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 leftcontinuous tnorm ⊙,
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}
∈
L^{X}
,
and
Lemma 1.3.
[3
,
5]
For each
x, y, z, x_{i}, y_{i}
∈
L
, we have the following properties.
Definition 1.4.
[8
,
9]
(1) A map
H
:
L^{X}
→
L^{X}
is called an
Lupper approximation operator
iff it satisfies the following conditions
(2) A map 𝒥 :
L^{X}
→
L^{X}
is called an
Llower approximation operator
iff it satisfies the following conditions
(3) A map
K
:
L^{X}
→
L^{X}
is called an
Ljoin meet approximation operator
iff it satisfies the following conditions
(4) A map
M
:
L^{X}
→
L^{X}
is called an
Lmeet join approximation operator
iff it satisfies the following conditions
Definition 1.5.
[6
,
9]
A subset 𝜏 ⊂
L^{X}
is called an
Alexandrov Ltopology
if it satisfies:
Theorem 1.6.
[8
,
9]
(1) 𝜏 is an Alexandrov topology on
X
iff 𝜏
_{⁎}
= {
A
* ∈
L^{X}

A
∈ 𝜏} is an Alexandrov topology on
X
.
(2) If
H
is an
L
upper approximation operator, then 𝜏
_{H}
= {
A
∈
L^{X}

H
(
A
) =
A
} is an Alexandrov topology on
X
.
(3) If 𝒥 is an
L
lower approximation operator, then 𝜏
_{𝒥}
= {
A
∈
L^{X}
 𝒥 (
A
) =
A
} is an Alexandrov topology on
X
.
(4) If
K
is an
L
join meet approximation operator, then 𝜏
_{K}
= {
A
∈
L^{X}

K
(
A
) =
A
*} is an Alexandrov topology on
X
.
(5) If
M
is an
L
meet join operator, then 𝜏
_{M}
= {
A
∈
L^{X}

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
Lfuzzy preorder
.
If
R
satisfies (R1), (R2) and (R3),
R
is called a
Lfuzzy equivalence relation
2. The Properties ofLlower Approximation Operators
Theorem 2.1.
Let 𝒥 :
L^{X}
→
L^{X}
be an
L
lower approximation operator. Then the following properties hold.
(1) For
A
∈
L^{X}
,
.
(2) Define
H_{J}
(
B
) = ∧{
A

B
≤ 𝒥 (
A
)}. Then
H_{J}
:
L^{X}
→
L^{X}
with
is an
L
upper approximation operator such that (
H_{J}
,𝒥 )
is a residuated connection;i.e.,
H_{J} (B) ≤ A iff B ≤ 𝒥 (A).
Moreover, 𝜏
_{𝒥}
= 𝜏
_{HJ}
.
(3) If 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
L^{X}
, then
H_{J}
(
H_{J}
(
A
)) =
H_{J}
(
A
) for
A
∈
L^{X}
such that 𝜏
_{𝒥}
= 𝜏
_{HJ}
with
(4) If 𝒥 (𝒥 *(
A
)) = 𝒥 *(
A
) for
A
∈
L^{X}
, then 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) such that
(5) Define
H_{s}
(
A
) = 𝒥 (
A
*)*. Then
Hs
:
L^{X}
→
L^{X}
with
is an
L
upper approximation operator. Moreover, 𝜏
_{Hs}
= (𝜏
_{𝒥}
)
_{*}
= (𝜏
_{HJ}
)
_{*}
.
(6) If 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
L^{X}
, then
H_{s}(H_{s}(A)) = H_{s}(A)
for
A
∈
L^{X}
such that 𝜏
_{Hs}
= (𝜏
_{𝒥}
)
_{*}
= (𝜏
_{HJ}
)
_{*}
. with
(7) If 𝒥 (𝒥 *(
A
)) = 𝒥 *(
A
) for
A
∈
L^{X}
, then
such that
(8) Define
K_{J}
(
A
) = 𝒥 (
A
*). Then
K_{J}
:
L^{X}
→
L^{X}
with
is an
L
join meet approximation operator.
(9) If 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
L^{X}
, then
for
A
∈
L^{X}
such that 𝜏
_{KJ}
= (𝜏
_{𝒥}
)
_{*}
with
(10) If 𝒥 (𝒥 *(
A
)) = 𝒥 *(
A
) for
A
∈
L^{X}
, then
such that
(11) Define
M_{J}
(
A
) = (𝒥 (
A
))*. Then
M_{J}
:
L^{X}
→
L^{X}
with
is an
L
meet join approximation operator. Moreover, 𝜏
_{MJ}
= 𝜏
_{𝒥}
.
(12) If 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
L^{X}
, then
for
A
∈
L^{X}
such that 𝜏
_{MJ}
= (𝜏
_{𝒥}
)
_{*}
with
(13) If 𝒥 (𝒥 *(
A
)) = 𝒥 *(
A
) for
A
∈
L^{X}
, then
such that
(14) Define
K_{HJ}
(
A
) = (
H_{J}
(
A
))*. Then
K_{HJ}
:
L^{X}
→
L^{X}
with
is an
L
meet join approximation operator. Moreover, 𝜏
_{KHJ}
= 𝜏
_{𝒥}
.
(15) If 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
L^{X}
, then
for
A
∈
L^{X}
such that 𝜏
_{KHJ}
= (𝜏
_{𝒥}
)
_{*}
with
(16) If
for
A
∈
L^{X}
, then
such that
(17) Define
M_{HJ}
(
A
) =
H_{J}
(
A
*). Then
M_{HJ}
:
L^{X}
→
L^{X}
with
is an
L
join meet approximation operator. Moreover, 𝜏
_{MHJ}
= (𝜏
_{𝒥}
)
_{*}
.
(18) If 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
L^{X}
, then
for
A
∈
L^{X}
such that 𝜏
_{MHJ}
= (𝜏
_{𝒥}
)
_{*}
with
(19) If
for
A
∈
L^{X}
, then
such that
(20) (
K_{HJ}
,
K_{J}
) is a Galois connection;i.e,
A ≤ K_{HJ} (B) iff B ≤ K_{J} (A).
Moreover, 𝜏
_{KJ}
= (𝜏
_{KHJ}
)
_{*}
.
(21) (
M_{J}
,
M_{HJ}
) is a dual Galois connection;i.e,
M_{HJ} (A) ≤ B iff M_{J} (B) ≤ A.
Moreover, 𝜏
_{MJ}
= (𝜏
_{MHJ}
)
_{*}
.
Proof.
(1) Since
, by (J2) and (J3),
(2) Since
iff
, we have
(H1) Since
H_{J}
(
A
) ≤
H_{J}
(
A
) iff A ≤ 𝒥 (
H_{J}
(
A
)), we have
A
≤ 𝒥 (
H_{J}
(
A
)) ≤
H_{J}
(
A
).
(H3) By the definition of
H_{J}
, since
H_{J}
(
A
) ≤
H_{J}
(
B
) for
B
≤
A
, we have
Since 𝒥 (∨
_{i∈𝚪}
H_{J}
(
A_{i}
)) ≥ 𝒥 (
H_{J}
(
A_{i}
)) ≥
A_{i}
, then
𝒥 (∨
_{i∈𝚪}
H_{J}
(
A_{i}
)) ≥ ∨
_{i∈𝚪}
A_{i}
. Thus
Thus
H_{J}
:
L^{X}
→
L^{X}
is an
L
upper approximation operator. By the definition of
H_{J}
, we have
H_{J} (B) ≤ A iff B ≤ 𝒥 (A).
Since
A
≤ 𝒥 (
A
) iff
H_{J}
(
A
) ≤
A
, we have 𝜏
_{HJ}
= 𝜏
_{𝒥}
.
(3) Let 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
L^{X}
. Since 𝒥 (
B
) ≥
H_{J}
(
A
) iff 𝒥 (𝒥 (
B
)) = 𝒥 (
B
) ≥
A
from the definition of
H_{J}
, 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
*,
H_{s}
(
A
) = 𝒥 (
A
*)* ≥
A
.
Hence
H_{s}
is an
L
upper approximation operator such that
Moreover, 𝜏
_{Hs}
= (𝜏
_{𝒥}
)
_{*}
from:
A = H_{s}(A) iff A = 𝒥 (A*)* iff A* = 𝒥 (A*).
(6) Let 𝒥 (𝒥 (
A
)) = 𝒥 (
A
) for
A
∈
L^{X}
. Then
Hence
(7) Let 𝒥 (𝒥 *(
A
)) = 𝒥 *(
A
) for
A
∈
L^{X}
. 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
∈
L^{X}
, then
(10) If 𝒥 (𝒥 *(
A
)) = 𝒥 *(
A
) for
A
∈
L^{X}
, then
Since
,
Hence 𝜏
_{KJ}
= {
K_{J}
(
A
) 
A
∈
L^{X}
} = (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
∈
L^{X}
, then
H_{J}
(
H_{J}
(
A
)) =
H_{J}
(
A
). Thus,
Since 𝒥 (
A
) =
A
iff
H_{J}
(
A
) =
A
iff
K_{HJ}
(
A
) =
A
*, 𝜏
_{KHJ}
= (𝜏
_{𝒥}
)
_{*}
with
(16) If
for A ∈
L^{X}
, then
(17) , (18) and (19) are similarly proved as (14), (15) and (16), respectively.
(20) (
K_{HJ}
,
K_{J}
) is a Galois connection;i.e,
A ≤ K_{HJ} (B) iff A ≤ (H_{J} (B))*
iff H_{J} (B) ≤ A* iff B ≤ 𝒥 (A*) = K_{J} (A)
Moreover, since
A
* ≤
K_{J}
(
A
) iff
A
≤
K_{HJ}
(
A
*), 𝜏
_{KJ}
= (𝜏
_{KHJ}
)
_{*}
.
(21) (
M_{J}
,
M_{HJ}
) is a dual Galois connection;i.e,
M_{HJ} (A) ≤ B iff H_{J} (A*) ≤ B
iff A* ≤ 𝒥 (B) iff M_{J} (B) = (𝒥 (B))* ≤ A.
Since
M_{HJ}
(
A
*) ≤
A
iff
M_{J}
(
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}
:
L^{X}
→
L^{X}
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
H_{JR}
(
B
) = ∨ {
A

B
≤ 𝒥
_{R}
(
A
)}. Since
then
By Theorem 2.1(2),
H_{JR}
=
H
_{R1}
is an
L
upper approximation operator such that (
H_{JR}
,𝒥
_{R}
) is a residuated connection;i.e.,
H_{JR}(A) ≤ B iff A ≤ 𝒥_{R}(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),
H_{JR}
(
H_{JR}
(
A
)) =
H_{JR}
(
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
∈
L^{X}
.
Thus, 𝜏
_{𝒥R}
= (𝜏
_{𝒥R}
)
_{*}
with
(5) Define
H_{s}
(
A
) = 𝒥
_{R}
(
A
*)*. By Theorem 2.1(5),
H_{s}
=
H_{R}
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
∈
L^{X}
. By Theorem 2.1(6), then
H_{s}
(
H_{s}
(
A
)) =
H_{s}
(
A
) for
A
∈
L^{X}
such that 𝜏
H_{s}
= (𝜏
_{𝒥R}
)
_{*}
= (𝜏
_{HJR}
)
_{*}
with
(7) If
R
is a reflexive and Euclidean
L
fuzzy relation, then
for
A
∈
L^{X}
. By Theorem 2.1(7),
such that
(8) Define
K_{JR}
(
A
) = 𝒥
_{R}
(
A
*). Then
K_{JR}
:
L^{X}
→
L^{X}
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
∈
L^{X}
. By Theorem 2.1(9),
for
A
∈
L^{X}
such that 𝜏
_{KJR}
= (𝜏
_{𝒥R}
)
_{*}
with
(10) If
R
is a reflexive and Euclidean
L
fuzzy relation, then
for
A
∈
L^{X}
. By Theorem 2.1(10),
such that
(11) Define
M_{JR}
(
A
) = (𝒥
_{R}
(
A
))*. Then
M_{JR}
:
L^{X}
→
L^{X}
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
∈
L^{X}
. By Theorem 2.1(12),
for
A
∈
L^{X}
such that 𝜏
_{MJR}
= 𝜏
_{𝒥R}
with
(13) If
R
is a reflexive and Euclidean
L
fuzzy relation, then
for
A
∈
L^{X}
. By Theorem 2.1(13),
such that
(14) Define
K_{HJR}
(
A
) = (
H_{JR}
(
A
))*. Then
K_{HJR} : L^{X} → L^{X}
with
is an
L
join meet approximation operator. Moreover, 𝜏
_{KR1}
= 𝜏
_{𝒥R}
= 𝜏
_{HR1}
.
(15) If
R
is an
L
fuzzy preorder, then 𝒥
_{R}
(𝒥
_{R}
(
A
)) = 𝒥
_{R}
(
A
) for
A
∈
L^{X}
. By Theorem 2.1(15),
for
A
∈
L^{X}
such that 𝜏
_{KR1}
= 𝜏
_{𝒥R}
= 𝜏
_{HR1}
with
(16) Let
R
^{1}
be a reflexive and Euclidean
L
fuzzy relation. Since
we have
Thus,
Hence
By (K1),
such that
(17) Define
M_{HJR}
(
A
) =
H_{JR}
(
A
*). Then
M_{HJR} : L^{X} → L^{X}
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
∈
L^{X}
. By Theorem 2.1(18),
for
A
∈
L^{X}
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) (
K_{HJR}
=
K
_{R1*}
,
K_{JR}
=
K
_{R*}
) is a Galois connection; i.e,
A
≤
K_{HJR}
(
B
) iff
B
≤
K_{JR}
(
A
): Moreover, 𝜏
_{KJR}
= (𝜏
_{KHJR}
)
_{*}
.
(21) (
M_{JR}
=
M_{R}
,
M_{HJR}
=
M
_{R1}
) is a dual Galois connection; i.e,
M_{HJR}
(
A
) ≤
B
iff
M_{JR}
(
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 GangneungWonju 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 GangneungWonju University, his research interests is a fuzzy topology and fuzzy logic.
Email: yck@gwnu.ac.kr
Pawlak Z.
1982
“Rough sets,”
International Journal of Computer & Information Sciences
http://dx.doi.org/10.1007/BF01001956
11
(5)
341 
356
DOI : 10.1007/BF01001956
Pawlak Z.
1984
“Rough probability,”
Bulletin of Polish Academy of Sciences: Mathematics
32
(910)
607 
615
Hájek P.
1998
Metamathematics of Fuzzy Logic
Kluwer
Dordrecht, The Netherlands
Radzikowska A. M.
,
Kerre E. E.
2002
“A comparative study of fuzzy rough sets,”
Fuzzy Sets and Systems
http://dx.doi.org/10.1016/S01650114(01)00032X
126
(2)
137 
155
DOI : 10.1016/S01650114(01)00032X
Bělohlávek R.
2002
Fuzzy Relational Systems: Foundations and Principles
Kluwer Academic/Plenum Publishers
New York, NY
Lai H.
,
Zhang D.
2006
“Fuzzy preorder and fuzzy topology,”
Fuzzy Sets and Systems
http://dx.doi.org/10.1016/j.fss.2006.02.013
157
(14)
1865 
1885
DOI : 10.1016/j.fss.2006.02.013
Lai H.
,
Zhang D.
2009
“Concept lattices of fuzzy contexts: formal concept analysis vs. rough set theory,”
International Journal of Approximate Reasoning
http://dx.doi.org/10.1016/j.ijar.2008.12.002
50
(5)
695 
707
DOI : 10.1016/j.ijar.2008.12.002
Kim Y. C.
2014
“Alexandrov Ltopologiesand Ljoin meet approximatin operators,”
International Journal of Pure and Applied Mathematics
http://dx.doi.org/10.12732/ijpam.v91i1.12
91
(1)
113 
129
DOI : 10.12732/ijpam.v91i1.12
Kim Y. C.
2014
“Alexandrov Ltopologies,”
International Journal of Pure and Applied Mathematics
She Y. H.
,
Wang G. J.
2009
“An axiomatic approach of fuzzy rough sets based on residuated lattices,”
Computers & Mathematics with Applications
http://dx.doi.org/10.1016/j.camwa.2009.03.100
58
(1)
189 
201
DOI : 10.1016/j.camwa.2009.03.100