In this paper, we investigate the properties of
L
-lower approximation operators as a generalization of fuzzy rough set in complete residuated lattices. We study relations lower (upper, join meet, meet join) approximation operators and Alexandrov
L
-topologies. Moreover, we give their examples as approximation operators induced by various
L
-fuzzy relations.
1. Introduction
Pawlak
[1
,
2]
introduced rough set theory as a formal tool to deal with imprecision and uncertainty in data analysis. Hájek
[3]
introduced a complete residuated lattice which is an algebraic structure for many valued logic. Radzikowska and Kerre
[4]
developed fuzzy rough sets in complete residuated lattice. Bělohlávek
[5]
investigated information systems and decision rules in complete residuated lattices. Lai and Zhang
[6
,
7]
introduced Alexandrov
L
-topologies induced by fuzzy rough sets. Kim
[8
,
9]
investigate relations between lower approximation operators as a generalization of fuzzy rough set and Alexandrov
L
-topologies. Algebraic structures of fuzzy rough sets are developed in many directions
[4
,
8
,
10]
In this paper, we investigate the properties of
L
-lower approximation operators as a generalization of fuzzy rough set in complete residuated lattices. We study relations lower (upper, join meet, meet join) approximation operators and Alexandrov
L
-topologies. Moreover, we give their examples as approximation operators induced by various
L
-fuzzy relations.
Definition 1.1.
[3
,
5]
An algebra (
L
,∧,∨,⊙,→,⊥,⊤) is called a complete residuated lattice if it satisfies the following conditions:
(C1)
L
= (
L
,≤,∨,∧,⊥,⊤) is a complete lattice with the greatest element ⊤ and the least element ⊥;
(C2) (
L
,⊙,⊤) is a commutative monoid;
C3)
x
⊙
y
≤
z
iff
x
≤
y
→
z
for
x
,
y
,
z
∈
L
Remark 1.2.
[3
,
5]
(1) A completely distributive lattice
L
= (
L
,≤,∨,∧ = ⊙,→, 1, 0) is a complete residuated lattice defined by
(2) The unit interval with a left-continuous t-norm ⊙,
is a complete residuated lattice defined by
In this paper, we assume (
L
,∧,∨,⊙,→,* ⊥,⊤) is a complete residuated lattice with the law of double negation;i.e.
x
** =
x
. For 𝛼 ∈
L
,
A
,⊤
x
∈
LX
,
and
Lemma 1.3.
[3
,
5]
For each
x, y, z, xi, yi
∈
L
, we have the following properties.
Definition 1.4.
[8
,
9]
(1) A map
H
:
LX
→
LX
is called an
L-upper approximation operator
iff it satisfies the following conditions
(2) A map 𝒥 :
LX
→
LX
is called an
L-lower approximation operator
iff it satisfies the following conditions
(3) A map
K
:
LX
→
LX
is called an
L-join meet approximation operator
iff it satisfies the following conditions
(4) A map
M
:
LX
→
LX
is called an
L-meet join approximation operator
iff it satisfies the following conditions
Definition 1.5.
[6
,
9]
A subset 𝜏 ⊂
LX
is called an
Alexandrov L-topology
if it satisfies:
Theorem 1.6.
[8
,
9]
(1) 𝜏 is an Alexandrov topology on
X
iff 𝜏
⁎
= {
A
* ∈
LX
|
A
∈ 𝜏} is an Alexandrov topology on
X
.
(2) If
H
is an
L
-upper approximation operator, then 𝜏
H
= {
A
∈
LX
|
H
(
A
) =
A
} is an Alexandrov topology on
X
.
(3) If 𝒥 is an
L
-lower approximation operator, then 𝜏
𝒥
= {
A
∈
LX
| 𝒥 (
A
) =
A
} is an Alexandrov topology on
X
.
(4) If
K
is an
L
-join meet approximation operator, then 𝜏
K
= {
A
∈
LX
|
K
(
A
) =
A
*} is an Alexandrov topology on
X
.
(5) If
M
is an
L
-meet join operator, then 𝜏
M
= {
A
∈
LX
|
M
(
A
) =
A
*} is an Alexandrov topology on
X
.
Definition 1.7.
[8
,
9]
Let
X
be a set. A function
R
:
X
×
X
→
L
is called:
If
R
satisfies (R1) and (R3),
R
is called a
L-fuzzy preorder
.
If
R
satisfies (R1), (R2) and (R3),
R
is called a
L-fuzzy equivalence relation
2. The Properties ofL-lower Approximation Operators
Theorem 2.1.
Let 𝒥 :
LX
→
LX
be an
L
-lower approximation operator. Then the following properties hold.
(1) For
A
∈
LX
,
.
(2) Define
HJ
(
B
) = ∧{
A
|
B
≤ 𝒥 (
A
)}. Then
HJ
:
LX
→
LX
with
is an
L
-upper approximation operator such that (
HJ
,𝒥 )
is a residuated connection;i.e.,
HJ (B) ≤ A iff B ≤ 𝒥 (A).
Moreover, 𝜏
𝒥
= 𝜏
HJ
.
(3) If 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
LX
, then
HJ
(
HJ
(
A
)) =
HJ
(
A
) for
A
∈
LX
such that 𝜏
𝒥
= 𝜏
HJ
with
(4) If 𝒥 (𝒥 *(
A
)) = 𝒥 *(
A
) for
A
∈
LX
, then 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) such that
(5) Define
Hs
(
A
) = 𝒥 (
A
*)*. Then
Hs
:
LX
→
LX
with
is an
L
-upper approximation operator. Moreover, 𝜏
Hs
= (𝜏
𝒥
)
*
= (𝜏
HJ
)
*
.
(6) If 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
LX
, then
Hs(Hs(A)) = Hs(A)
for
A
∈
LX
such that 𝜏
Hs
= (𝜏
𝒥
)
*
= (𝜏
HJ
)
*
. with
(7) If 𝒥 (𝒥 *(
A
)) = 𝒥 *(
A
) for
A
∈
LX
, then
such that
(8) Define
KJ
(
A
) = 𝒥 (
A
*). Then
KJ
:
LX
→
LX
with
is an
L
-join meet approximation operator.
(9) If 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
LX
, then
for
A
∈
LX
such that 𝜏
KJ
= (𝜏
𝒥
)
*
with
(10) If 𝒥 (𝒥 *(
A
)) = 𝒥 *(
A
) for
A
∈
LX
, then
such that
(11) Define
MJ
(
A
) = (𝒥 (
A
))*. Then
MJ
:
LX
→
LX
with
is an
L
-meet join approximation operator. Moreover, 𝜏
MJ
= 𝜏
𝒥
.
(12) If 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
LX
, then
for
A
∈
LX
such that 𝜏
MJ
= (𝜏
𝒥
)
*
with
(13) If 𝒥 (𝒥 *(
A
)) = 𝒥 *(
A
) for
A
∈
LX
, then
such that
(14) Define
KHJ
(
A
) = (
HJ
(
A
))*. Then
KHJ
:
LX
→
LX
with
is an
L
-meet join approximation operator. Moreover, 𝜏
KHJ
= 𝜏
𝒥
.
(15) If 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
LX
, then
for
A
∈
LX
such that 𝜏
KHJ
= (𝜏
𝒥
)
*
with
(16) If
for
A
∈
LX
, then
such that
(17) Define
MHJ
(
A
) =
HJ
(
A
*). Then
MHJ
:
LX
→
LX
with
is an
L
-join meet approximation operator. Moreover, 𝜏
MHJ
= (𝜏
𝒥
)
*
.
(18) If 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
LX
, then
for
A
∈
LX
such that 𝜏
MHJ
= (𝜏
𝒥
)
*
with
(19) If
for
A
∈
LX
, then
such that
(20) (
KHJ
,
KJ
) is a Galois connection;i.e,
A ≤ KHJ (B) iff B ≤ KJ (A).
Moreover, 𝜏
KJ
= (𝜏
KHJ
)
*
.
(21) (
MJ
,
MHJ
) is a dual Galois connection;i.e,
MHJ (A) ≤ B iff MJ (B) ≤ A.
Moreover, 𝜏
MJ
= (𝜏
MHJ
)
*
.
Proof.
(1) Since
, by (J2) and (J3),
(2) Since
iff
, we have
(H1) Since
HJ
(
A
) ≤
HJ
(
A
) iff A ≤ 𝒥 (
HJ
(
A
)), we have
A
≤ 𝒥 (
HJ
(
A
)) ≤
HJ
(
A
).
(H3) By the definition of
HJ
, since
HJ
(
A
) ≤
HJ
(
B
) for
B
≤
A
, we have
Since 𝒥 (∨
i∈𝚪
HJ
(
Ai
)) ≥ 𝒥 (
HJ
(
Ai
)) ≥
Ai
, then
𝒥 (∨
i∈𝚪
HJ
(
Ai
)) ≥ ∨
i∈𝚪
Ai
. Thus
Thus
HJ
:
LX
→
LX
is an
L
-upper approximation operator. By the definition of
HJ
, we have
HJ (B) ≤ A iff B ≤ 𝒥 (A).
Since
A
≤ 𝒥 (
A
) iff
HJ
(
A
) ≤
A
, we have 𝜏
HJ
= 𝜏
𝒥
.
(3) Let 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
LX
. Since 𝒥 (
B
) ≥
HJ
(
A
) iff 𝒥 (𝒥 (
B
)) = 𝒥 (
B
) ≥
A
from the definition of
HJ
, we have
(4) Let 𝒥 *(
A
) ∈ 𝜏
𝒥
. Since 𝒥 (𝒥 *(
A
)) = 𝒥 *(
A
),
𝒥 (𝒥 (
A
)) = 𝒥 (𝒥 *(𝒥 *(
A
))) = (𝒥 (𝒥 *(
A
)))* = 𝒥 (
A
).
Hence 𝒥 (
A
) ∈ 𝜏
𝒥
; i.e. 𝒥 *(
A
) ∈ (𝜏
𝒥
)
*
. Thus, 𝜏
𝒥
⊂ (𝜏
𝒥
)
*
.
Let
A
∈ (𝜏
𝒥
)
*
. Then
A
* = 𝒥 (
A
*). Since 𝒥 (
A
) = 𝒥 (𝒥 *(
A
*)) = 𝒥 *(
A
*) =
A
, then
A
∈ 𝜏
𝒥
. Thus, (𝜏
𝒥
)
*
⊂ 𝜏
𝒥
.
(5) (H1) Since 𝒥 (
A
*) ≤
A
*,
Hs
(
A
) = 𝒥 (
A
*)* ≥
A
.
Hence
Hs
is an
L
-upper approximation operator such that
Moreover, 𝜏
Hs
= (𝜏
𝒥
)
*
from:
A = Hs(A) iff A = 𝒥 (A*)* iff A* = 𝒥 (A*).
(6) Let 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
LX
. Then
Hence
(7) Let 𝒥 (𝒥 *(
A
)) = 𝒥 *(
A
) for
A
∈
LX
. Then
Hence
By a similar method in (4), 𝜏
Hs
= (𝜏
Hs
)
*
.
(8) It is similarly proved as (5).
(9) If 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
LX
, then
(10) If 𝒥 (𝒥 *(
A
)) = 𝒥 *(
A
) for
A
∈
LX
, then
Since
,
Hence 𝜏
KJ
= {
KJ
(
A
) |
A
∈
LX
} = (𝜏
KJ
)
*
.
(11) , (12), (13) and (14) are similarly proved as (5), (9), (10) and (5), respectively.
(15) If 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
LX
, then
HJ
(
HJ
(
A
)) =
HJ
(
A
). Thus,
Since 𝒥 (
A
) =
A
iff
HJ
(
A
) =
A
iff
KHJ
(
A
) =
A
*, 𝜏
KHJ
= (𝜏
𝒥
)
*
with
(16) If
for A ∈
LX
, then
(17) , (18) and (19) are similarly proved as (14), (15) and (16), respectively.
(20) (
KHJ
,
KJ
) is a Galois connection;i.e,
A ≤ KHJ (B) iff A ≤ (HJ (B))*
iff HJ (B) ≤ A* iff B ≤ 𝒥 (A*) = KJ (A)
Moreover, since
A
* ≤
KJ
(
A
) iff
A
≤
KHJ
(
A
*), 𝜏
KJ
= (𝜏
KHJ
)
*
.
(21) (
MJ
,
MHJ
) is a dual Galois connection;i.e,
MHJ (A) ≤ B iff HJ (A*) ≤ B
iff A* ≤ 𝒥 (B) iff MJ (B) = (𝒥 (B))* ≤ A.
Since
MHJ
(
A
*) ≤
A
iff
MJ
(
A
) ≤
A
*, 𝜏
MJ
= (𝜏
MHJ
)
*
.
Let
R
∈
L
X × X
be an
L
-fuzzy relation. Define operators as follows
Example 2.2.
Let
R
be a reflexive
L
-fuzzy relation. Define 𝒥
R
:
LX
→
LX
as follows:
(1) (J1) 𝒥
R
(
A
)(
y
) ≤
R
(
y, y
) →
A
(
y
) =
A
(
y
): 𝒥
R
satisfies the conditions (J1) and (J2) from:
Hence 𝒥
R
is an
L
-lower approximation operator.
(2) Define
HJR
(
B
) = ∨ {
A
|
B
≤ 𝒥
R
(
A
)}. Since
then
By Theorem 2.1(2),
HJR
=
H
R-1
is an
L
-upper approximation operator such that (
HJR
,𝒥
R
) is a residuated connection;i.e.,
HJR(A) ≤ B iff A ≤ 𝒥R(B).
Moreover, 𝜏
HJR
= 𝜏
𝒥R
.
(3) If
R
is an
L
-fuzzy preorder, then
R
-1
is an
L
-fuzzy preorder. Since
By Theorem 2.1(3),
HJR
(
HJR
(
A
)) =
HJR
(
A
): By Theorem 2.1(3), 𝜏
HJR
= 𝜏
𝒥R
with
(4) Let
R
be a reflexive and Euclidean
L
-fuzzy relation. Since
R
(
x, z
) ⊙
R
(
y, z
) ⊙
A
*(
x
) ≤
R
(
x, y
) ⊙
A
*(
x
) iff
R
(
x, z
) ⊙
A
*(
x
) ≤
R
(
y, z
) →
R
(
x, y
) ≤
A
*(
x
),
Thus,
.
By Theorem 2.1(4), 𝒥
R
(𝒥
R
(
A
)) = 𝒥
R
(
A
) for
A
∈
LX
.
Thus, 𝜏
𝒥R
= (𝜏
𝒥R
)
*
with
(5) Define
Hs
(
A
) = 𝒥
R
(
A
*)*. By Theorem 2.1(5),
Hs
=
HR
is an
L
-upper approximation operator such that
Moreover, 𝜏
Hs
= 𝜏
HR
= (𝜏
HJR
)
*
.
(6) If
R
is an
L
-fuzzy preorder, then 𝒥
R
(𝒥
R
(
A
)) = 𝒥
R
(
A
) for
A
∈
LX
. By Theorem 2.1(6), then
Hs
(
Hs
(
A
)) =
Hs
(
A
) for
A
∈
LX
such that 𝜏
Hs
= (𝜏
𝒥R
)
*
= (𝜏
HJR
)
*
with
(7) If
R
is a reflexive and Euclidean
L
-fuzzy relation, then
for
A
∈
LX
. By Theorem 2.1(7),
such that
(8) Define
KJR
(
A
) = 𝒥
R
(
A
*). Then
KJR
:
LX
→
LX
with
is an
L
-join meet approximation operator. Moreover, 𝜏
KJR
= (𝜏
𝒥R
)
*
.
(9)
R
is an
L
-fuzzy preorder, then 𝒥
R
(𝒥
R
(
A
)) = 𝒥
R
(
A
) for
A
∈
LX
. By Theorem 2.1(9),
for
A
∈
LX
such that 𝜏
KJR
= (𝜏
𝒥R
)
*
with
(10) If
R
is a reflexive and Euclidean
L
-fuzzy relation, then
for
A
∈
LX
. By Theorem 2.1(10),
such that
(11) Define
MJR
(
A
) = (𝒥
R
(
A
))*. Then
MJR
:
LX
→
LX
with
is an
L
-join meet approximation operator. Moreover, 𝜏
MJR
= 𝜏
𝒥R
.
(12) If
R
is an
L
-fuzzy preorder, then 𝒥
R
(𝒥
R
(
A
)) = 𝒥
R
(
A
) for
A
∈
LX
. By Theorem 2.1(12),
for
A
∈
LX
such that 𝜏
MJR
= 𝜏
𝒥R
with
(13) If
R
is a reflexive and Euclidean
L
-fuzzy relation, then
for
A
∈
LX
. By Theorem 2.1(13),
such that
(14) Define
KHJR
(
A
) = (
HJR
(
A
))*. Then
KHJR : LX → LX
with
is an
L
-join meet approximation operator. Moreover, 𝜏
KR-1
= 𝜏
𝒥R
= 𝜏
HR-1
.
(15) If
R
is an
L
-fuzzy preorder, then 𝒥
R
(𝒥
R
(
A
)) = 𝒥
R
(
A
) for
A
∈
LX
. By Theorem 2.1(15),
for
A
∈
LX
such that 𝜏
KR-1
= 𝜏
𝒥R
= 𝜏
HR-1
with
(16) Let
R
-1
be a reflexive and Euclidean
L
-fuzzy relation. Since
we have
Thus,
Hence
By (K1),
such that
(17) Define
MHJR
(
A
) =
HJR
(
A
*). Then
MHJR : LX → LX
is an
L
-meet join approximation operator as follows:
Moreover, 𝜏
MHJR
= (𝜏
𝒥R
)
*
.
(18) If
R
is an
L
-fuzzy preorder, then 𝒥
R
(𝒥
R
(
A
)) = 𝒥
R
(
A
) for
A
∈
LX
. By Theorem 2.1(18),
for
A
∈
LX
such that 𝜏
MHJR
= (𝜏
𝒥
)
*
with
(19) Let
R
-1
be a reflexive and Euclidean
L
-fuzzy relation.
Since
then (
R
(
y, x
) →
A
(
x
)) ⊙
R
(
z, y
) ≤
R
(
z, x
) →
A
(
x
).
Thus,
By (M1),
such that
(20) (
KHJR
=
K
R-1*
,
KJR
=
K
R*
) is a Galois connection; i.e,
A
≤
KHJR
(
B
) iff
B
≤
KJR
(
A
): Moreover, 𝜏
KJR
= (𝜏
KHJR
)
*
.
(21) (
MJR
=
MR
,
MHJR
=
M
R-1
) is a dual Galois connection; i.e,
MHJR
(
A
) ≤
B
iff
MJR
(
B
) ≤
A
. Moreover, 𝜏
MJR
= (𝜏
MHJR
)
*
.
3. Conclusions
In this paper,
L
-lower approximation operators induce
L
-upper approximation operators by residuated connection. We study relations lower (upper, join meet, meet join) approximation operators, Galois (dual Galois, residuated, dual residuated) connections and Alexandrov
L
-topologies. Moreover, we give their examples as approximation operators induced by various
L
-fuzzy relations.
Conflict of Interest
No potential conflict of interest relevant to this article was reported.
Acknowledgements
This work was supported by the Research Institute of Natural Science of Gangneung-Wonju National University.
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 Gangneung-Wonju University, his research interests is a fuzzy topology and fuzzy logic.
E-mail: yck@gwnu.ac.kr
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