In this paper, we investigate the properties of Alexandrov fuzzy topologies and meetjoin approximation operators. We study fuzzy preorder, Alexandrov topologies and meetjoin approximation operators induced by Alexandrov fuzzy topologies. We give their examples.
1. Introduction
Hájek
[2]
introduced a complete residuated lattice which is an algebraic structure for many valued logic. Höhle
[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. Bělohlávek
[1]
investigated information systems and decision 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 meetjoin approximation operators in a sense as Höhle
[3]
. We study fuzzy preorder, Alexandrov topologies and meetjoin approximation operators induced by Alexandrov fuzzy topologies. We give their examples.
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.
An operator * :
L
→
L
defined by
a
* =
a
→ ⊥ is called
strong negations
if
a
** =
a
.
In this paper, we assume that (
L
, ∨, ∧, ⊙, →, *, ⊥, ⊤) be a complete residuated 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 ifeX(x, x) = 1 for allx∈X,

(E2) transitive ifeX(x, y) ⊙eX(y, z) ≤eX(x, z), for allx, y, z∈ X’
Example 2.3.
(1) We define a function
e_{L}
:
L × L → L
as
e_{L}
(
x, y
) =
x → y.
Then
e_{L}
is a fuzzy preorder on
L
.

(2) We define a functioneLX:LX× LX→LasTheneLXis 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, 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= ⊤iff x≤y.

(4)x→ ⊤ = ⊤ and ⊤ →x = x.

(5)x⊙y≤x∧y.

(6)and.

(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.
Definition 2.5
(
[5]
). A map
:
L^{X} → L^{Y}
is called an
meetjoin approximation operator
if it satisfies the following conditions, for all
A, A_{i} ∈ L^{X}
, and
α ∈ L
,

(M1)where (α → A)(x) =α →A(x) for eachx ∈ X,

(M2)

(M3)A* ≤(A),

(M4)(*(A))≤(A).
Definition 2.6
(
[4]
). An operator
T
:
L^{X} → L
is called an
Alexandrov fuzzy topology
on
X
iff it satisfies the following conditions, for all
A, A_{i}
∈
L^{X}
, and
α
∈
L
,

(T1)T(αX) = ⊤, whereαX(x) =αfor eachx∈X,

(T2)T(Ai) ≥T(Ai) andT(Ai) ≥T(Ai),

(T3)T(α⊙A) ≥T(A), where (α⊙A)(x) =α⊙A(x) for eachx∈X,

(T4)T(α→A) ≥T(A).
Definition 2.7
(
[5]
). A subset
τ
⊂
L^{X}
is called an
Alexandrov topology
if it satisfies satisfies the following conditions.

(O1)αX∈τ.

(O2) IfAi∈τfori∈ Γ,Ai,Ai∈τ.

(O3)α⊙A∈τfor allα∈LandA∈τ.

(O4)α→A∈τfor allα∈LandA∈τ.
Remark 2.8.
(1) If T :
L^{X} → L
is an Alexandrov fuzzy topology. Define
T
*(
A
) =
T
(
A
*). Then
T
* is an Alexandrov fuzzy topology.

(2) IfTbe an Alexandrov fuzzy topology onX,= {A∈LXT(A) ≥r} is an Alexandrov topology onXandfors≤r∈L.
3. Structures Induced by Alexandrov Fuzzy Topologies
Theorem 3.1
.
If
is a meetjoin approximation operator, then
= {A ∈
L^{X}

(
A
) =
A
*}
is an Alexandrov topology on X.
Proof.
(O1) Since ⊤
_{X}
≤
(⊥
_{X}
) and
(⊤
_{X}
) =
(⊥
_{X}
→ A) = ⊥
_{X}
⊙
(
A
) = ⊥, ⊥
_{X}
=
(⊤
_{X}
) and ⊤
_{X}
=
(⊤
_{X}
). Then ⊥
_{X}
;⊤
_{X}
∈
.

(O2) ForAi∈for eachi∈ Γ , by (M2),So,Ai∈. SinceThus,Ai∈

(O3) ForA∈, sinceα⊙(α⊙A) =(α→ (α⊙A)) ≥(A),(α⊙A) ≥α→(A) = (α⊙A)*. Thenα⊙A∈.

(O4) ForA∈, by (M4),(α→A) =α⊙(A) =α⊙A*. Henceα→A∈.
Theorem 3.2.
Let
T
be an Alexandorv fuzzy topology on X. Define
We have the following properties
.

(1)is a fuzzy preorder with≤for each s≤r.

(2)is a fuzzy preorder with≤for each s≤r and= (x, y) =*(x, y)

(3)Defineas follows

Then
is a meetjoin approximation operator on X with
for each s
≤
r.

(4)

(5)is a meetjoin approximation operator on X such that

(6)

(7)for all A∈LXand r∈L.

Moreover,,for each x, y∈X.

(8)for all A∈LXand r∈L.

Moreover, for each x, y∈X.

(9)If=B for all i∈ Γ≠,thenwith s=ri.

(10)Iffor all i∈ Γ≠,thenwith s=ri.
Proof.
(1) Since
T
(
B
) ≥
r
* iff
then
Since
and
Hence
is a fuzzy preorder.

Fors≤r, sinceT(B) ≥s* ≥r*, we have≤

(2) By a similar method as (1),is a fuzzy preorder. Moreover,


(3) (M1)

(M2)


(M3)

(M4)


Fors≤r, since≤r, since, then

(4) Since; i.e.T(A) ≥r*,=⊙A*(x) ≤ (A*(x) →A*(y)) ⊙A*(x) ≤A*(y), by M(3),So,ThusLet; i.e. LetThen

Since
and
we have
. Hence
⊂
^{*}
.

(5) It is similarly proved as (4).

(6) Letsince A ∈*,

Hence
(
A
) =
A
*; i.e.
. Thus
Since
and
we have
Hence

(7) For eachA∈LXwithA* ≤Ai,T(Ai) ≥r*, sincethen

So,
Since
A
≥
Since
and
So, ,
≤
. Hence
=
for all
A
∈
L^{X}
and
r
∈
L
.

(8) It is proved in a similar way as (7).

(9) Let=Bfor alli∈ Γ ≠. Since

then
where
Since
then
So,
Thus
Since
s
≤
r_{i}
,
Thus
⧠
Theorem 3.3.
Let
T
be an Alexandorv fuzzy topology on X. We have the following properties.
Then
=
T
*
is an Alexandrov fuzzy topology on X
.
Then
=
T
*
is an Alexandrov fuzzy topology on X
.

(3)for all A, B∈LX.

(4)There exists an Alexandrov fuzzy topology Trsuch that

If r ≤ s, then
T
^{r}
≤
T
^{s}
f
or all A
∈
L^{X}
.

(5)There exists an Alexandrov fuzzy topology T*rsuch that

T*r(A) =eLX((A) ,A*).
Moreover,
T
*
^{r}
(
A
) =
T
^{r}
(
A
*)
for all A
∈
L^{X}
.
If r
≤
s
,
then
T
*
^{r}
≤
T
*
^{s}
for all A
∈
L^{X}
.
Then
T
_{M}
=
T
* =
T
_{MT}
is an Alexandrov fuzzy topology on X.
Then
T
_{M*}
=
T
=
T
_{MT*}
is an Alexandrov fuzzy topology on X.
Proof. (1) We only show that
T
_{MT}
=
T
*. Let
=
A
*. Then
form Theorem 3.3 (6). So
T
* (
A
) =
T
(
A
*) =
Thus,
Since
T
*(
A
) ≥ (
T
(
A
))* then
with s = T(
A
). Thus,
Hence
T
_{MT}
=
T
*.

(4) (T1) Since

(T2) Sincewe have


(T3) SincethenThus


(T4)

Hence
T
^{r}
is an Alexandrov fuzzy topology. Since
for
r ≤ s
,
T
^{s}
(
A
) =
=
T
^{r}
(
A
).

(5) From a similar method as (4),T*ris an Alexandrov fuzzy topology. By (3),Tr(A*) ==T*r(A) for allA∈LX.

(6) SinceTr(A) =iffby (9),


(2) and (7) are similarly proved as (1) and (6), respectively. ⧠
Example 3.4.
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
Trivially,
T
(
α_{X}
) = 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(
A_{i}
) ≥
T
(
A_{i}
) and
T
(
A_{i}
) ≥
T
(
A_{i}
). Hence
T
is an Alexandrov fuzzy topology.
If
T
(
A
) =
A
(
x
) →
A
(
z
) ≥ r*, then
A
(
z
) ≥
A
(
x
) ⊙ r*. Put
A
(
x
) = 1,
A
(
y
) = 0. So,
and
similarly, we can obtain
By Theorem 3.2(3), we obtain
such that
If
A
*(
x
) ⊙
r
* ≤
A
*(
z
), then
Thus
. Moreover, since
T
*(
A
) =
A
*(
x
)→
A
*(
z
) ≥
r
* iff A*(z) ≥
A
*(
x
) ⊙
r
*,
iff
. So,
From Theorem 3.3(1), we have
Moreover, we obtain
Hence
T
_{M}
=
T
_{MT}
=
T
*.
Since
B
(
x
) = 1 and
T
(
B
) = 1 →
B
(
z
) = B(
z
) ≥
r
*, then
Since
B
(
z
) = 1 and
T
(
B
) =
B
(
x
) → 1= 1, then
Then
(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
) = 1,
A
(
y
) = 0. So,
and
Moreover,
for all
x, y
∈
X
.
If
A
*(
z
) ⊙
r
* ≤
A
*(
x
), then
If
, then
A
*(
z
) ⊙ r* ≤
A
*(
x
). Moreover, since
T
(
A
) =
A
(
x
) →
A
(
z
) ≥
r
* iff
A
*(
z
) ⊙
r
* ≤
A
*(
z
),
iff
Thus
Moreover, we obtain
Hence
T
_{M*}
=
T
_{MT*}
=
T
.
Since
B
(
x
) = 1 and
T
*(
B
) =
B
(
z
) →1 = 1, then
Since
B
(
z
) = 1 and
T
*(
B
) = 1 →
B
(
x
) =
B
(
x
) ≥
r
*, then
B
(
x
) ≥
r
*. We have
Then
(3) Let (
L
= [0, 1], ⊙, →, * ) be a complete residuated lattice with a strong negation defined by, for each
n
∈
N
,
By (1) and (2), we obtain
Since
we have
Bělohlávek R.
2002
Fuzzy Relational Systems
Kluwer Academic Publishers
New York
Hájek P.
1998
Metamathematices of Fuzzy Logic
Kluwer Academic Publishers
Dordrecht
Höhle U.
,
Rodabaugh S.E.
1999
Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series 3
Kluwer Academic Publishers
Boston
Kim Y.C.
2014
Alexandrov Ltopologies and Ljoin meet approximation operators
International Journal of Pure and Applied Mathematics
91
(1)
113 
129
Lai H.
,
Zhang D.
2009
Concept lattices of fuzzy contexts: Formal concept analysis vs. rough set theory
Int. J. Approx. Reasoning
50
695 
707
DOI : 10.1016/j.ijar.2008.12.002
Pawlak Z.
1984
Rough probability
Bull. Pol. Acad. Sci. Math.
32
607 
615
She Y.H.
,
Wang G.J.
2009
An axiomatic approach of fuzzy rough sets based on residuated lattices
Computers and Mathematics with Applications
58
189 
201
DOI : 10.1016/j.camwa.2009.03.100
Ma Zhen Ming
,
Hu Bao Qing
2013
Topological and lattice structures of Lfuzzy rough set determined by lower and upper sets
Information Sciences
218
194 
204
DOI : 10.1016/j.ins.2012.06.029