Some Properties of Alexandrov Topologies
 Author: Kim Yong Chan, Kim Young Sun
 Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 15, Issue1, p72~78, 25 March 2015

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 [37]. 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, 1012] 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 . Forα ∈L ,A , 𝖳_{x} ∈L ^{X} , (α →A )(x ) =α →A (x ), (α ⊙A )(x ) =α ⊙A (x ) and 𝖳_{x} (x ) = 𝖳, 𝖳_{x} (x ) = ⊥, otherwise.Lemma 1.2. [3, 4] For eachx ,y ,z ,x_{i} ,y_{i} ∈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 subsetτ ⊂L ^{X} is called anAlexandrov 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 ⊂L^{X} satisfying (T1), (T3) and (T4) is called astrong topology if it satisfies:(ST) If
A_{i} ∈τ fori ∈ Γ, for each finite index Λ ⊂ Γ.A subset
τ ⊂L^{X} satisfying (T1), (T3) and (T4) is called astrong cotopology if it satisfies:(SC) If
A_{i} ∈τ fori ∈ 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] LetX be a set. A functione_{X} :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 ,e_{X} ) is a fuzzy preordered set. Ife satisfies (E1), (E2) and (E3), (X ,e_{X} ) is a fuzzy partially ordered set.Example 1.6. (1) We define a functione_{LX} :L ^{X} ×L ^{X} →L as . Then (L ^{X} ,e_{LX} ) is a fuzzy partially ordered set from Lemma 1.2 (8).(2) Let
τ be an Alexandrov topology. We define a functione_{τ} :τ ×τ → . Then (τ ,e_{τ} ) is a fuzzy partially ordered set.Definition 1.7. [8, 9] Let (X ,e_{X} ) be a fuzzy partially ordered set andA ∈L^{X} .(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 ,e_{X} ) be a fuzzy partially ordered set andA ∈L^{X} .(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} ,e_{LX} ) be a fuzzy partially ordered and Φ ∈L^{LX} .(1) Since then .(2) We have because
2. Some Properties of Alexandrov Topologies
Theorem 2.1. (1) A subsetτ ⊂L^{X} is an Alexandrov topology onX iff for each Φ :τ →L , ⊔Φ ∈τ and ⊓Φ ∈τ .(2)
τ is an Alexandrov topology onX iffτ ^{∗} = {A ^{∗} ∈L^{X} A ∈τ } is an Alexandrov topology onX .Proof. (1) (⇒) For each Φ :τ →L , we defineSince
τ is an Alexandrov topology onX , (Φ(A ) ⊙A ) ∈τ . ThusP ∈τ . ThenP = ⊔Φ from:For each Φ :
τ →L , we define . Sinceτ is an Alexandrov topology onX , (Φ(A ) →A ) ∈τ . ThusQ ∈τ . ThenQ = ⊓Φ from:(⇒) (T1) For Φ(
A ) = ⊥ for allA ∈τ , and .(T2) Let Φ(
A_{i} ) = 𝖳 for all {A_{i} i ∈ Γ} ⊂τ , otherwise Φ(A ) = ⊥. We have(T3) Let Φ(
A ) = ⊥ forA =B ∈τ , otherwise Φ(A ) =α ifA ≠B . We have(2) Let
A ^{∗} ∈τ ^{∗} forA ∈τ . Sinceα ⊙A ^{∗} = (α →A )^{∗} andα →A ^{∗} = (α ⊙A )^{∗},τ ^{∗} is an Alexandrov topology onX .Theorem 2.2. Letτ be an Alexandrov topology onX . DefineI_{τ} :L^{X} →L^{X} 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
e_{LX} (C ,A )⊙C ≤A from Lemma 1.2 (9),I_{τ} (A ) ≤A .(3) Since
I_{τ} (A ) ∈τ , thenI_{τ} (I_{τ} (A )) ≥e_{LX} (I_{τ} (A ),I_{τ} (A )) ⊙I_{τ} (A ) =I_{τ} (A ).By (2),
I_{τ} (I_{τ} (A )) =I_{τ} (A ).(4) Since
α →I_{τ} (A ) ≤α →A andα →I_{τ} (A ) ∈τ ,(5) By (1), since
I_{τ} (A ) ≤I_{τ} (B ) for . Since and , we have(6) For each Φ :
L^{X} →L , put . Since is a map, we haveand from:
(7) . Since
I (A ) ≤A andI (A ) ∈τ , we haveSince
I_{τ} (A ) ≤A andI_{τ} (A ) ∈τ , we haveI (A ) ≥I_{τ} (A ).(8) It follows from
A ∈τ iffI_{τ} (A ) =A iffA ∈τ_{Iτ} .(9) Since , by (4) and (5), . Put . Then
Hence
e_{X} is a fuzzy preorder.Theorem 2.3. Letτ be an Alexandrov topology onX . DefineC_{τ} :L^{X} →L^{X} 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
e_{LX} (A ,B ) ⊙A ≤B iffA ≤e_{LX} (A ,B ) →B , thenA ≤C_{τ} (A ).(3) Since
C_{τ} (A ) ∈τ , thenC_{τ} (C_{τ} (A )) ≤e_{LX} (C_{τ} (A ),C_{τ} (A )) →C_{τ} (A ) =C_{τ} (A ). By (2),C_{τ} (C_{τ} (A )) =C_{τ} (A ).(4) Since
α ⊙A ≤α ⊙C_{τ} (A ) andα ⊙C_{τ} (A ) ∈τ ,C_{τ} (α ⊙A ) ≤e_{LX} (α ⊙A ,α ⊙C_{τ} (A )) →α ⊙C_{τ} (A ) =α ⊙C_{τ} (A ).(5) By (1), since
C_{τ} (A ) ≤C_{τ} (B ) forA ≤B , . Sinceand
we have
(6) For each Φ :
L^{X} →L , put . Since is a map, we haveand from:
(7) Put . Since
A ≤C (A ) andC (A ) ∈τ , we haveSince
A ≤C_{τ} (A ) andC_{τ} (A ) ∈τ , we haveC (A ) ≤C_{τ} (A ).(8) It follows from
A ∈τ iffC_{τ} (A ) =A iffA ∈τ_{Cτ} .(9)
(10) Since , by (4) and (5), . Put
e_{X} (x ,y ) =C_{τ} (𝖳_{x} )(y ). ThenHence
e_{X} 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 byx ⊙y = (x +y −1)∨0,x →y = (1−x +y )∧1,x ^{∗} = 1−x .Let
X = {x ,y ,z } be a set andA _{1} = (1, 0.8, 0.6),A _{2} = (0.7, 1, 0.7),A _{3} = (0.5, 0.7, 1).(1) We define
where
(T1) For ⊥
_{X} ∈L^{X} ,e_{X} (⊥_{X} ) = ⊥_{X} ∈τ . For 𝖳_{X} ∈L^{X} ,e_{X} (𝖳_{X} ) = 𝖳_{X} ∈τ .(T2) For
e_{X} (A_{i} ) ∈τ for eachi ∈ Γ, . Moreover, sincee_{X} (A )(x ) ≥e_{X} (x ,x ) ⊙A (x ) =A (x ) ande_{X} (e_{X} (A )) =e_{X} (A ),Hence .
(T3) For
e_{X} (A ) ∈τ ,α ⊙e_{X} (A ) =e_{X} (α ⊙A ) ∈τ .(T4) Since
α ⊙e_{X} (α →e_{X} (A )) ≤e_{X} (e_{X} (A )) =e_{X} (A ), we haveα →e_{X} (A ) ≤e_{X} (α →e_{X} (A )) ≤α →e_{X} (A )Hence, for
e_{X} (A ) ∈τ ,α →e_{X} (A ) =e_{X} (α →e_{X} (A )) ∈τ . Henceτ is an Alexandrov topology onX .(2) For
B _{1} = (0.7, 0.3, 0.6),B _{1} = (0.5, 0.9, 0.3), we obtainIτ (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 :
L^{X} →L as followsThus, .
Thus, .
(3) We define
For
B _{1},B _{2} and Φ in (2), we obtainSince ⊓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.

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