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). L RT

If r ≤ s, then T∗r ≤ T∗s for all A ∈ LX . X (13) Define TT : L → L as

∗ r TT (A)= {r ∈ L | T (A)= ⊤}. Then TT = T = THT is an Alexandrov fuzzy topology on X. X (14) Define TT ∗ : L → L as

∗ ∗r TT ∗ (A)= {r ∈ L | T (A)= ⊤}. ∗ ∗ Then TT = T = THT ∗ is an Alexandrov fuzzy topology on X. 250 Yong Chan Kim

∗ r∗ r Proof T B r B τ R x, y r∗ B x (1) Since ( ) ≥ iff ∈ T , then T ( ) = B∈τT ( ( ) → r B(y)). Since R (x, x)= ∈ r∗ (B(x) → B(x)) = ⊤ and T B τT r r R x, y R y, z r∗ B x B y r∗ B y B z T ( ) ⊙ T ( )= B∈τT ( ( ) → ( )) ⊙ B∈τT ( ( ) → ( )) r∗ B x B y B y B z ≤ B∈τT ( ( ) → ( )) ⊙ ( ( ) → ( )) r r∗ B x B z R x, y . ≤ B∈τT ( ( ) → ( )) = T ( ) r Hence RT is a fuzzy preorder. ∗ ∗ r s For s ≤ r, since T(B) ≥ s ≥ r , we have RT ≤ RT . −r (2) By a similar method as (1), RT is a fuzzy preorder. Moreover, −r ∗ RT (x, y) = {B(y) → B(x) | T(B) ≥ r } ∗ ∗ ∗ ∗ ∗ = {B (x) → B (y) | T(B )= T (B) ≥ r } r = RT ∗ (x, y).

r (3) (H1) and (H2) follows from the definition of HRT . r r r (H3) HRT (A)(y)= x∈X (A(x) ⊙ RT (x, y)) ≥ A(y) ⊙ RT (y,y)= A(y). (H4)

r r r r HRT (HRT (A))(x) = y∈X (HRT (A)(y) ⊙ RT (y, x)) r r = y∈X ( z∈X (A(z) ⊙ RT (z,y)) ⊙ RT (y, x)) r r = z∈X (A(z) ⊙ y∈X (RT (z,y) ⊙ RT (y, x))) r ≤ z∈X (A(z) ⊙ RT (z, x)) = H r (A)(x). RT

r s r s For s ≤ r, since RT ≤ RT , then HRT ≤ HRT . r∗ ∗ r A τ T A r R x, y A x r∗ B x B y Since ∈ T ;i.e. ( ) ≥ , T ( ) ⊙ ( ) = B∈τT ( ( ) → ( )) ⊙ r A(x) ≤ (A(x) → A(y)) ⊙ A(x) ≤ A(y), by H(3), HR (A) = A ∈ τHRr . Thus T T r∗ r τ ⊂ τHRr . Let HR (A)= A ∈ τHRr . Then T T T T

r r A(x) = HRT (A)(x)= y∈X (A(y) ⊙ RT (y, x)) = ∈ (A(y) ⊙ ∈ r∗ (B(y) → B(x))) y X B τT r∗ r∗ r∗ B y B τ A y r∗ B y B τ Since B∈τT ( ( ) → ) ∈ T and y∈X ( ( ) ⊙ B∈τT ( ( ) → )) ∈ T , r∗ r∗ we have A ∈ τ . Hence τHRr ⊂ τ . T T T r∗ −r ∗ ∗ ∗ ∗ A τ R x, y A x r∗ B x B y A x (4) Since ∈ T , T ( ) ⊙ ( )= B∈τT ( ( ) → ( )) ⊙ ( ) ≤ ∗ ∗ ∗ ∗ ∗ ∗ −r (A (x) → A (y)) ⊙ A (x) ≤ A (y). Hence H (A )= A ∈ τH −r . RT R T −r Let H (A)= A ∈ τH −r . Then RT R T

−r A(x) = H −r (A)(x)= ∈ (A(y) ⊙ R (y, x)) RT y X T ∗ ∗ A y r∗ B y B x = y∈X ( ( ) ⊙ B∈τT ( ( ) → ( )))

∗ ∗ r∗ ∗ ∗ ∗ r∗ B y B τ A y B y B Since B∈τT ( ( ) → ) ∈ ( T ) and y∈X ( ( ) ⊙ B∈τ ( ( ) → )) ∈ r∗ ∗ r∗ ∗ r∗ ∗ τ A τ τ τH − ( T ) , we have ∈ ( T ) . Hence ( T ) = R r . T Alexandrov fuzzy topologies and fuzzy preorders 251

X ∗ r∗ (5) For each A ∈ L with A ≤ Ai, T(Ai) ≥ r , since Ai ∈ τ = τHRr with T T A ≤ Ai, Ai ∈ τ, then

r r Ai ≤ HRT ( Ai) ≤ Ai = HRT (Ai). i i

r So, HRT ( i Ai)= i Ai. Thus

∗ r r HRT (A) ≤ HRT ( Ai)= Ai = {Ai | A ≤ Ai, T(Ai) ≥ r }. i i

r r r r r∗ Since HR (HR (A)) = HR (A) ≥ A and HR (A) ∈ τHRr = τ . So,, {Ai | T T T T T T ∗ r ∗ A ≤ Ai, T(Ai) ≥ r } ≤ HRT (A). Hence {Ai | A ≤ Ai, T(Ai) ≥ r } = r A A LX r L HRT ( ) for all ∈ and ∈ . (6) It is proved in a similar way as (5). (7) Let H ri (A)= B for all i ∈ Γ = ∅. Since RT

∗ ri ri H ri (A)= (A(x) ⊙ R (x, −)) = (A(x) ⊙ (D(x) → D)) ∈ τ RT T T r∗ x∈X x∈X i D∈τT

r ri HR i (A)= x∈X (A(x) ⊙ RT (x, −)) T ∗ ri r∗ = x∈X (A(x) ⊙ i (D(x) → D)) ∈ τT D∈τT ∗ ∗ ∗ ∗ T(B) = T(H ri (A)) ≥ r , then T(B) ≥ ∈Γ r = ( ∈Γ r ) = s where RT i i i i i s s s = i∈Γ ri. Since HRT (B)(y) = x∈X (B(x) ⊙ RT (x, y)) ≤ x∈X (B(x) ⊙ (B(x)→ B(y))) ≤ B(y),

s s B = HRT (B) ≥ HRT (A).

Since s ≤ r , H s (A) ≥ H ri (A)= B. Thus H s (A)= B. i RT RT RT ∗ (9) Since T(A)= T(H ri (A)) ≥ r , for H ri (A)= A, RT i RT

∗ r T T (A)= {r ∈ L | H i (A)= A} ≤ T(A). H i RT ∗ ∗ s ∗ Since T(A) ≥ (T (A)) , then HRT (A) where T (A)= s. Thus

∗ T (A)= {r ∈ L | H ri (A)= A} ≥ T(A). HT i RT Hence THT = T. r r (11) (T1) T (⊤X )= T (⊥X )= ⊤. (T2) r X r T ( i∈Γ Ai)= eL (HRT ( i∈Γ Ai), i∈Γ Ai) e X Γ r A , Γ A = L ( i∈ HRT ( i) i∈ i) r Γ e X r A , A Γ T A ≥ i∈ L (HRT ( i) i)= i∈ ( i) 252 Yong Chan Kim

r r Since HRT ( i∈Γ Ai) ≤ i∈Γ HRT (Ai), we have r X r T ( i∈Γ Ai)= eL (HRT ( i∈Γ Ai), i∈Γ Ai) e X Γ r A , Γ A ≥ L ( i∈ HRT ( i) i∈ i) r Γ e X r A , A Γ T A ≥ i∈ L (HRT ( i) i)= i∈ ( i) (T3) r X r T (α ⊙ A)= eL (HRT (α ⊙ A),α ⊙ A) X r = eL (α ⊙ HRT (A),α ⊙ A) X r r ≥ eL (HRT (A), A)= T (A)

r r r (T4) Since α ⊙ HRT (α → A) = HRT (α ⊙ (α → A)) ≤ HRT (A), then r r HRT (α → A) ≤ α → HRT (A). Thus

r X r T (α → A)= eL (HRT (α → A),α → A) X r = eL (α → HRT (A),α → A) X r r ≥ eL (HRT (A), A)= T (A)

r s r Hence T is an Alexandrov fuzzy topology. Since HRT ≤ HRT for r ≤ s, s X s X r r T (A)= eL (HRT , A) ≥ eL (HRT , A)= T (A). r X r r (13) Since T (A)= eL (HRT (A), A)= ⊤ iff A = HRT (A), by (9),

∗ r TT (A) = {r ∈ L | T (A)= ⊤} ∗ r L r A A = { ∈ | HRT ( )= } = T HT (A)= T(A). (8), (10), (12) and (14) are similarly proved.

Example 3.3. 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

T(A)= A(x) → A(z).

Trivially, T(αX )=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( i∈Γ Ai) ≥ i∈Γ T(Ai) and T( i∈Γ Ai) ≥ i∈Γ T(Ai). Hence T is an Alexandrov fuzzy topology. ∗ ∗ Since T(A) = A(x) → A(z) ≥ r , then A(z) ≥ A(x) ⊙ r . Put A(x) = r ∗ r 1, A(y) = 0. So, RT (x, y)= {A(x) → A(y) | T(A) ≥ r } = 0 and RT (x, z)= ∗ ∗ {A(x) → A(z) | T(A) ≥ r } = r r r r ∗ RT (x, x)=1 RT (x, y)=0 RT (x, z)= r r r r  RT (y, x)=0 RT (y,y)=1 RT (y, z)=0  Rr (z, x)=0 Rr (z,y)=0 Rr (z, z)=1  T T T    Alexandrov fuzzy topologies and fuzzy preorders 253

r r By Theorem 3.1(3), we obtain HRT (A)(y)= x∈X (A(x) ⊙ RT (x, y)) such that ∗ r HRT (A)=(A(x), A(y), A(z) ∨ (A(x) ⊙ r ))

∗ r If A(x) ⊙ r ≤ A(z), then HR (A) = A. Thus A ∈ τHRr . Moreover, since T T ∗ ∗ r∗ T(A)= A(x) → A(z) ≥ r iff A(z) ≥ A(x) ⊙ r , A ∈ τ iff A ∈ τHRr . T T r∗ So, τ = τHRr . T T

∗ r THT (A) = {r ∈ L | HRT (A)= A} = A(x) → A(z)= T(A). Moreover, we obtain

r r T (A) = x∈X (HRT (A)(x) → A(x)) ∗ ∗ =(A(x) ⊙ r ) → A(z)= r → (A(x) → A(z)). ∗ r TT (A) = {r ∈ L | T (A)=1} = A(x) → A(z)

Hence TT = THT = T. ∗ r HRT (1x)(z)= {B(z) | B ≥ 1x, T(B) ≥ r }

∗ r ∗ Since B(x) = 1 and T(B)=1 → B(z)= B(z) ≥ r , then HRT (1x)(z)= r .

∗ r HRT (1x)(x)= {B(x) | B ≥ 1x, T(B) ≥ r } =1

∗ r HRT (1x)(y)= {B(y) | B ≥ 1x, T(B) ≥ r } =0 ∗ r HRT (1z)(x)= {B(x) | B ≥ 1z, T(B) ≥ r }

r Since B(z) = 1 and T(B)= B(x) → 1 = 1, then HRT (1z)(x)=0.

r r r ∗ HRT (1x)(x)=1 HRT (1x)(y)=0 HRT (1x)(z)= r r x r y r z  HRT (1y)( )=0 HRT (1y)( )=1 HRT (1y)( )=0  H r (1 )(x)=0 H r (1 )(y)=0 H r (1 )(z)=1  RT z RT z RT z    r r Then RT (x, y)= HRT (1x)(y). (2) By (1), we obtain a map T∗ : [0, 1]Y → [0, 1] as T∗(A)= A∗(x) → A∗(z)= A(z) → A(x). Since T∗(A) = A(z) → A(x) ≥ r∗, then A(x) ≥ A(z) ⊙ r∗. Put A(z) = r ∗ ∗ 1, A(y) = 0. So, RT ∗ (z,y) = {A(z) → A(y) | T (A) ≥ r } = 0 and r ∗ ∗ RT ∗ (z, x)= {A(z) → A(x) | T(A) ≥ r } = r r r r RT ∗ (x, x)=1 RT ∗ (x, y)=0 RT ∗ (x, z)=0 r r r  RT ∗ (y, x)=0 RT ∗ (y,y)=1 RT ∗ (y, z)=0  r ∗ r r R ∗ (z, x)= r R ∗ (z,y)=0 R ∗ (z, z)=1  T T T    254 Yong Chan Kim

r −r r Moreover, RT ∗ (x, y)= RT (x, y)= RT (y, x) for all x, y ∈ X.

r r HRT ∗ (A)(y)= (A(x) ⊙ RT ∗(x, y)). ∈ xX ∗ r HRT ∗ (A)=(A(x) ∨ (A(z) ⊙ r ), A(y), A(z))

∗ r ∗ If A(z) ⊙ r ≤ A(x), then HRT ∗ (A) = A. If A(x) ⊙ r ≤ A(z), then ∗ ∗ r r HRT (A) = A. Thus A ∈ τHR . Moreover, since T (A) = A(z) → A(x) ≥ r T ∗ ∗ r∗ ∗ iff A(x) ≥ A(z) ⊙ r , A ∈ τT iff A ∈ τHRr . Thus T ∗

∗ ∗ r THT (A) = {r ∈ L | HRT ∗ (A)= A} ∗ ∗ ∗ ∗ = A(z) → A(x)= T (A)= A (x) → A (z)= T(A ). Moreover, we obtain

∗r r T (A) = x∈X (HRT ∗ (A)(x) → A(x)) ∗ ∗ =(A(z) ⊙ r ) → A(x)= r → (A(z) → A(x)).

∗ ∗r TT ∗ (A) = {r ∈ L | T (A)=1} ∗ = A(z) → A(x)= T ∗ ∗ Hence TT = THT ∗ = T .

∗ ∗ r HRT ∗ (1x)(z)= {B(z) | B ≥ 1x, T (B) ≥ r }

∗ r Since B(x) = 1 and T (B)= B(z) → 1 = 1, then HRT ∗ (1x)(z)=0.

∗ ∗ r X HRT ∗ (1z)(y)= {B(y) ∈ L | B ≥ 1z, T (B) ≥ r } =0 ∗ ∗ r X HRT ∗ (1y)(y)= {B(y) ∈ L | B ≥ 1y, T (B) ≥ r } =1 ∗ ∗ r X HRT ∗ (1z)(x)= {B(x) ∈ L | B ≥ 1z, T (B) ≥ r } Since B(z) = 1 and T∗(B)=1 → B(x) = B(x) ≥ r∗, then B(x) ≥ r∗. We r ∗ have HRT ∗ (1z)(x)= r .

r r r HRT ∗ (1x)(x)=1 HRT ∗ (1x)(y)=0 HRT ∗ (1x)(z)=0 r x r y r z  HRT ∗ (1y)( )=0 HRT ∗ (1y)( )=1 HRT ∗ (1y)( )=0  ∗ H r (1 )(x)= r H r (1 )(y)=0 H r (1 )(z)=1  RT ∗ z RT ∗ z RT ∗ z    r ∗ r Then RT (x, y)= HRT ∗ (1x)(y). (3) Let (L = [0, 1], ⊙, →,∗ ) be a complete residuated lattice with a strong negation defined by, for each n ∈ N,

1 1 ∗ 1 x ⊙ y = ((xn + yn − 1) ∨ 0) n , x → y = (1 − xn + yn) n ∧ 1, x = (1 − xn) n . Alexandrov fuzzy topologies and fuzzy preorders 255

By (1) and (2), we obtain

1 ∗ 1 T(A)=(1 − A(x)n + A(z)n) n ∧ 1, T (A)=(1 − A(z)n + A(x)n) n ∧ 1.

1 1 0 (1 − rn) n 1 00 r r R = 01 0 R ∗ = 0 10 T   T  1  00 1 (1 − rn) n 0 1        1  r n n n HRT (A)=(A(x), A(y), A(z) ∨ ((A(x) +1 − r ) ∨ 0) ) 1 r n n n HRT ∗ (A)=(A(x) ∨ ((A(z) +1 − r ) ∨ 0) , A(y), A(z)) 1 1 Since T(A)=(1 − A(x)n + A(z)n) n ∧ 1 ≥ (1 − xn) n , we have

∗ r r X n n n τT = τRT = {A ∈ L | A (x) − A (z) ≤ r } r∗ X n n n ∗ r τT = τRT ∗ = {A ∈ L | A (z) − A (x) ≤ r }.

∗ 1 Tr(A) =(A(x) ⊙ r ) → A(z)=(rn − A(x)n + A(z)n) n ∧ 1 ∗ ∗ 1 T r(A) =(A(z) ⊙ r ) → A(x)=(rn − A(z)n + A(x)n) n ∧ 1.

References

[1] R. Bˇelohl´avek, Fuzzy Relational Systems, Kluwer Academic Publishers, New York , 2002.

[2] P. H´ajek, Metamathematices of Fuzzy Logic,Kluwer Academic Publishers, Dordrecht (1998).

[3] U. H¨ohle, S.E. Rodabaugh,(1999), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series 3, Kluwer Academic Publishers, Boston.

[4] Fang Jinming, I-fuzzy Alexandrov topologies and specialization orders, Fuzzy Sets and Systems, 158(2007), 2359-2374.

[5] Y.C. Kim, Alexandrov L-topologies and L-join meet approximation opera- tors International Journal of Pure and Applied Mathematics, 91(1)(2014), 113-129.

[6] H. Lai, D. Zhang, Fuzzy preorder and fuzzy topology,Fuzzy Sets and Sys- tems, 157 (2006),1865-1885.

[7] H. Lai, D. Zhang, Concept lattices of fuzzy contexts: Formal concept analysis vs. rough set theory,Int. J. Approx. Reasoning, 50(2009),695-707. 256 Yong Chan Kim

[8] Z. Pawlak, Rough sets, Int. J. Comput. Inf. Sci., 11 (1982), 341-356.

[9] Z. Pawlak, Rough probability, Bull. Pol. Acad. Sci. Math., 32(1984), 607- 615.

[10] A. M. Radzikowska, E.E. Kerre, A comparative study of fuzy rough sets, Fuzzy Sets and Systems, 126(2002), 137-155.

[11] Y.H. She, G.J. Wang, An axiomatic approach of fuzzy rough sets based on residuated lattices,Computers and Mathematics with Applications, 58 (2009), 189-201.

[12] Zhen Ming Ma, Bao Qing Hu, Topological and lattice structures of L- fuzzy rough set determined by lower and upper sets, Information Sciences, 218(2013), 194-204.

Received: March, 2014