Some Properties of Alexandrov Topologies

Abstract

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.

1. Introduction

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:

(C1) L = (L, ≤, ∨, ∧, ⊥, 𝖳) is a complete lattice with the greatest element 𝖳 and the least element ⊥; (C2) (L, ⊙, 𝖳) is a commutative monoid; (C3) x ⊙ y ≤ z iff x ≤ y → z for x, y, z ∈ L.

In this paper, we assume (L, ∧, ∨, ⊙, →, ∗ ⊥, 𝖳) is a complete residuated lattice with a negation; i.e., x∗∗ = x. For α ∈ L, A, 𝖳x ∈ L X, (α → A)(x) = α → A(x), (α ⊙A)(x) = α ⊙ A(x) and 𝖳x(x) = 𝖳, 𝖳x(x) = ⊥, otherwise.

Lemma 1.2. [3, 4] 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 = 𝖳 iff x ≤ y. (4) x → 𝖳 = 𝖳 and 𝖳 → x = x. (5) x ⊙ y ≤ x ∧ y. (6). (7) and . (8) and . (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). (11) and . (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. (15) x → y ⊙ z ≥ (x → y) ⊙ z and (x → y) → z ≥ x ⊙ (y → z).

Definition 1.3. [7, 10, 12, 13] A subset τ ⊂ L X is called an Alexandrov topology if it satisfies:

(T1) ⊥X, 𝖳X ∈ τ where 𝖳X(x) = 𝖳 and ⊥X(x) = ⊥ for x ∈ X. (T2) If Ai ∈ τ for i ∈ Γ, . (T3) α ⊙ A ∈ τ for all α ∈ L and A ∈ τ. (T4) α → A ∈ τ for all α ∈ L and A ∈ τ.

A subset τ ⊂ LX satisfying (T1), (T3) and (T4) is called a strong topology if it satisfies:

(ST) If Ai ∈ τ for i ∈ Γ, for each finite index Λ ⊂ Γ.

A subset τ ⊂ LX satisfying (T1), (T3) and (T4) is called a strong cotopology if it satisfies:

(SC) If Ai ∈ 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 eX : X×X → L is called:

(E1) reflexive if eX (x, x) = 𝖳 for all x ∈ X, (E2) transitive if eX(x, y) ⊙ eX(y, z) ≤ eX(x, z), for all x, y, z ∈ X, (E3) if eX(x, y) = eX(y, x) = 𝖳, then x = y.

If e satisfies (E1) and (E2), (X, eX) is a fuzzy preordered set. If e satisfies (E1), (E2) and (E3), (X, eX) is a fuzzy partially ordered set.

Example 1.6. (1) We define a function eLX : L X × L X → L as . Then (L X, eLX ) is a fuzzy partially ordered set from Lemma 1.2 (8).

(2) Let τ be an Alexandrov topology. We define a function eτ : τ × τ → . Then (τ, eτ ) is a fuzzy partially ordered set.

Definition 1.7. [8, 9] Let (X, eX) be a fuzzy partially ordered set and A ∈ LX.

(1) A point x0 is called a join of A, denoted by x0 = ⊔A if it satisfies (J1) A(x) ≤ eX(x, x0), (J2) . A point x1 is called a meet of A, denoted by x1 = ⊓A, if it satisfies (M1) A(x) ≤ eX(x1, x), (M2) .

Remark 1.8. [8, 9] Let (X, eX) be a fuzzy partially ordered set and A ∈ LX.

(1) x0 is a join of A iff. (2) x1 is a meet of A iff . (3) If x0 is a join of A, then it is unique because eX(x0, y) = eX(y0, y) for all y ∈ X, put y = x0 or y = y0, then eX(x0, y0) = eX(y0, x0) = 𝖳 implies x0 = y0. Similarly, if a meet of A exist, then it is unique.

Remark 1.9. [8, 9] Let (L X, eLX ) be a fuzzy partially ordered and Φ ∈ LLX.

(1) Since then .(2) We have because

 

2. Some Properties of Alexandrov Topologies

Theorem 2.1. (1) A subset τ ⊂ LX 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∗ ∈ LX | 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(Ai) = 𝖳 for all {Ai | 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 Iτ : LX → LX as follows:

Then the following properties hold.

(1) eLX (A, B) ≤ eLX (Iτ (A), Iτ (B)), for A, B ∈ LX. (2) Iτ (A) ≤ A for all A ∈ LX. (3) Iτ (Iτ (A)) = Iτ (A) for all A ∈ LX. (4) Iτ (α → A) = α → Iτ (A) for all α ∈ L, A ∈ LX. (5) for all Ai ∈ LX. (6) for each Φ : L X → L where defined as . (7) . (8) Define τIτ = {A | A = Iτ (A)}. Then τ = τIτ . (9) There exists a fuzzy preorder eX : X × X → L such that

Proof. (1) By Lemma 1.2 (8,10,14), we have

(2) Since eLX (C, A)⊙C ≤ A from Lemma 1.2 (9), Iτ (A) ≤ A.

(3) Since Iτ (A) ∈ τ , then

Iτ (Iτ (A)) ≥ eLX (Iτ (A), Iτ (A)) ⊙ Iτ (A) = Iτ (A).

By (2), Iτ (Iτ (A)) = Iτ (A).

(4) Since α → Iτ (A) ≤ α → A and α → Iτ (A) ∈ τ ,

(5) By (1), since Iτ (A) ≤ Iτ (B) for . Since and , we have

(6) For each Φ : LX → L, put . Since is a map, we have

and from:

(7) . Since I(A) ≤ A and I(A) ∈ τ , we have

.

Since Iτ (A) ≤ A and Iτ (A) ∈ τ , we have I(A) ≥ Iτ (A).

(8) It follows from A ∈ τ iff Iτ (A) = A iff A ∈ τIτ.

(9) Since , by (4) and (5), . Put . Then

Hence eX is a fuzzy preorder.

Theorem 2.3. Let τ be an Alexandrov topology on X. Define Cτ : LX → LX as follows:

Then the following properties hold.

(1) eLX (A, B) ≤ eLX (Cτ (A), Cτ (B)), for all A, B ∈ LX. (2) A ≤ Cτ (A) for all A ∈ LX. (3) Cτ (Cτ (A)) = Cτ (A) for all A ∈ LX. (4) Cτ (α ⊙ A) = α ⊙ Cτ (A) for all α ∈ L, A ∈ LX. (5) for all Ai ∈ LX. (6) for each Φ : LX → L where defined as . (7) . (8) Define τCτ = {A | A = Cτ (A)}. Then τ = τCτ. (9) (Cτ (A∗))∗ = Iτ∗ (A) for all A ∈ LX. (10) There exists a fuzzy preorder eX : X × X → L such that

Proof. (1) By Lemma 1.2 (8,10), we have

(2) Since eLX (A, B) ⊙ A ≤ B iff A ≤ eLX (A, B) → B , then A ≤ Cτ (A).

(3) Since Cτ (A) ∈ τ , then Cτ (Cτ (A)) ≤ eLX (Cτ (A), Cτ (A)) → Cτ (A) = Cτ (A). By (2), Cτ (Cτ (A)) = Cτ (A).

(4) Since α ⊙ A ≤ α ⊙ Cτ (A) and α ⊙ Cτ (A) ∈ τ ,

Cτ (α ⊙ A) ≤ eLX (α ⊙ A, α ⊙ Cτ (A)) → α ⊙ Cτ (A) = α ⊙ Cτ (A).

(5) By (1), since Cτ (A) ≤ Cτ (B) for A ≤ B, . Since

and

we have

(6) For each Φ : LX → L, put . Since is a map, we have

and from:

(7) Put . Since A ≤ C(A) and C(A) ∈ τ , we have

Since A ≤ Cτ (A) and Cτ (A) ∈ τ , we have C(A) ≤ Cτ (A).

(8) It follows from A ∈ τ iff Cτ (A) = A iff A ∈ τCτ.

(9)

(10) Since , by (4) and (5), . Put eX(x, y) = Cτ (𝖳x)(y). Then

Hence eX 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 A1 = (1, 0.8, 0.6), A2 = (0.7, 1, 0.7), A3 = (0.5, 0.7, 1).

(1) We define

where

(T1) For ⊥X ∈ LX, eX(⊥X) = ⊥X ∈ τ . For 𝖳X ∈ LX, eX(𝖳X) = 𝖳X ∈ τ .

(T2) For eX(Ai) ∈ τ for each i ∈ Γ, . Moreover, since eX(A)(x) ≥ eX(x, x) ⊙ A(x) = A(x) and eX(eX(A)) = eX(A),

Hence .

(T3) For eX(A) ∈ τ , α ⊙ eX(A) = eX(α ⊙ A) ∈ τ.

(T4) Since α ⊙ eX(α → eX(A)) ≤ eX(eX(A)) = eX(A), we have

α → eX(A) ≤ eX(α → eX(A)) ≤ α → eX(A)

Hence, for eX(A) ∈ τ , α → eX(A) = eX(α → eX(A)) ∈ τ . Hence τ is an Alexandrov topology on X.

(2) For B1 = (0.7, 0.3, 0.6), B1 = (0.5, 0.9, 0.3), we obtain

Iτ (B1) = (0.5, 0.3, 0.6), Iτ (B2) = (0.5, 0.6, 0.3), Cτ (B1) = (0.7, 0.5, 0.6), Cτ (B2) = (0.6, 0.9, 0.6).

Let Φ : LX → L as follows

Thus, .

Thus, .

(3) We define

For B1, B2 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 .

 

3. Conclusions

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.