In this paper, we investigate the relationships between fuzzy relations and Alexandrov L -topologies in complete residuated lattices. Moreover, we give their examples. Pawlak [9 , 10] introduced rough set theory as a formal tool to deal with impre- cision and uncertainty in data analysis. Hájek [3] introduced a complete residuated lattice which is an algebraic structure for many valued logic. Radzikowska [11] de- veloped fuzzy rough sets in complete residuated lattice. Bělohlávek [1] investigated information systems and decision rules in complete residuated lattices. Lai [7 , 8] in- troduced Alexandrov L -topologies induced by fuzzy rough sets. Algebraic structures of fuzzy rough sets are developed in many directions [1 - 13] . In this paper, we investigate the relationships between fuzzy relations and Alexan- drov L -topologies in complete residuated lattices. Moreover, we give their examples. In this paper, we assume ( L ,∧,∨,⊙, →, *⊥,⊤) is a complete residuated lattice with the law of negation;i.e. x** = x . For α ∈ L , A ,⊤ _x ∈ L^X , ( α → A )( x ) = α → A ( x ), ( α ⊙ A )( x ) = α ⊙ A ( x ) and ⊤ _x ( x ) = ⊤,⊤ _x ( x ) = ⊥, otherwise. where R^r ( x, y ) = Δ∨ R ( x, y ) and Δ ( x, y ) = ⊤ if x = y and Δ( x, y ) = ⊥ if x ≠ y . Moreover, Proof. (1) Let A * ∈ τ * for A ∈ τ . Since α ⊙ A * = ( α →A )* and α → A * = ( α ⊙ A )*, τ * is an Alexandrov topology on X . (2) (T1) Since ⊤ _X ( x ) ⊙ R ( x, y ) ≤ ⊤ _X ( y ) = ⊤ and ⊥ _X ( x ) ⊙ R ( x, y ) = ⊥ = ⊥ _X ( y ), Then⊥ _X ,⊤ _X ∈ τ_R . (T2) For A_i ∈ τ_R for each i ∈ Γ, since Similarly, (T3) For A ∈ τ_R , α ⊙ A ∈ τ_R . (T4) For A ∈ τ_R , by Lemma 2.3(5), since α ⊙ ( α → A ( x )) ⊙ R ( x, y ) ≤ A ( x ) ⊙ R ( x, y ) ≤ A ( y ), ( α → A ( x )) ⊙ R ( x, y ) ≤ α → A ( y ). Then α → A ∈ τ_R . Moreover A ∈ τ_R iff A * ∈ τ _R−1 from: (3) Define Then Rτ_R is a fuzzy preorder. Since B ∈ τ_R and B ( x ) ⊙ R ( x, y ) ≤ B ( y ), then R ( x, y ) ≤ B ( x ) → B ( y ). Hence R (x, y ) ≤ R_τR . If P is a fuzzy preorder with R ≤ P , for P_w ( x ) = P ( w, x ), then P_w ( x ) ⊙ R ( x, y ) ≤ P_w ( x ) ⊙ P ( x, y ) ≤ P_w ( y ). Hence P_w ∈ τ_R . Thus R_τR ( x, y ) = Thus, Since R^r ( x, y ) = Δ∨ R ( x , y ), we have ( R^r ) ⁿ ( x , x ) = ⊤ for each n ∈ N . So Since then Hence is a fuzzy preorder. If R ≤ P and P is fuzzy preorder, then R^r ≤ P and ( R^r ) ⁿ ≤ Pⁿ = P , thus, Hence (4) Put and Since A ∈ τ_R , Hence . Since Thus, A ∈ τ_R . Let A ∈ τ . Since Thus, A ∈ τ_R . Let A ∈ τ . Then Put A ( x ) = a_x . Then Let Then Thus, D ∈ τ . Hence τ_R = τ = τ ₁ . (5) Put and Since A ∈ τ_R , R_τR ( x, y ) → A ( y ) = → A ( y ) ≥ ( A ( x ) → A ( y )) → A ( y ) ≥ A ( x ). Hence Since , Thus, A ∈ η . Let A ∈ η . Since Thus, R ( x, y ) → A ( y ) ≥ A ( x ) iff A ( x ) ⊙ R ( x, y ) ≤ A ( y ). So, A ∈ τ_R . Let A ∈ η Then Put A ( y ) = b_y . Then Let Then Thus, A ∈ η . Hence τ_R = η = η ₁ . (6) It follows from iff (7) It follows from iff (8) Put Then B ∈ τ_R from: If A ≤ E and E ∈ τ_R , then B ≤ E from: Hence C_τR = B . (9) Let from If E ≤ A and E ∈ τ_R , then E ≤ B from: Hence I_τR = B . (11) Theorem 3.2. Let R_X and R_Y be fuzzy relations and f : X → Y a map with R_X ( x , y ) ≤ R_Y ( f ( x ), f ( y )) for all x , y ∈ X. Then the following equivalent conditions hold. Proof. (1) For all B ∈ τ_RY , f ⁻¹ ( B ) ∈ τ_RX from: (1) ⇔ (2) It follows from (1) and Theorem 3.1(2). (1) ⇒ (3) (1) ⇒ (5) (3) ⇒ (5) (5) ⇒ (7) By (5), put A = f ⁻¹ ( B ). Since f ( C_τRX ( f ⁻¹ ( B ))) ≤ C_τRY ( f ( f ⁻¹ ( B ))) ≤ C_τRY ( B ), we have C_τRX ( f ⁻¹ ( B )) ≤ f ⁻¹ ( C_τRX ( B )). (7) ⇒ (1) For all B ∈ τ_RY , C_τY ( B ) = B . Since C_τRX ( f ⁻¹ (B)) ≤ f ⁻¹ ( C_τRX ( B )) = f ⁻¹ ( B ), f ⁻¹ ( B ) ∈ τ_RX . (1) ⇒ (9) (9) ⇒ (1) For all B ∈ τ_RY , I_τY ( B ) = B . Since I_τRX ( f ⁻¹ ( B )) ≥ f ⁻¹ ( I_τRX ( B )) = f ⁻¹ ( B ), f ⁻¹ ( B ) ∈ τ_RX . Other cases are similarly proved. Example 3.3. Let ( L = [0, 1],⊙,→,* ) be a complete residuated lattice with the law of double negation defined by Let X = { a , b , c }, Y = { x , y , z } be sets and f : X → Y as follows: (1) Define R_X ∈ L^X×X , R_Y ∈ L^{Y ×Y} as follows Then R_X ( a , b ) ≤ R_Y ( f ( a ), f ( b )) for all a , b ∈ X . For n ≥ 2, and as follows: Then Moreover, Then R_τRX ( a, b ) ≤ R_τRX ( f ( a ), f ( b )) for all a , b ∈ X . (2) where a_i ∈ L and For For where b_i ∈ L and For For (3) where a_i ∈ L and where b_i ∈ L and (4) For A = (0.2. 0.8, 0.6) ∈ L^X ,