Mathematica Aeterna, Vol. 4, 2014, no. 3, 245 - 256 Alexandrov fuzzy topologies and fuzzy preorders Yong Chan Kim Department of Mathematics, Gangneung-Wonju National University, Gangneung, Gangwondo 210-702, Korea [email protected] Abstract In this paper, we investigate the properties of Alexandrov fuzzy topologies and upper approximation operators. We study fuzzy pre- order, Alexandrov topologies upper approximation operators induced by Alexandrov fuzzy topologies. We give their examples. Mathematics Subject Classification: 03E72, 03G10, 06A15, 54F05 Keywords: Complete residuated lattices, fuzzy preorder, upper approximation opera- tors, Alexandrov (fuzzy) topologies 1 Introduction H´ajek [2] introduced a complete residuated lattice which is an algebraic struc- ture for many valued logic. H¨ohle [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. Bˇelohl´avek [1] investigated information systems and deci- sion 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 upper approximation operators in a sense as H¨ohle [3]. We study fuzzy preorder, Alexandrov topologies upper approximation operators induced by Alexandrov fuzzy topologies. We give their examples. 246 Yong Chan Kim 2 Preliminaries Definition 2.1. [1-3] A structure (L, ∨, ∧, ⊙, →, ⊥, ⊤) is called a complete residuated lattice iff it satisfies the following properties: (L1) (L, ∨, ∧, ⊥, ⊤) is a complete lattice where ⊥ is the bottom element and ⊤ is the top element; (L2) (L, ⊙, ⊤) is a monoid; (L3) It has an adjointness,i.e. x ≤ y → z iff x ⊙ y ≤ z. An operator ∗ : L → L defined by a∗ = a → ⊥ is called strong negations if a∗∗ = a. ⊤, if y = x, ∗ ⊥, if y = x, ⊤x(y)= ⊤x(y)= ⊥, otherwise. ⊤, otherwise. In this paper, we assume that (L, ∨, ∧, ⊙, →,∗ , ⊥, ⊤) be a complete resid- uated 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 (E1) reflexive if eX (x, x) = 1 for all x ∈ X, (E2) transitive if eX (x, y) ⊙ eX (y, z) ≤ eX (x, z), for all x, y, z ∈ X’ Example 2.3. (1) We define a function eL : L × L → L as eL(x, y)= x → y. Then eL is a fuzzy preorder on L. X X (2) We define a function eLX : L ×L → L as eLX (A, B)= x∈X (A(x) → X B(x)). Then eL is a fuzzy preorder from Lemma 2.4 (9). Lemma 2.4. [1,2] Let (L, ∨, ∧, ⊙, →,∗ , ⊥, ⊤) be a complete residuated ∗ lattice with a strong negation . For each x,y,z,xi,yi ∈ L, the following properties hold. (1) If y ≤ z, then x ⊙ y ≤ x ⊙ z. (2) If y ≤ z, then x → y ≤ x → z and z → x ≤ y → x. (3) x → y = ⊤ iff x ≤ y. (4) x → ⊤ = ⊤ and ⊤→ x = x. (5) x ⊙ y ≤ x ∧ y. (6) x ⊙ ( i∈Γ yi)= i∈Γ(x ⊙ yi) and ( i∈Γ xi) ⊙ y = i∈Γ(xi ⊙ y). (7) x → ( i∈Γ yi)= i∈Γ(x → yi) and ( i∈Γ xi) → y = i∈Γ(xi → y). (8) i∈Γ xi → i∈Γ yi ≥ i∈Γ(xi → yi) and i∈Γ xi → i∈Γ yi ≥ i∈Γ(xi → yi). (9) (x → y) ⊙ x ≤ y and (y → z) ⊙ (x → y) ≤ (x → z). (10) x → y ≤ (y → z) → (x → z) and x → y ≤ (z → x) → (z → y). Alexandrov fuzzy topologies and fuzzy preorders 247 ∗ ∗ ∗ ∗ (11) i∈Γ xi =( i∈Γ xi) and i∈Γ xi =( i∈Γ xi) . ∗ ∗ (12) (x ⊙ y) → z= x → (y →z)= y → (x → z) and (x ⊙ y) = x → y . (13) x∗ → y∗ = y → x and (x → y)∗ = x ⊙ y∗. (14) y → z ≤ x ⊙ y → x ⊙ z. Definition 2.5. [5] A map H : LX → LY is called an upper approximation X operator if it satisfies the following conditions, for all A, Ai ∈ L , and α ∈ L, (H1) H(α ⊙ A)= α ⊙ H(A), (H2) H( i∈I Ai)= i∈I H(Ai), (H3) A ≤ H(A), (H4) H(H(A)) ≤ H(A). Definition 2.6. [4] An operator T : LX → L is called an Alexandrov fuzzy X topology on X iff it satisfies the following conditions, for all A, Ai ∈ L , and α ∈ L, (T1) T(α)= ⊤, (T2) T( i∈Γ Ai) ≥ i∈Γ T(Ai) and T( i∈Γ Ai) ≥ i∈Γ T(Ai), (T3) T(α ⊙ A) ≥ T(A), (T4) T(α → A) ≥ T(A). Definition 2.1 [5] A subset τ ⊂ LX is called an Alexandrov topology if it satisfies satisfies the following conditions. (O1) ⊥X , ⊤X ∈ τ where ⊤X (x)= ⊤ and ⊥X (x)= ⊥ for x ∈ X. (O2) If Ai ∈ τ for i ∈ Γ, i∈Γ Ai, i∈Γ Ai ∈ τ. (O3) α ⊙ A ∈ τ for all α ∈ L andA ∈ τ. (O4) α → A ∈ τ for all α ∈ L and A ∈ τ. 3 Alexandrov fuzzy topologies and fuzzy pre- orders Theorem 3.1 (1) If T : LX → L is an Alexander fuzzy topology. Define T∗(A)= T(A∗). Then T∗ is an Alexander fuzzy topology. r X (2) If T be an Alexandrov fuzzy topology on X, τT = {A ∈ L | T(A) ≥ r} r s is an Alexandrov topology on X and τT ⊂ τT for s ≤ r ∈ L. X (3) If H is an L-upper approximation operator, then τH = {A ∈ L | H(A)= A} is an Alexandrov topology on X. Proof. (1) (T1) T∗(α)= T((α∗)= ⊤. ∗ ∗ ∗ ∗ ∗ (T2) T ( i∈Γ Ai)= T( i∈Γ Ai ) ≥ i∈Γ T(Ai )= i∈Γ T (Ai) and T ( i∈Γ Ai)= ∗ ∗ ∗ T( i∈Γ Ai ) ≥ i∈Γ T(Ai )= i∈Γ T (Ai). 248 Yong Chan Kim (T3) T∗(α ⊙ A) = T((α ⊙ A)∗)= T(α → A∗) ≥ T(A∗)= T∗(A). (T4) T∗(α → A) = T((α → A)∗)= T((α ⊙ A∗)∗∗) = T(α ⊙ A∗) ≥ T(A∗)= T∗(A). Hence T∗ is an Alexander fuzzy topology. r (2) (O1) Since T(⊤X )= T(⊥X )= ⊤ ≥ r, ⊥X , ⊤X ∈ τT . r (O2) For Ai ∈ τT for each i ∈ Γ, by (H3), T( i∈Γ Ai) ≥ i∈Γ T(Ai) ≥ r and T( i∈Γ Ai) ≥ i∈Γ T(Ai) ≥ r. r Thus, i∈Γ Ai, i∈Γ Ai ∈ τT . r (O3) and (O4), For A ∈ τT , by (H2), T(α ⊙ A) ≥ T(A) and T(α → A) ≥ r T(A). So, α ⊙ A, α → A ∈ τT . r r Hence τT is an Alexandrov topology on X. Moreover, for A ∈ τT with s r s s ≤ r, T(A) ≥ r ≥ s. Then A ∈ τT . Thus τT ⊂ τT . (3) (O1) Since ⊤X ≤ H(⊤X ) and H(⊥X ) = H(⊥X ⊙ A) = ⊥X ⊙ H(A), ⊤X = H(⊤X ) and ⊤X = H(⊤X ). Then ⊥X , ⊤X ∈ τH. (O2) For Ai ∈ τH for each i ∈ Γ, by (H3), i∈Γ Ai ∈ τH. Since i∈Γ Ai ≤ H( i∈Γ Ai) ≤ i∈Γ H(Ai)= i∈Γ Ai, Thus, i∈ΓAi ∈ τH. (O3) For A∈ τH, by (H2), α ⊙ A ∈ τH. (O4) For A ∈ τH, since α ⊙ H(α → A) = H(α ⊙ (α → A)) ≤ H(A), H(α → A) ≤ α → H(A)= α → A. Then α → A ∈ τH. Theorem 3.2 Let T be an Alexandorv fuzzy topology on X. Define r ∗ RT (x, y)= {A(x) → A(y) | T(A) ≥ r } −r ∗ RT (x, y)= {B(y) → B(x) | T(B) ≥ r }. r r s (1) RT is a fuzzy preorder with RT ≤ RT for each s ≤ r. −r −r −s (2) RT is a fuzzy preorder with RT ≤ RT for each s ≤ r and −r r RT (x, y)= RT ∗ (x, y). r X X (3) Define HRT : L → L as follows r r HRT (A)(y)= (A(x) ⊙ RT (x, y)). ∈ xX r r s Then HRT is an upper approximation operator on X with HRT ≤ HRT for each r∗ s ≤ r such that τ = τHRr . T T Alexandrov fuzzy topologies and fuzzy preorders 249 (4) H −r is an upper approximation operator on X such that RT −r H −r (A)(y)= (A(x) ⊙ R (x, y)) = (A(x) ⊙ R ∗ (x, y)). RT T T ∈ ∈ xX xX r∗ ∗ Moreover, τ τH − . ( T ) = R r T ∗ r X (5) HRT (A) = {Ai | A ≤ Ai, T(Ai) ≥ r } for all A ∈ L and r ∈ L. Moreover, Rr x, y r y , for each x, y X. T ( )= HRT (⊤x)( ) ∈ r ∗ ∗ X (6) HRT (A) = {Ai | A ≤ Ai, T (Ai) ≥ r } for all A ∈ L and r ∈ L. −r r Moreover, R x, y R ∗ x, y r y , for each x, y X. T ( )= T ( )= HRT ∗ (⊤x)( ) ∈ (7) If H ri (A)= B for all i ∈ Γ = ∅, then H s (A)= B with s = ∈Γ r . RT RT i i (8) If H −ri (A)= B for all i ∈ Γ = ∅, then H −s (A)= B with s = i∈Γ ri. RT RT X (9) Define THT : L → L as ∗ r T T (A)= {r ∈ L | H i (A)= A}. H i RT Then THT = T is an Alexandrov fuzzy topology on X. X (10) Define THT ∗ : L → L as ∗ −r THT ∗ (A)= {r ∈ L | H i (A)= A} i RT ∗ Then THT ∗ = T is an Alexandrov fuzzy topology on X. (11) There exists an Alexandrov fuzzy topology Tr such that r X r T (A)= eL (HRT (A), A). If r ≤ s, then Tr ≤ Ts for all A ∈ LX . (12) There exists an Alexandrov fuzzy topology T∗r such that ∗r T (A)= e X (H −r (A), A).