In this paper, we investigate the properties of L -lower approximation operators as a generalization of fuzzy rough set in complete residuated lattices. We study relations lower (upper, join meet, meet join) approximation operators and Alexandrov L -topologies. Moreover, we give their examples as approximation operators induced by various L -fuzzy relations. Pawlak [1 , 2] introduced rough set theory as a formal tool to deal with imprecision and uncertainty in data analysis. Hájek [3] introduced a complete residuated lattice which is an algebraic structure for many valued logic. Radzikowska and Kerre [4] developed fuzzy rough sets in complete residuated lattice. Bělohlávek [5] investigated information systems and decision rules in complete residuated lattices. Lai and Zhang [6 , 7] introduced Alexandrov L -topologies induced by fuzzy rough sets. Kim [8 , 9] investigate relations between lower approximation operators as a generalization of fuzzy rough set and Alexandrov L -topologies. Algebraic structures of fuzzy rough sets are developed in many directions [4 , 8 , 10] In this paper, we investigate the properties of L -lower approximation operators as a generalization of fuzzy rough set in complete residuated lattices. We study relations lower (upper, join meet, meet join) approximation operators and Alexandrov L -topologies. Moreover, we give their examples as approximation operators induced by various L -fuzzy relations. Definition 1.1. [3 , 5] An algebra ( L ,∧,∨,⊙,→,⊥,⊤) is called a complete residuated lattice if it satisfies the following conditions: (C1) L = ( L ,≤,∨,∧,⊥,⊤) is a complete lattice with the greatest element ⊤ and the least element ⊥; (C2) ( L ,⊙,⊤) is a commutative monoid; C3) x ⊙ y ≤ z iff x ≤ y → z for x , y , z ∈ L Remark 1.2. [3 , 5] (1) A completely distributive lattice L = ( L ,≤,∨,∧ = ⊙,→, 1, 0) is a complete residuated lattice defined by (2) The unit interval with a left-continuous t-norm ⊙, is a complete residuated lattice defined by In this paper, we assume ( L ,∧,∨,⊙,→,* ⊥,⊤) is a complete residuated lattice with the law of double negation;i.e. x ** = x . For 𝛼 ∈ L , A ,⊤ _x ∈ L^X , and Lemma 1.3. [3 , 5] For each x, y, z, x_i, y_i ∈ L , we have the following properties. Definition 1.4. [8 , 9] (1) A map H : L^X → L^X is called an L-upper approximation operator iff it satisfies the following conditions (2) A map 𝒥 : L^X → L^X is called an L-lower approximation operator iff it satisfies the following conditions (3) A map K : L^X → L^X is called an L-join meet approximation operator iff it satisfies the following conditions (4) A map M : L^X → L^X is called an L-meet join approximation operator iff it satisfies the following conditions Definition 1.5. [6 , 9] A subset 𝜏 ⊂ L^X is called an Alexandrov L-topology if it satisfies: Theorem 1.6. [8 , 9] (1) 𝜏 is an Alexandrov topology on X iff 𝜏 _⁎ = { A * ∈ L^X | A ∈ 𝜏} is an Alexandrov topology on X . (2) If H is an L -upper approximation operator, then 𝜏 _H = { A ∈ L^X | H ( A ) = A } is an Alexandrov topology on X . (3) If 𝒥 is an L -lower approximation operator, then 𝜏 _𝒥 = { A ∈ L^X | 𝒥 ( A ) = A } is an Alexandrov topology on X . (4) If K is an L -join meet approximation operator, then 𝜏 _K = { A ∈ L^X | K ( A ) = A *} is an Alexandrov topology on X . (5) If M is an L -meet join operator, then 𝜏 _M = { A ∈ L^X | M ( A ) = A *} is an Alexandrov topology on X . Definition 1.7. [8 , 9] Let X be a set. A function R : X × X → L is called: If R satisfies (R1) and (R3), R is called a L-fuzzy preorder . If R satisfies (R1), (R2) and (R3), R is called a L-fuzzy equivalence relation Theorem 2.1. Let 𝒥 : L^X → L^X be an L -lower approximation operator. Then the following properties hold. (1) For A ∈ L^X , . (2) Define H_J ( B ) = ∧{ A | B ≤ 𝒥 ( A )}. Then H_J : L^X → L^X with is an L -upper approximation operator such that ( H_J ,𝒥 ) is a residuated connection;i.e., H_J (B) ≤ A iff B ≤ 𝒥 (A). Moreover, 𝜏 _𝒥 = 𝜏 _HJ . (3) If 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A ∈ L^X , then H_J ( H_J ( A )) = H_J ( A ) for A ∈ L^X such that 𝜏 _𝒥 = 𝜏 _HJ with (4) If 𝒥 (𝒥 *( A )) = 𝒥 *( A ) for A ∈ L^X , then 𝒥 (𝒥 ( A )) = 𝒥 ( A ) such that (5) Define H_s ( A ) = 𝒥 ( A *)*. Then Hs : L^X → L^X with is an L -upper approximation operator. Moreover, 𝜏 _Hs = (𝜏 _𝒥 ) _* = (𝜏 _HJ ) _* . (6) If 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A ∈ L^X , then H_s(H_s(A)) = H_s(A) for A ∈ L^X such that 𝜏 _Hs = (𝜏 _𝒥 ) _* = (𝜏 _HJ ) _* . with (7) If 𝒥 (𝒥 *( A )) = 𝒥 *( A ) for A ∈ L^X , then such that (8) Define K_J ( A ) = 𝒥 ( A *). Then K_J : L^X → L^X with is an L -join meet approximation operator. (9) If 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A ∈ L^X , then for A ∈ L^X such that 𝜏 _KJ = (𝜏 _𝒥 ) _* with (10) If 𝒥 (𝒥 *( A )) = 𝒥 *( A ) for A ∈ L^X , then such that (11) Define M_J ( A ) = (𝒥 ( A ))*. Then M_J : L^X → L^X with is an L -meet join approximation operator. Moreover, 𝜏 _MJ = 𝜏 _𝒥 . (12) If 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A ∈ L^X , then for A ∈ L^X such that 𝜏 _MJ = (𝜏 _𝒥 ) _* with (13) If 𝒥 (𝒥 *( A )) = 𝒥 *( A ) for A ∈ L^X , then such that (14) Define K_HJ ( A ) = ( H_J ( A ))*. Then K_HJ : L^X → L^X with is an L -meet join approximation operator. Moreover, 𝜏 _KHJ = 𝜏 _𝒥 . (15) If 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A ∈ L^X , then for A ∈ L^X such that 𝜏 _KHJ = (𝜏 _𝒥 ) _* with (16) If for A ∈ L^X , then such that (17) Define M_HJ ( A ) = H_J ( A *). Then M_HJ : L^X → L^X with is an L -join meet approximation operator. Moreover, 𝜏 _MHJ = (𝜏 _𝒥 ) _* . (18) If 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A ∈ L^X , then for A ∈ L^X such that 𝜏 _MHJ = (𝜏 _𝒥 ) _* with (19) If for A ∈ L^X , then such that (20) ( K_HJ , K_J ) is a Galois connection;i.e, A ≤ K_HJ (B) iff B ≤ K_J (A). Moreover, 𝜏 _KJ = (𝜏 _KHJ ) _* . (21) ( M_J , M_HJ ) is a dual Galois connection;i.e, M_HJ (A) ≤ B iff M_J (B) ≤ A. Moreover, 𝜏 _MJ = (𝜏 _MHJ ) _* . Proof. (1) Since , by (J2) and (J3), (2) Since iff , we have (H1) Since H_J ( A ) ≤ H_J ( A ) iff A ≤ 𝒥 ( H_J ( A )), we have A ≤ 𝒥 ( H_J ( A )) ≤ H_J ( A ). (H3) By the definition of H_J , since H_J ( A ) ≤ H_J ( B ) for B ≤ A , we have Since 𝒥 (∨ _i∈𝚪 H_J ( A_i )) ≥ 𝒥 ( H_J ( A_i )) ≥ A_i , then 𝒥 (∨ _i∈𝚪 H_J ( A_i )) ≥ ∨ _i∈𝚪 A_i . Thus Thus H_J : L^X → L^X is an L -upper approximation operator. By the definition of H_J , we have H_J (B) ≤ A iff B ≤ 𝒥 (A). Since A ≤ 𝒥 ( A ) iff H_J ( A ) ≤ A , we have 𝜏 _HJ = 𝜏 _𝒥 . (3) Let 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A ∈ L^X . Since 𝒥 ( B ) ≥ H_J ( A ) iff 𝒥 (𝒥 ( B )) = 𝒥 ( B ) ≥ A from the definition of H_J , we have (4) Let 𝒥 *( A ) ∈ 𝜏 _𝒥 . Since 𝒥 (𝒥 *( A )) = 𝒥 *( A ), 𝒥 (𝒥 ( A )) = 𝒥 (𝒥 *(𝒥 *( A ))) = (𝒥 (𝒥 *( A )))* = 𝒥 ( A ). Hence 𝒥 ( A ) ∈ 𝜏 _𝒥 ; i.e. 𝒥 *( A ) ∈ (𝜏 _𝒥 ) _* . Thus, 𝜏 _𝒥 ⊂ (𝜏 _𝒥 ) _* . Let A ∈ (𝜏 _𝒥 ) _* . Then A * = 𝒥 ( A *). Since 𝒥 ( A ) = 𝒥 (𝒥 *( A *)) = 𝒥 *( A *) = A , then A ∈ 𝜏 _𝒥 . Thus, (𝜏 _𝒥 ) _* ⊂ 𝜏 _𝒥 . (5) (H1) Since 𝒥 ( A *) ≤ A *, H_s ( A ) = 𝒥 ( A *)* ≥ A . Hence H_s is an L -upper approximation operator such that Moreover, 𝜏 _Hs = (𝜏 _𝒥 ) _* from: A = H_s(A) iff A = 𝒥 (A*)* iff A* = 𝒥 (A*). (6) Let 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A ∈ L^X . Then Hence (7) Let 𝒥 (𝒥 *( A )) = 𝒥 *( A ) for A ∈ L^X . Then Hence By a similar method in (4), 𝜏 _Hs = (𝜏 _Hs ) _* . (8) It is similarly proved as (5). (9) If 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A ∈ L^X , then (10) If 𝒥 (𝒥 *( A )) = 𝒥 *( A ) for A ∈ L^X , then Since , Hence 𝜏 _KJ = { K_J ( A ) | A ∈ L^X } = (𝜏 _KJ ) _* . (11) , (12), (13) and (14) are similarly proved as (5), (9), (10) and (5), respectively. (15) If 𝒥 (𝒥 ( A )) = 𝒥 ( A ) for A ∈ L^X , then H_J ( H_J ( A )) = H_J ( A ). Thus, Since 𝒥 ( A ) = A iff H_J ( A ) = A iff K_HJ ( A ) = A *, 𝜏 _KHJ = (𝜏 _𝒥 ) _* with (16) If for A ∈ L^X , then (17) , (18) and (19) are similarly proved as (14), (15) and (16), respectively. (20) ( K_HJ , K_J ) is a Galois connection;i.e, A ≤ K_HJ (B) iff A ≤ (H_J (B))* iff H_J (B) ≤ A* iff B ≤ 𝒥 (A*) = K_J (A) Moreover, since A * ≤ K_J ( A ) iff A ≤ K_HJ ( A *), 𝜏 _KJ = (𝜏 _KHJ ) _* . (21) ( M_J , M_HJ ) is a dual Galois connection;i.e, M_HJ (A) ≤ B iff H_J (A*) ≤ B iff A* ≤ 𝒥 (B) iff M_J (B) = (𝒥 (B))* ≤ A. Since M_HJ ( A *) ≤ A iff M_J ( A ) ≤ A *, 𝜏 _MJ = (𝜏 _MHJ ) _* . Let R ∈ L ^{X × X} be an L -fuzzy relation. Define operators as follows Example 2.2. Let R be a reflexive L -fuzzy relation. Define 𝒥 _R : L^X → L^X as follows: (1) (J1) 𝒥 _R ( A )( y ) ≤ R ( y, y ) → A ( y ) = A ( y ): 𝒥 _R satisfies the conditions (J1) and (J2) from: Hence 𝒥 _R is an L -lower approximation operator. (2) Define H_JR ( B ) = ∨ { A | B ≤ 𝒥 _R ( A )}. Since then By Theorem 2.1(2), H_JR = H _R^-1 is an L -upper approximation operator such that ( H_JR ,𝒥 _R ) is a residuated connection;i.e., H_JR(A) ≤ B iff A ≤ 𝒥_R(B). Moreover, 𝜏 _HJR = 𝜏 _𝒥R . (3) If R is an L -fuzzy preorder, then R ^-1 is an L -fuzzy preorder. Since By Theorem 2.1(3), H_JR ( H_JR ( A )) = H_JR ( A ): By Theorem 2.1(3), 𝜏 _HJR = 𝜏 _𝒥R with (4) Let R be a reflexive and Euclidean L -fuzzy relation. Since R ( x, z ) ⊙ R ( y, z ) ⊙ A *( x ) ≤ R ( x, y ) ⊙ A *( x ) iff R ( x, z ) ⊙ A *( x ) ≤ R ( y, z ) → R ( x, y ) ≤ A *( x ), Thus, . By Theorem 2.1(4), 𝒥 _R (𝒥 _R ( A )) = 𝒥 _R ( A ) for A ∈ L^X . Thus, 𝜏 _𝒥R = (𝜏 _𝒥R ) _* with (5) Define H_s ( A ) = 𝒥 _R ( A *)*. By Theorem 2.1(5), H_s = H_R is an L -upper approximation operator such that Moreover, 𝜏 _Hs = 𝜏 _HR = (𝜏 _HJR ) _* . (6) If R is an L -fuzzy preorder, then 𝒥 _R (𝒥 _R ( A )) = 𝒥 _R ( A ) for A ∈ L^X . By Theorem 2.1(6), then H_s ( H_s ( A )) = H_s ( A ) for A ∈ L^X such that 𝜏 H_s = (𝜏 _𝒥R ) _* = (𝜏 _HJR ) _* with (7) If R is a reflexive and Euclidean L -fuzzy relation, then for A ∈ L^X . By Theorem 2.1(7), such that (8) Define K_JR ( A ) = 𝒥 _R ( A *). Then K_JR : L^X → L^X with is an L -join meet approximation operator. Moreover, 𝜏 _KJR = (𝜏 _𝒥R ) _* . (9) R is an L -fuzzy preorder, then 𝒥 _R (𝒥 _R ( A )) = 𝒥 _R ( A ) for A ∈ L^X . By Theorem 2.1(9), for A ∈ L^X such that 𝜏 _KJR = (𝜏 _𝒥R ) _* with (10) If R is a reflexive and Euclidean L -fuzzy relation, then for A ∈ L^X . By Theorem 2.1(10), such that (11) Define M_JR ( A ) = (𝒥 _R ( A ))*. Then M_JR : L^X → L^X with is an L -join meet approximation operator. Moreover, 𝜏 _MJR = 𝜏 _𝒥R . (12) If R is an L -fuzzy preorder, then 𝒥 _R (𝒥 _R ( A )) = 𝒥 _R ( A ) for A ∈ L^X . By Theorem 2.1(12), for A ∈ L^X such that 𝜏 _MJR = 𝜏 _𝒥R with (13) If R is a reflexive and Euclidean L -fuzzy relation, then for A ∈ L^X . By Theorem 2.1(13), such that (14) Define K_HJR ( A ) = ( H_JR ( A ))*. Then K_HJR : L^X → L^X with is an L -join meet approximation operator. Moreover, 𝜏 _KR^-1 = 𝜏 _𝒥R = 𝜏 _HR^-1 . (15) If R is an L -fuzzy preorder, then 𝒥 _R (𝒥 _R ( A )) = 𝒥 _R ( A ) for A ∈ L^X . By Theorem 2.1(15), for A ∈ L^X such that 𝜏 _KR^-1 = 𝜏 _𝒥R = 𝜏 _HR^-1 with (16) Let R ^-1 be a reflexive and Euclidean L -fuzzy relation. Since we have Thus, Hence By (K1), such that (17) Define M_HJR ( A ) = H_JR ( A *). Then M_HJR : L^X → L^X is an L -meet join approximation operator as follows: Moreover, 𝜏 _MHJR = (𝜏 _𝒥R ) _* . (18) If R is an L -fuzzy preorder, then 𝒥 _R (𝒥 _R ( A )) = 𝒥 _R ( A ) for A ∈ L^X . By Theorem 2.1(18), for A ∈ L^X such that 𝜏 _MHJR = (𝜏 _𝒥 ) _* with (19) Let R ^-1 be a reflexive and Euclidean L -fuzzy relation. Since then ( R ( y, x ) → A ( x )) ⊙ R ( z, y ) ≤ R ( z, x ) → A ( x ). Thus, By (M1), such that (20) ( K_HJR = K _R^-1* , K_JR = K _R* ) is a Galois connection; i.e, A ≤ K_HJR ( B ) iff B ≤ K_JR ( A ): Moreover, 𝜏 _KJR = (𝜏 _KHJR ) _* . (21) ( M_JR = M_R , M_HJR = M _R^-1 ) is a dual Galois connection; i.e, M_HJR ( A ) ≤ B iff M_JR ( B ) ≤ A . Moreover, 𝜏 _MJR = (𝜏 _MHJR ) _* . In this paper, L -lower approximation operators induce L -upper approximation operators by residuated connection. We study relations lower (upper, join meet, meet join) approximation operators, Galois (dual Galois, residuated, dual residuated) connections and Alexandrov L -topologies. Moreover, we give their examples as approximation operators induced by various L -fuzzy relations. No potential conflict of interest relevant to this article was reported. Yong Chan Kim received the B.S., M.S. and Ph.D. degrees in Mathematics from Yonsei University, Seoul, Korea, in 1982, 1984 and 1991, respectively. He is currently Professor of Gangneung-Wonju University, his research interests is a fuzzy topology and fuzzy logic. E-mail: yck@gwnu.ac.kr