A), R (A), R (A) as three kinds of different lower approximation operators. But they have I II III the following relationships: Theorem 3.1: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space, then β β β β (i) R (A) ≤ R (A), R (A) ≤ R (A); I II II I β β β β (ii) R (A) ≤ R (A), R (A) ≤ R (A). III II II III Proof: (i) Since A(y) ≤∼ N (y) ∨ A(y), β β β R (A)(x) =∧ β A(y) ≤∧ β (∼ N (y) ∨ A(y)) = R (A)(x). I II y∈N y∈N x x β β Similarly, we have R (A) ≤ R (A). II I (ii) Since N ⊆ U,wehave β β β β R (A)(x) =∧ (∼ N (y) ∨ A(y)) ≤∧ (∼ N (y) ∨ A(y)) = R (A)(x). y∈U x x III II y∈N β β Similarly, we have R (A) ≤ R (A). II III 28 P. YU ET AL. Theorem 3.2: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. β β (i) if β = 1 , then R (A) = R (A); I II β β (ii) if β = 0 , then R (A) = R (A). II III Proof: β β (i) If β = 1 , then ∀y ∈ N ;wehave N (y) = 1 . Hence R (A)(x) = L L x x II β β ∧ (¬N (y) ∨ A(y)) =∧ (0 ∨ A(y)) =∧ A(y) = R (A)(x). β β β x L y∈N y∈N y∈N x x x β β β (ii) If β = 0 , then N = U;wehave R (A)(x) =∧ β (¬N (y) ∨ A(y)) =∧ (¬N (y) ∨ L x y∈U x x II y∈N A(y)) = R (A)(x). III Example 3.3: In Example 3.1, let (1, 0) (0, e) (d, a) (0, b) C = + + + , x x x x 1 2 3 4 (b,0) (1, 0) (1, 0) (0, e) C = + + + , x x x x 1 2 3 4 (0, c) (1, 0) (a, d) (1, 0) C = + + + , x x x x 1 2 3 4 (0, c) (1, 0) (a, e) (1, 0) C = + + + . x x x x 1 2 3 4 β = (1, 0) ∈ L, then C , C , C , C being an L-fuzzy β-covering of U. Suppose that A = 1 2 3 4 β β (e, a)/x + (b, a)/x + (a, c)/x + (0, e)/x ,wehave R (A)(x ) = (e, a) = (a, a) = R (A) 1 2 3 4 1 I III β β (x ).If β = (0, 1) ∈ L,wehave R (A)(x ) = (0, e) = (e, a) = R (A)(x ). 1 1 1 I III β β Theorem 3.3: For the operators R (A) and R (A), the following properties hold: I I β β (i) R (0 ) = 0 , R (0 ) = 0 ; ¯ ¯ ¯ ¯ L L L L I I β β (ii) R (1 ) = 1 , R (1 ) = 1 ; ¯ ¯ ¯ ¯ L L L L I I β β β β (iii) R (A) =∼ R (∼ A) and R (A) =∼ R (∼ A); I I I I β β β β β β (iv) R (A ∩ B) = R (A) ∩ R (B), R (A ∪ B) = R (A) ∪ R (B); I I I I I I β β β β β β (v) R (A ∪ B) ⊇ R (A) ∪ R (B), R (A ∩ B) ⊆ R (A) ∩ R (B); I I I I I I β β (vi) ∀A ∈ ILF(U), R (A) ⊆ A ⊆ R (A); I I β β β β (vii) If A ⊆ B, then R (A) ⊆ R (B), R (A) ⊆ R (B); I I I I β β β β β β (viii) R (R (A)) = R (A), R (R (A)) = R (A). I I I I I I Proof: Straightforward.  FUZZY INFORMATION AND ENGINEERING 29 β β Theorem 3.4: For the operators R (A) and R (A), the following properties hold: II II β β (i) R (1 ) = 1 , R (0 ) = 0 ; ¯ ¯ ¯ ¯ L L L L II II β β β β (ii) R (A) =∼ R (∼ A) and R (A) =∼ R (∼ A); II II II II β β β β β β (iii) R (A ∩ B) = R (A) ∩ R (B), R (A ∪ B) = R (A) ∪ R (B); II II II II II II β β β β β β (iv) R (A ∪ B) ⊇ R (A) ∪ R (B), R (A ∩ B) ⊆ R (A) ∩ R (B); II II II II II II β β β β (v) If A ⊆ B, then R (A) ⊆ R (B), R (A) ⊆ R (B); II II II II β β β (vi) R (A) ≤∼ β ∨ Aand R (A) ≥ β ∧ A. Particularly, when β = 1 , we have that R (A) ⊆ II II II A ⊆ R (A). II Proof: β β β β (i) R (1 ) =∧ (∼ N (y) ∨ (1, 0)) = 1 , R (0 ) =∨ (N (y) ∧ (0, 1)) = 0 . β β ¯ x ¯ ¯ x ¯ II L L II L L y∈N y∈N x x (ii) β β R (∼ A)(x) =∧ (∼ N (y)∨∼ A(y)) II y∈N =∧ (∼ (N (y) ∧ A(y)) y∈N =∼ ∨ (N (y) ∧ A(y)) y∈N =∼ R (A)(x). II β β Similarly, we have R (A) =∼ R (∼ A). II II (iii) For each x ∈ U, β β R (A ∩ B)(x) =∧ (∼ N (y) ∨ (A ∩ B)(y)) II y∈N β β =∧ ((∼ N (y) ∨ A(y)) ∧ (∼ N (y) ∨ B(y))) x x y∈N β β =∧ β (∼ N (y) ∨ A(y)) ∧ (∧ β (∼ N (y) ∨ B(y))) x x y∈N y∈N x x β β = R (A)(x) ∩ R (B)(x). II II β β β Similarly, we have R (A ∪ B) = R (A) ∪ R (B). II II II (iv) For each x ∈ U, β β R (A ∪ B)(x) =∧ (N (y) ∧ (A ∪ B)(y)) II y∈N β β =∧ ((N (y) ∧ A(y)) ∨ (N (y) ∧ B(y))) x x y∈N β β ≥∧ (N (y) ∧ A(y)) ∨ (∧ (N (y) ∧ B(y))) β β x x y∈N y∈N x x β β = R (A)(x) ∪ R (B)(x). II II 30 P. YU ET AL. β β β Similarly, we have R (A ∩ B) ⊆ R (A) ∩ R (B). II II II (v) Since A ⊆ B,wehave A(x) ≤ B(x) for every x ∈ U. Furthermore, β β R (A)(x) =∧ β (∼ N (y) ∨ A(y)) II y∈N ≤∧ β (∼ N (y) ∨ B(y)) y∈N = R (B)(x). II β β β (vi) For each x ∈ U, R (A)(x) =∧ β (∼ N (y) ∨ A(y)) ≤∼ N (x) ∨ A(x) ≤∼ β ∨ A(x), x x II y∈N β β then R (A) ≤∼ β ∨ A(x). Similarly, we have R (A) ≥ β ∧ A. Particularly, when β = 1 , II II L β β we have R (A) ⊆ A ⊆ R (A). II II β β β β β Remark 3.2: In Example 3.2, R (R (A))(x ) = (a, a) = (0, a) = R (A)(x ), R (R (A))(x ) = 2 2 2 II II II II II β β β β β β β (a, b) = (a, a) = R (A)(x ).Wehave R (R (A)) = R (A), R (R (A)) = R (A),thatistosay II II II II II II II β β the idempotent of lower and upper approximate operators R ,R does not hold. II II β β Theorem 3.5: For the III-type approximate operators R and R , the following properties III III hold: β β (i) R (0 ) = 0 , R (1 ) = 1 ; ¯ ¯ ¯ ¯ III L L III L L β β β β (ii) R (A) =∼ R (∼ A) and R (A) =∼ R (∼ A); III I III III β β β β β β (iii) R (A ∩ B) = R (A) ∩ R (B), R (A ∪ B) = R (A) ∪ R (B); III III III III III III β β β β β β (iv) R (A ∪ B) ⊇ R (A) ∪ R (B), R (A ∩ B) ⊆ R (A) ∩ R (B); III III III III III III β β β β (v) If A ⊆ B, then R (A) ⊆ R (B), R (A) ⊆ R (B). III III III III Definition 3.6: A subset τ ⊆ ILF(U) is called an ILF topology if it satisfies: (i) 1 ,0 ∈ τ; ¯ ¯ L L (ii) A, B ∈ τ implies A B ∈ τ ; (iii) {A |i ∈ }∈ τ implies A ∈ τ. i i i∈ Theorem 3.6: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. Then τ ={A ∈ ILF(U)|R (A) = A} is an ALexandrov topology. Proof: FUZZY INFORMATION AND ENGINEERING 31 (i) By (ii) of Theorem 3.3, we have R (1 ) = 1 ,1 ∈ τ . By (i) of Theorem 3.3, we have ¯ ¯ ¯ C I L L L β β R (0 ) ⊆ 0 , R (0 ) = 0 ,0 ∈ τ . ¯ ¯ ¯ ¯ ¯ I L L I L L L β β β (ii) Let A, B ∈ τ , by (iii) of Theorem 3.3, we have R (A ∩ B) = R (A) ∩ R (B) = A ∩ B, A ∩ I I I B ∈ τ . β β (iii) Let {A |i ∈ }∈ τ, by (iv) of Theorem 3.3, we have A = R (A ) ⊆ R ( A ). Hence, i i i i i∈ I I β β A ⊆ R ( A ).At the same time, by (vi) of Theorem 3.3, we have R ( A ) ⊆ i i i i∈ I i∈ I i∈ A . Hence, A = R ( A ). τ is an ALexandrov topology. i i i i∈ i∈ I i∈ Corollary 3.1: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. If β = 1 , then τ ={A ∈ ILF(U)|R (A) = A} is an ALexandrov topology. ¯ C II β β Proof: By Theorem 3.2(i), when β = 1 ,wehave R = R , hence τ is an ALexandrov L I II topology. 3.3. Some Other Rough Approximate Operators Based on I,II,III-type Rough Approximate Operators The Pawlak approximation operators satisfy many properties [[4],[5]]. When generaliz- ing Pawlak approximations, one task is to specify a subset of these properties that new approximation operators are required to preserve. Another task is to search for possible generalizations of the various notions used in the three definitions. Furthermore, one must make sure that the generalized approximation operators satisfy the required properties. In this paper, we consider the following properties suggested by PomykaIa [10]: for all A ∈ ILF (i) R (∼ A) =∼ R (A), (ii) R (A) ⊆ A ⊆ R (A). Definition 3.7: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space with C ={C , C , ... , C } being a fuzzy β-covering of U for some β ∈ L.For each A ∈ ILFS,we 1 2 m define other three kinds of new lower approximation operators and upper approximation operators of A as β β β β β β R (A)(x) = R (A)(x) ∧ R (A)(x), R (A)(x) = R (A)(x) ∨ R (A)(x), IV I II IV I II β β β β β β R (A)(x) = R (A)(x) ∧ R (A)(x), R (A)(x) = R (A)(x) ∨ R (A)(x), V I III V I III β β β β β β R (A)(x) = R (A)(x) ∧ R (A)(x), R (A)(x) = R (A)(x) ∨ R (A)(x). VI IV V VI IV V β β β β β β Then (R , R (A)), (R , R (A)),(R , R (A)) are called IV,V,VI-type rough operators of A, IV IV IV IV IV IV respectively. Theorem 3.7: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. Then 32 P. YU ET AL. β β (i) R (∼ A) =∼ R (A); IV IV β β (ii) R (∼ A) =∼ R (A); V V β β (iii) R (∼ A) =∼ R (A). VI VI Proof: We only prove (i), (ii) and (iii) can be proved similarly. β β β R (∼ A) = R (∼ A) ∧ R (∼ A) IV I II β β =∼ R (A)∧∼ R (A) I II β β =∼ (R (A) ∨ R (A)) I II =∼ R (A). IV Theorem 3.8: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. Then β β (i) R (A) ⊆ A ⊆ R (A); IV IV β β (ii) R (A) ⊆ A ⊆ R (A); V V β β (iii) R (A) ⊆ A ⊆ R (A). VI VI β β Proof: (i) By (vi) of Theorem 3.3, for every x ∈ U,wehave R (A)(x) ≤ A(x) ≤ R (A)(x). Then I I β β β β R (A)(x) = R (A)(x) ∧ R (A)(x) ≤ A(x) ∧ R (A)(x) ≤ A(x) IV I II II and β β β β R (A)(x) = R (A)(x) ∨ R (A)(x) ≥ A(x) ∨ R (A)(x) ≥ A(x). IV I II II β β Hence, R (A) ⊆ A ⊆ R (A). IV IV β β β β β β The relationships between R , R , R , R , R , R can be described as follows (Figure 2): IV IV V V VI VI Theorem 3.9: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. Then β β β β (i) R ≤ R , R ≤ R ; IV I V I β β β β (ii) R ≤ R , R ≤ R ; VI V VI IV β β β β (iii) R ≤ R , R ≤ R ; IV VI V VI β β β β (iv) R ≤ R , R ≤ R . I IV I V β β Remark 3.3: In Example 3.2, R (A) = (0, d)/x + (0, e)/x + (0, d)/x + (c, a)/x , R (A) = 1 2 3 4 V IV β β (0, e)/x + (0, e)/x + (0, e)/x + (b, b)/x . R (A)(x ) and R (A)(x ) are not comparable, 1 2 3 4 4 4 V Iv FUZZY INFORMATION AND ENGINEERING 33 Figure 2. The relationships between three new lower and upper approximation operators based on I,II,III-type Rough Approximate Operators. β β β hence R (A) and R (A) are incomparable. At the same time, in Example 3.2, R (A)(x ) = V Iv IV β β β β β (b, a) ≥ (b, c) = R (A)(x ) and R (A)(x ) = (b, e) ≤ (b, a) = R (A)(x ). R (A) and R (A) 3 4 4 V IV V V IV are also incomparable. 3.4. Matrix Representations of Lower and Upper Approximation Operators In this subsection, we will present matrix representations of the lower and upper approxi- mation operators defined in Definitions 3.3–3.5. The matrix representations of the approx- imation operators make it possible to calculate the lower and upper approximations of subsets through the operations on matrices, which is algorithmic, and can easily be implemented through the computer. 34 P. YU ET AL. Definition 3.8: Let A = (a ) and B = (b ) be two matrices. We define C = A B = ij n×m kj m×l (c ) and D = A B = (d ) as follows: ij n×l ij n×l c = (∼ a ∨ b ), i = 1, 2, ... , n, j = 1, 2, ... , l, ij ik kj k=1 d = (a ∧ b ), i = 1, 2, ... , n, j = 1, 2, ... , l. ij ik kj k=1 Obviously, for arbitrary A, B, C,if B ≤ C, then A B ≤ A C, B A ≥ C A, A B ≤ A C and B A ≤ C A. Meanwhile, A B = B A and A B = B A. Definition 3.9: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space. We β β β define matrix M , M , M as follows: I II III 1 , y ∈ N , β L M (x, y) = 0 , otherwise, β β N (y), y ∈ N , β x M (x, y) = II 0 , otherwise, β β M (x, y) = N (y). III Here N is defined as in Definition 3.2. β β β β From Definition 3.9, we can see that M  M and M  M . II I II III Theorem 3.10: Let (U, C) be an intuitionistic L-fuzzy β-covering approximation space, A ∈ β β β β T T ILF(U). Then R (A) = M A , R (A) = M A (i ∈{I, II, II}). i i i i Example 3.4: In Example 3.2, take β = (a, b), then ⎛ ⎞ ⎛ ⎞ 1 0 1 0 (a, b) 0 (a, b) 0 L L L L L L ⎜ ⎟ ⎜ ⎟ 1 1 1 0 (b,0)(b, a)(a, b) 0 β L L L L β L ⎜ ⎟ ⎜ ⎟ M = , M = , I II ⎝ ⎠ ⎝ ⎠ 1 0 1 0 (a, b) 0 (a, b) 0 L L L L L L 0 0 0 1 0 0 0 (b, b) L L L L L L L ⎛ ⎞ (a, b)(0, e)(a, b)(0, e) ⎜ ⎟ (b,0)(b, a)(a, b)(0, e) ⎜ ⎟ M = . III ⎝ ⎠ (a, b)(0, e)(a, b)(0, e) (0, c)(a, c)(a, d)(b, b) (0, d) (a, c) (b, a) (e, a) If A = + + + ,we have x x x x 1 2 3 4 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 1 0 (0, d) (0, d) L L L L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1 1 1 0 (a, c) (0, e) β β L L L L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ R (A) = M A = = , I I ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 1 0 1 0 (b, a) (0, d) L L L L 0 0 0 1 (e, a) (e, a) L L L L FUZZY INFORMATION AND ENGINEERING 35 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ (a, b) 0 (a, b) 0 (0, d) (b, a) L L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (b,0)(b, a)(a, b) 0 (a, c) (0, a) β β T L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = , R (A) = M A = II II ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (a, b) 0 (a, b) 0 (b, a) (b, a) L L 0 0 0 (b, b) (e, a) (e, a) L L L ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ (a, b)(0, e)(a, b)(0, e) (0, d) (b, a) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (b,0)(b, a)(a, b)(0, e) (a, c) (0, a) β β ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ R (A) = M A = = , III II ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (a, b)(0, e)(a, b)(0, e) (b, a) (b, a) (0, c)(a, c)(a, d)(b, b) (e, a) (c, a) ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 1 0 (0, d) (b, a) L L L L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 1 1 1 0 (a, c) (b, a) β β L L L L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ R (A) = M A = = , I I ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 1 0 1 0 (b, a) (b, a) L L L L 0 0 0 1 (e, a) (e, a) L L L L ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ (a, b) (a, b) 0 (a, b) 0 (0, d) L L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (b,0)(b, a)(a, b) 0 (a, c) (a, a) β β L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ R (A) = M A = = , II II ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (a, b) 0 (a, b) 0 (b, a) (a, b) L L 0 0 0 (b, b) (e, a) (b, b) L L L ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ (a, b)(0, e)(a, b)(0, e) (0, d) (a, b) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (b,0)(b, a)(a, b)(0, e) (a, c) (a, a) β β ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ R (A) = M A = = . III III ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (a, b)(0, e)(a, b)(0, e) (b, a) (a, b) (0, c)(a, c)(a, d)(b, b) (e, a) (b, a) 4. Conclusions The theories of covering rough sets, intuitionistic L-fuzzy sets, are important for dealing with uncertainty and inaccuracy problems. In order to handle these uncertainty and inaccu- racy problems more effectively, the combination of the covering and intuitionistic L-fuzzy sets is further researched in this paper. The concepts of intuitionistic L-fuzzy β-covering and intuitionistic L-fuzzy β-neighborhood are proposed. Several intuitionistic L-fuzzy β- covering rough set models are discussed. Some properties of these models are proved and demonstrated by some examples. The research results fill in the blanks of the study of the intuitionistic L-fuzzy rough set with the method of covering. We will investigate the applications of the presented models and construct multi-granulation intuitionistic L-fuzzy covering rough sets, which will be part of the future research directions considered by our group. Disclosure statement No potential conflict of interest was reported by the author(s). Notes on contributors Peng YU was born in 1981 in Ningxia, China. He received the Ph.D. degree in Mathematic from Shaanxi Normal University, xian, china in 2019. He is currently an associate professor of Shaanxi University of Science and Technology. He has published 10 research papers on fuzzy reasoning and rough set in 36 P. YU ET AL. international journals. His current research interests include Non-classic logic, fuzzy reasoning and rough set. Xiao-gang An was born in 1975 in Shaanxi, China. He received the master’s degree in computer science from Southwest Jiaotong University, Chongqing, china in 2005. He is currently a lecturer of Shaanxi University of Science and Technology. He has published three research papers on Logic algebra in international journals. His current research interests include Logic algebra and rough set. Xiao-hong Zhang was born in 1965 in Shaanxi, China. He received the Ph.D. degree in computer sci- ence from Northwestern Polytechnical University, xian, china in 2005. He is currently an fuzz professor of Shaanxi University of Science and Technology. He has published 60 research papers on logic alge- bra and rough set in international journals. His current research interests include The logical basis of artificial intelligence, approximate reasoning, Logic algebra and rough set. Funding Scientific Research Program Funded by Shaanxi Provincial Education Department [No. 18JK0099] and the Scientific Research Start-up Fund of Shaanxi University of Science and Technology [2019BJ-41]. References [1] Pawlak Z. Rough set. Int J Comput Inf Sci II. 1982;11(5):341–356. [2] Greco S, Matarazzo B, Slowinski R. 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Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Jan 2, 2020

Keywords: Intuitionistic L -fuzzy set; covering rough set; rough set; Alexandrov topolgy

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