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≤ziffx≤y→zforx,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
, 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 each
x
,
y
,
z
,
x_{i}
,
y_{i}
∈
L
, the following properties hold.

(1) Ify≤z, thenx⊙y≤x⊙z.

(2) Ify≤z, thenx→y≤x→zandz→x≤y→x.

(3)x→y= 𝖳 iffx≤y.

(4)x→ 𝖳 = 𝖳 and 𝖳 →x=x.

(5)x⊙y≤x∧y.

(6).

(7)and.

(8)and.

(9) (x→y) ⊙x≤yand (y→z) ⊙ (x→y) ≤ (x→z).

(10)x→y≤ (y→z) → (x→z) andx→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→xand (x→y)∗=x⊙y∗.

(14)y→z≤x⊙y→x⊙z.

(15)x→y⊙z≥ (x→y) ⊙zand (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∈τwhere 𝖳X(x) = 𝖳 and ⊥X(x) = ⊥ forx∈X.

(T2) IfAi∈τfori∈ Γ,.

(T3)α⊙A∈τfor allα∈LandA∈τ.

(T4)α→A∈τfor allα∈LandA∈τ.
A subset
τ
⊂
L^{X}
satisfying (T1), (T3) and (T4) is called a
strong topology
if it satisfies:
(ST) If
A_{i}
∈
τ
for
i
∈ Γ,
for each finite index Λ ⊂ Γ.
A subset
τ
⊂
L^{X}
satisfying (T1), (T3) and (T4) is called a
strong cotopology
if it satisfies:
(SC) If
A_{i}
∈
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
e_{X}
:
X
×
X
→
L
is called:

(E1) reflexive ifeX(x,x) = 𝖳 for allx∈X,

(E2) transitive ifeX(x,y) ⊙eX(y,z) ≤eX(x,z), for allx,y,z∈X,

(E3) ifeX(x,y) =eX(y,x) = 𝖳, thenx=y.
If
e
satisfies (E1) and (E2), (
X
,
e_{X}
) is a fuzzy preordered set. If
e
satisfies (E1), (E2) and (E3), (
X
,
e_{X}
) is a fuzzy partially ordered set.
Example 1.6.
(1) We define a function
e_{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 function
e_{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 and
A
∈
L^{X}
.

(1) A pointx0is called a join ofA, denoted byx0= ⊔Aif it satisfies

(J1)A(x) ≤eX(x,x0),

(J2).

A pointx1is called a meet ofA, denoted byx1= ⊓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 and
A
∈
L^{X}
.

(1)x0is a join ofAiff

.

(2)x1is a meet ofAiff.

(3) Ifx0is a join ofA, then it is unique becauseeX(x0,y) =eX(y0,y) for ally∈X, puty=x0ory=y0, theneX(x0,y0) =eX(y0,x0) = 𝖳 impliesx0=y0. Similarly, if a meet ofAexist, 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 havebecause

2. Some Properties of Alexandrov Topologies
Theorem 2.1.
(1) A subset
τ
⊂
L^{X}
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
^{∗}
∈
L^{X}

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(
A_{i}
) = 𝖳 for all {
A_{i}

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
I_{τ}
:
L^{X}
→
L^{X}
as follows:
Then the following properties hold.

(1)eLX(A,B) ≤eLX(Iτ(A),Iτ(B)), forA,B∈LX.

(2)Iτ(A) ≤Afor allA∈LX.

(3)Iτ(Iτ(A)) =Iτ(A) for allA∈LX.

(4)Iτ(α→A) =α→Iτ(A) for allα∈L,A∈LX.

(5)for allAi∈LX.

(6)for each Φ :LX→Lwheredefined as.

(7).

(8) DefineτIτ= {AA=Iτ(A)}. Thenτ=τIτ.

(9) There exists a fuzzy preordereX:X×X→Lsuch 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
, then
I_{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
and
α
→
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 have
and
from:
(7)
. Since
I
(
A
) ≤
A
and
I
(
A
) ∈
τ
, we have
.
Since
I_{τ}
(
A
) ≤
A
and
I_{τ}
(
A
) ∈
τ
, we have
I
(
A
) ≥
I_{τ}
(
A
).
(8) It follows from
A
∈
τ
iff
I_{τ}
(
A
) =
A
iff
A
∈
τ_{Iτ}
.
(9) Since
, by (4) and (5),
. Put
. Then
Hence
e_{X}
is a fuzzy preorder.
Theorem 2.3.
Let
τ
be an Alexandrov topology on
X
. Define
C_{τ}
:
L^{X}
→
L^{X}
as follows:
Then the following properties hold.

(1)eLX(A,B) ≤eLX(Cτ(A),Cτ(B)), for allA,B∈LX.

(2)A≤Cτ(A) for allA∈LX.

(3)Cτ(Cτ(A)) =Cτ(A) for allA∈LX.

(4)Cτ(α⊙A) =α⊙Cτ(A) for allα∈L,A∈LX.

(5)for allAi∈LX.

(6)for each Φ :LX→Lwheredefined as.

(7).

(8) DefineτCτ= {AA=Cτ(A)}. Thenτ=τCτ.

(9) (Cτ(A∗))∗=Iτ∗(A) for allA∈LX.

(10) There exists a fuzzy preordereX:X×X→Lsuch that
Proof
. (1) By Lemma 1.2 (8,10), we have
(2) Since
e_{LX}
(
A
,
B
) ⊙
A
≤
B
iff
A
≤
e_{LX}
(
A
,
B
) →
B
, then
A
≤
C_{τ}
(
A
).
(3) Since
C_{τ}
(
A
) ∈
τ
, then
C_{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
) and
α
⊙
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
) for
A
≤
B
,
. Since
and
we have
(6) For each Φ :
L^{X}
→
L
, put
. Since
is a map, we have
and
from:
(7) Put
. Since
A
≤
C
(
A
) and
C
(
A
) ∈
τ
, we have
Since
A
≤
C_{τ}
(
A
) and
C_{τ}
(
A
) ∈
τ
, we have
C
(
A
) ≤
C_{τ}
(
A
).
(8) It follows from
A
∈
τ
iff
C_{τ}
(
A
) =
A
iff
A
∈
τ_{Cτ}
.
(9)
(10) Since
, by (4) and (5),
. Put
e_{X}
(
x
,
y
) =
C_{τ}
(𝖳
_{x}
)(
y
). Then
Hence
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 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
A
_{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 each
i
∈ Γ,
. Moreover, since
e_{X}
(
A
)(
x
) ≥
e_{X}
(
x
,
x
) ⊙
A
(
x
) =
A
(
x
) and
e_{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 have
α
→
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
. Hence
τ
is an Alexandrov topology on
X
.
(2) For
B
_{1}
= (0.7, 0.3, 0.6),
B
_{1}
= (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 :
L^{X}
→
L
as follows
Thus,
.
Thus,
.
(3) We define
For
B
_{1}
,
B
_{2}
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.
Conflict of InterestNo potential conflict of interest relevant to this article was reported.
BIO
Yong Chan Kim received the B.S., M.S. and Ph.D. degrees in Mathematics from Yonsei University, Seoul, Korea, in 1982, 1984 and 1991, respectively. He is currently professor of GangneungWonju University. His research interests is a fuzzy topology and fuzzy logic.
Email: yck@gwnu.ac.kr
Young Sun Kim received the B.S., M.S. and Ph.D. degrees in Mathematics from Yonsei University, Seoul, Korea, in 1981, 1985 and 1991, respectively. He is currently Professor of Pai Chai University. His research interests is a fuzzy topology and fuzzy logic.
Email: yskim@pcu.ac.kr
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