In this paper, we investigate the properties of Alexandrov fuzzy topologies and meet-join approximation operators. We study fuzzy preorder, Alexandrov topologies and meet-join approximation operators induced by Alexandrov fuzzy topologies. We give their examples. Hájek [2] introduced a complete residuated lattice which is an algebraic structure for many valued logic. Höhle [3] introduced L -fuzzy topologies and L -fuzzy interior operators on complete residuated lattices. Pawlak [8 , 9] introduced rough set theory as a formal tool to deal with imprecision and uncertainty in data analysis. Radzikowska [10] developed fuzzy rough sets in complete residuated lattice. Bělohlávek [1] investigated information systems and decision rules in complete residuated lattices. Zhang [6 , 7] introduced Alexandrov L -topologies induced by fuzzy rough sets. Kim [5] investigated the properties of Alexandrov topologies in complete residuated lattices. In this paper, we investigate the properties of Alexandrov fuzzy topologies and meet-join approximation operators in a sense as Höhle [3] . We study fuzzy preorder, Alexandrov topologies and meet-join approximation operators induced by Alexandrov fuzzy topologies. We give their examples. Definition 2.1 ( [1 - 3] ). A structure ( L ,∨,∧,⊙, →, ⊥,⊤) is called a complete residuated lattice iff it satisfies the following properties: An operator * : L → L defined by a * = a → ⊥ is called strong negations if a ** = a . In this paper, we assume that ( L , ∨, ∧, ⊙, →, *, ⊥, ⊤) be a complete residuated lattice with a strong negation *. Definition 2.2 ( [6 , 7] ). Let X be a set. A function eX : X × X → L is called a fuzzy preorder if it satisfies the following conditions Example 2.3. (1) We define a function e_L : L × L → L as e_L ( x, y ) = x → y. Then e_L is a fuzzy preorder on L . Lemma 2.4 ( [1 , 2] ). Let ( L ,∨,∧,⊙, →,*, ⊥,⊤) be a complete residuated lattice with a strong negation *. For each x, y, z, x_i, y_i ∈ L, the following properties hold. Definition 2.5 ( [5] ). A map : L^X → L^Y is called an meet-join approximation operator if it satisfies the following conditions, for all A, A_i ∈ L^X , and α ∈ L , Definition 2.6 ( [4] ). An operator T : L^X → L is called an Alexandrov fuzzy topology on X iff it satisfies the following conditions, for all A, A_i ∈ L^X , and α ∈ L , Definition 2.7 ( [5] ). A subset τ ⊂ L^X is called an Alexandrov topology if it satisfies satisfies the following conditions. Remark 2.8. (1) If T : L^X → L is an Alexandrov fuzzy topology. Define T *( A ) = T ( A *). Then T * is an Alexandrov fuzzy topology. Theorem 3.1 . If is a meet-join approximation operator, then = {A ∈ L^X | ( A ) = A *} is an Alexandrov topology on X. Proof. (O1) Since ⊤ _X ≤ (⊥ _X ) and (⊤ _X ) = (⊥ _X → A) = ⊥ _X ⊙ ( A ) = ⊥, ⊥ _X = (⊤ _X ) and ⊤ _X = (⊤ _X ). Then ⊥ _X ;⊤ _X ∈ . Theorem 3.2.   Let   T   be an Alexandorv fuzzy topology on X. Define We have the following properties . Then   is a meet-join approximation operator on X with   for each s ≤ r. Proof. (1) Since T ( B ) ≥ r * iff then Since and Hence is a fuzzy preorder. Since and we have . Hence ⊂ ^* . Hence ( A ) = A *; i.e. . Thus Since and we have Hence So, Since A ≥ Since and So, , ≤ . Hence = for all A ∈ L^X and r ∈ L . then where Since then So, Thus Since s ≤ r_i , Thus                            ⧠ Theorem 3.3. Let   T be an Alexandorv fuzzy topology on X. We have the following properties. Then = T * is an Alexandrov fuzzy topology on X . Then = T * is an Alexandrov fuzzy topology on X . If r ≤ s, then   T ^r ≤ T ^s f or all A ∈ L^X . Moreover, T * ^r ( A ) = T ^r ( A *) for all A ∈ L^X . If r ≤ s , then   T * ^r ≤ T * ^s   for all A ∈ L^X . Then T _M = T * = T _MT   is an Alexandrov fuzzy topology on X. Then T _M* = T = T _MT* is an Alexandrov fuzzy topology on X. Proof. (1) We only show that T _MT = T *. Let = A *. Then form Theorem 3.3 (6). So T * ( A ) = T ( A *) = Thus, Since T *( A ) ≥ ( T ( A ))* then with s = T( A ). Thus, Hence T _MT = T *. Hence T ^r is an Alexandrov fuzzy topology. Since for r ≤ s , T ^s ( A ) = = T ^r ( A ). Example 3.4. Let ( L = [0, 1], ⊙, →, * ) be a complete residuated lattice with a strong negation. (1) Let X = { x, y, z } be a set. Define a map T : [0, 1] ^X → [0, 1] as Trivially, T ( α_X ) = 1 Since α ⊙ A ( x ) → α ⊙ A ( z ) ≥ A ( x ) → A ( z ) from Lemma 2.4 (14), T ( α ⊙ A ) ≥ T ( A ). Since ( α → A ( x )) → ( α → A ( z )) ≥ A ( x ) → A ( z ) from Lemma 2.4 (10), T ( α → A ) ≥ T(A). By Lemma 2.4 (8), T( A_i ) ≥ T ( A_i ) and T ( A_i ) ≥ T ( A_i ). Hence T is an Alexandrov fuzzy topology. If T ( A ) = A ( x ) → A ( z ) ≥ r*, then A ( z ) ≥ A ( x ) ⊙ r*. Put A ( x ) = 1, A ( y ) = 0. So, and similarly, we can obtain By Theorem 3.2(3), we obtain such that If A *( x ) ⊙ r * ≤ A *( z ), then Thus . Moreover, since T *( A ) = A *( x )→ A *( z ) ≥ r * iff A*(z) ≥ A *( x ) ⊙ r *, iff . So, From Theorem 3.3(1), we have Moreover, we obtain Hence T _M = T _MT = T *. Since B ( x ) = 1 and T ( B ) = 1 → B ( z ) = B( z ) ≥ r *, then Since B ( z ) = 1 and T ( B ) = B ( x ) → 1= 1, then Then (2) By (1), we obtain a map T* : [0, 1] ^Y → [0, 1] as Since T *( A ) = A ( z ) → A ( x ) ≥ r *, then A ( x ) ≥ A ( z ) ⊙ r *. Put A ( z ) = 1, A ( y ) = 0. So, and Moreover, for all x, y ∈ X . If A *( z ) ⊙ r * ≤ A *( x ), then If , then A *( z ) ⊙ r* ≤ A *( x ). Moreover, since T ( A ) = A ( x ) → A ( z ) ≥ r * iff A *( z ) ⊙ r * ≤ A *( z ), iff Thus Moreover, we obtain Hence T _M* = T _MT* = T . Since B ( x ) = 1 and T *( B ) = B ( z ) →1 = 1, then Since B ( z ) = 1 and T *( B ) = 1 → B ( x ) = B ( x ) ≥ r *, then B ( x ) ≥ r *. We have Then (3) Let ( L = [0, 1], ⊙, →, * ) be a complete residuated lattice with a strong negation defined by, for each n ∈ N , By (1) and (2), we obtain Since we have