Some Properties of Alexandrov Topologies

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  • 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.


  • KEYWORD

    Complete residuated lattices , Alexandrov topologies , Fuzzy partially ordered set , Meet and join

  • 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 αL, A, 𝖳xL X, (αA)(x) = αA(x), (αA)(x) = αA(x) and 𝖳x(x) = 𝖳, 𝖳x(x) = ⊥, otherwise.

    Lemma 1.2. [3, 4] For each x, y, z, xi , yiL, 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 τL 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 τLX satisfying (T1), (T3) and (T4) is called a strong topology if it satisfies:

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

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

    (SC) If Aiτ 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×XL 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 XL 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 ALX.

    (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 ALX.

    (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 τLX is an Alexandrov topology on X iff for each Φ : τL, ⊔Φ ∈ τ and ⊓Φ ∈ τ.

    (2) τ is an Alexandrov topology on X iff τ = {ALX | Aτ} is an Alexandrov topology on X.

    Proof. (1) (⇒) For each Φ : τL, we define

    image

    Since τ is an Alexandrov topology on X, (Φ(A) ⊙ A) ∈ τ . Thus Pτ . Then P = ⊔Φ from:

    image

    For each Φ : τL, we define . Since τ is an Alexandrov topology on X, (Φ(A) → A) ∈ τ. Thus Qτ. Then Q = ⊓Φ from:

    image

    (⇒) (T1) For Φ(A) = ⊥ for all Aτ , and .

    (T2) Let Φ(Ai) = 𝖳 for all {Ai | i ∈ Γ} ⊂ τ , otherwise Φ(A) = ⊥. We have

    image

    (T3) Let Φ(A) = ⊥ for A = Bτ , otherwise Φ(A) = α if AB. We have

    image

    (2) Let Aτ for Aτ . Since αA = (αA) and αA = (αA), τ is an Alexandrov topology on X.

    Theorem 2.2. Let τ be an Alexandrov topology on X. Define Iτ : LXLX as follows:

    image

    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

    image

    (2) Since eLX (C, A)⊙CA 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 αIτ (A) ≤ αA and αIτ (A) ∈ τ ,

    image

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

    image

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

    image

    and from:

    image

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

    image

    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 AτIτ.

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

    image

    Hence eX is a fuzzy preorder.

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

    image

    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

    image

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

    image

    (2) Since eLX (A, B) ⊙ AB iff AeLX (A, B) → B , then ACτ (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 αAαCτ (A) and αCτ (A) ∈ τ ,

    Cτ (αA) ≤ eLX (αA, α Cτ (A)) → αCτ (A) = αCτ (A).

    image

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

    image

    and

    image

    we have

    image

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

    image

    and from:

    image

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

    image

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

    (8) It follows from Aτ iff Cτ (A) = A iff AτCτ.

    (9)

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

    image

    Hence eX is a fuzzy preorder. Since , by Theorem 2.2(9),

    image

    Example 2.4. Let (L = [0, 1], ⊙, →, ) be a complete residuated lattice with a negation defined by

    xy = (x+y−1)∨0, xy = (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

    image

    where

    image

    (T1) For ⊥XLX, eX(⊥X) = ⊥Xτ . For 𝖳XLX, eX(𝖳X) = 𝖳Xτ .

    (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),

    image

    Hence .

    (T3) For eX(A) ∈ τ , αeX(A) = eX(αA) ∈ τ.

    (T4) Since αeX(αeX(A)) ≤ eX(eX(A)) = eX(A), we have

    αeX(A) ≤ eX(αeX(A)) ≤ αeX(A)

    Hence, for eX(A) ∈ τ , αeX(A) = eX(αeX(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 : LXL as follows

    image
    image

    Thus, .

    image

    Thus, .

    (3) We define

    image

    For B1, B2 and Φ in (2), we obtain

    image

    Since ⊓Φ = (0.7, 0.4, 0.5) and

    image

    we have .

    Since ⊔Φ = (0.6, 0.7, 0.5) and

    image

    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.

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