Alexandrov topologies are the topologies induced by relations. This paper addresses the properties of Alexandrov topologies as the extensions of strong topologies and strong cotopologies in complete residuated lattices. With the concepts of Zhang’s completeness, the notions are discussed as extensions of interior and closure operators in a sense as Pawlak’s the rough set theory. It is shown that interior operators are meet preserving maps and closure operators are join preserving maps in the perspective of Zhang’s definition. Pawlak [1 , 2] introduced the rough set theory as a formal tool to deal with imprecision and uncertainty in the data analysis. Hájek [3] introduced a complete residuated lattice which is an algebraic structure for many valued logic. By using the concepts of lower and upper approximation operators, information systems and decision rules are investigated in complete residuated lattices [3 - 7] . Zhang and Fan [8] and Zhang et al. [9] introduced the fuzzy complete lattice which is defined by join and meet on fuzzy partially ordered sets. Alexandrov topologies [7 , 10 - 12] were introduced the extensions of fuzzy topology and strong topology [13] . In this paper, we investigate the properties of Alexandrov topologies as the extensions of strong topologies and strong cotopologies in complete residuated lattices. Moreover, we study the notions as extensions of interior and closure operators. We give their examples. Definition 1.1. [3 , 4] An algebra ( L , ∧, ∨, ⊙, →, ⊥, 𝖳) is called a complete residuated lattice if it satisfies the following conditions: In this paper, we assume ( L , ∧, ∨, ⊙, →, ^∗ ⊥, 𝖳) is a complete residuated lattice with a negation; i.e., x ^∗∗ = x . For α ∈ L , A , 𝖳 _x ∈ L ^X , ( α → A )( x ) = α → A ( x ), ( α ⊙ A )( x ) = α ⊙ A ( x ) and 𝖳 _x ( x ) = 𝖳, 𝖳 _x ( x ) = ⊥, otherwise. Lemma 1.2. [3 , 4] For each x , y , z , x_i , y_i ∈ L , the following properties hold. Definition 1.3. [7 , 10 , 12 , 13] A subset τ ⊂ L ^X is called an Alexandrov topology if it satisfies: A subset τ ⊂ L^X satisfying (T1), (T3) and (T4) is called a strong topology if it satisfies: (ST) If A_i ∈ τ for i ∈ Γ, for each finite index Λ ⊂ Γ. A subset τ ⊂ L^X satisfying (T1), (T3) and (T4) is called a strong cotopology if it satisfies: (SC) If A_i ∈ τ for i ∈ Γ, for each finite index Λ ⊂ Γ. Remark 1.4. Each Alexandrov topology is both strong topology and strong cotopology. Definition 1.5. [8 , 9] Let X be a set. A function e_X : X × X → L is called: If e satisfies (E1) and (E2), ( X , e_X ) is a fuzzy preordered set. If e satisfies (E1), (E2) and (E3), ( X , e_X ) is a fuzzy partially ordered set. Example 1.6. (1) We define a function e_L^X : L ^X × L ^X → L as . Then ( L ^X , e_L^X ) is a fuzzy partially ordered set from Lemma 1.2 (8). (2) Let τ be an Alexandrov topology. We define a function e_τ : τ × τ → . Then ( τ , e_τ ) is a fuzzy partially ordered set. Definition 1.7. [8 , 9] Let ( X , e_X ) be a fuzzy partially ordered set and A ∈ L^X . Remark 1.8. [8 , 9] Let ( X , e_X ) be a fuzzy partially ordered set and A ∈ L^X . Remark 1.9. [8 , 9] Let ( L ^X , e_L^X ) be a fuzzy partially ordered and Φ ∈ L^LX . Theorem 2.1. (1) A subset τ ⊂ L^X is an Alexandrov topology on X iff for each Φ : τ → L , ⊔Φ ∈ τ and ⊓Φ ∈ τ . (2) τ is an Alexandrov topology on X iff τ ^∗ = { A ^∗ ∈ L^X | A ∈ τ } is an Alexandrov topology on X . Proof. (1) (⇒) For each Φ : τ → L , we define Since τ is an Alexandrov topology on X , (Φ( A ) ⊙ A ) ∈ τ . Thus P ∈ τ . Then P = ⊔Φ from: For each Φ : τ → L , we define . Since τ is an Alexandrov topology on X , (Φ( A ) → A ) ∈ τ . Thus Q ∈ τ . Then Q = ⊓Φ from: (⇒) (T1) For Φ( A ) = ⊥ for all A ∈ τ , and . (T2) Let Φ( A_i ) = 𝖳 for all { A_i | i ∈ Γ} ⊂ τ , otherwise Φ( A ) = ⊥. We have (T3) Let Φ( A ) = ⊥ for A = B ∈ τ , otherwise Φ( A ) = α if A ≠ B . We have (2) Let A ^∗ ∈ τ ^∗ for A ∈ τ . Since α ⊙ A ^∗ = ( α → A ) ^∗ and α → A ^∗ = ( α ⊙ A ) ^∗ , τ ^∗ is an Alexandrov topology on X . Theorem 2.2. Let τ be an Alexandrov topology on X . Define I_τ : L^X → L^X as follows: Then the following properties hold. Proof . (1) By Lemma 1.2 (8,10,14), we have (2) Since e_L^X ( C , A )⊙ C ≤ A from Lemma 1.2 (9), I_τ ( A ) ≤ A . (3) Since I_τ ( A ) ∈ τ , then I_τ ( I_τ ( A )) ≥ e_L^X ( I_τ ( A ), I_τ ( A )) ⊙ I_τ ( A ) = I_τ ( A ). By (2), I_τ ( I_τ ( A )) = I_τ ( A ). (4) Since α → I_τ ( A ) ≤ α → A and α → I_τ ( A ) ∈ τ , (5) By (1), since I_τ ( A ) ≤ I_τ ( B ) for . Since and , we have (6) For each Φ : L^X → L , put . Since is a map, we have and from: (7) . Since I ( A ) ≤ A and I ( A ) ∈ τ , we have . Since I_τ ( A ) ≤ A and I_τ ( A ) ∈ τ , we have I ( A ) ≥ I_τ ( A ). (8) It follows from A ∈ τ iff I_τ ( A ) = A iff A ∈ τ_Iτ . (9) Since , by (4) and (5), . Put . Then Hence e_X is a fuzzy preorder. Theorem 2.3. Let τ be an Alexandrov topology on X . Define C_τ : L^X → L^X as follows: Then the following properties hold. Proof . (1) By Lemma 1.2 (8,10), we have (2) Since e_L^X ( A , B ) ⊙ A ≤ B iff A ≤ e_L^X ( A , B ) → B , then A ≤ C_τ ( A ). (3) Since C_τ ( A ) ∈ τ , then C_τ ( C_τ ( A )) ≤ e_L^X ( C_τ ( A ), C_τ ( A )) → C_τ ( A ) = C_τ ( A ). By (2), C_τ ( C_τ ( A )) = C_τ ( A ). (4) Since α ⊙ A ≤ α ⊙ C_τ ( A ) and α ⊙ C_τ ( A ) ∈ τ , C_τ ( α ⊙ A ) ≤ e_L^X ( α ⊙ A , α ⊙ C_τ ( A )) → α ⊙ C_τ ( A ) = α ⊙ C_τ ( A ). (5) By (1), since C_τ ( A ) ≤ C_τ ( B ) for A ≤ B , . Since and we have (6) For each Φ : L^X → L , put . Since is a map, we have and from: (7) Put . Since A ≤ C ( A ) and C ( A ) ∈ τ , we have Since A ≤ C_τ ( A ) and C_τ ( A ) ∈ τ , we have C ( A ) ≤ C_τ ( A ). (8) It follows from A ∈ τ iff C_τ ( A ) = A iff A ∈ τ_Cτ . (9) (10) Since , by (4) and (5), . Put e_X ( x , y ) = C_τ (𝖳 _x )( y ). Then Hence e_X is a fuzzy preorder. Since , by Theorem 2.2(9), Example 2.4. Let ( L = [0, 1], ⊙, →, ^∗ ) be a complete residuated lattice with a negation defined by x ⊙ y = ( x + y −1)∨0, x → y = (1− x + y )∧1, x ^∗ = 1− x . Let X = { x , y , z } be a set and A ₁ = (1, 0.8, 0.6), A ₂ = (0.7, 1, 0.7), A ₃ = (0.5, 0.7, 1). (1) We define where (T1) For ⊥ _X ∈ L^X , e_X (⊥ _X ) = ⊥ _X ∈ τ . For 𝖳 _X ∈ L^X , e_X (𝖳 _X ) = 𝖳 _X ∈ τ . (T2) For e_X ( A_i ) ∈ τ for each i ∈ Γ, . Moreover, since e_X ( A )( x ) ≥ e_X ( x , x ) ⊙ A ( x ) = A ( x ) and e_X ( e_X ( A )) = e_X ( A ), Hence . (T3) For e_X ( A ) ∈ τ , α ⊙ e_X ( A ) = e_X ( α ⊙ A ) ∈ τ . (T4) Since α ⊙ e_X ( α → e_X ( A )) ≤ e_X ( e_X ( A )) = e_X ( A ), we have α → e_X ( A ) ≤ e_X ( α → e_X ( A )) ≤ α → e_X ( A ) Hence, for e_X ( A ) ∈ τ , α → e_X ( A ) = e_X ( α → e_X ( A )) ∈ τ . Hence τ is an Alexandrov topology on X . (2) For B ₁ = (0.7, 0.3, 0.6), B ₁ = (0.5, 0.9, 0.3), we obtain Let Φ : L^X → L as follows Thus, . Thus, . (3) We define For B ₁ , B ₂ and Φ in (2), we obtain Since ⊓Φ = (0.7, 0.4, 0.5) and we have . Since ⊔Φ = (0.6, 0.7, 0.5) and then . The fuzzy complete lattice is defined with join and meet operators on fuzzy partially ordered sets. Alexandrov topologies are the extensions of fuzzy topology and strong topology. Several properties of join and meet operators induced by Alexandrov topologies in complete residuated lattices have been elicited and proved. In addition, with the concepts of Zhang’s completeness, some extensions of interior and closure operators are investigated in the sense of Pawlak’s rough set theory on complete residuated lattices. It is expected to find some interesting functorial relationships between Alexandrov topologies and two operators. 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 Young Sun Kim received the B.S., M.S. and Ph.D. degrees in Mathematics from Yonsei University, Seoul, Korea, in 1981, 1985 and 1991, respectively. He is currently Professor of Pai Chai University. His research interests is a fuzzy topology and fuzzy logic. E-mail: yskim@pcu.ac.kr