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In-Depth Information
Definition 15
([
16
,
49
]) AQ-reduct inDRSMis aminimal condition attribute subset
A
ↆ
C
satisfying the following condition:
ʳ
A
(
d
)
=
ʳ
C
(
d
).
(DQ)
Now, we introduce structure-based reducts in DRSM. Lower and upper approx-
imations and boundary regions of upward and downward unions can be considered
as a structure over a given object set
U
. From this point, we define 7 union-structure-
preserving reducts. The following reducts are conceivable.
Definition 16
([
25
,
52
]) We define 7 types of reducts as follows.
An L
≥
-reduct in DRSM is a minimal condition attribute subset
A
•
ↆ
C
satisfying
the following condition:
X
i
X
i
(DL
≥
)
LA
A
(
)
=
LA
C
(
)
for all
i
∈
V
d
.
An L
≤
-reduct in DRSM is a minimal condition attribute subset
A
•
ↆ
C
satisfying
the following condition:
X
i
X
i
(DL
≤
)
LA
A
(
)
=
LA
C
(
)
for all
i
∈
V
d
.
AU
≥
-reduct in DRSM is a minimal condition attribute subset
A
•
ↆ
C
satisfying
the following condition:
X
i
X
i
(DU
≥
)
UA
A
(
)
=
UA
C
(
)
for all
i
∈
V
d
.
AU
≤
-reduct in DRSM is a minimal condition attribute subset
A
•
ↆ
C
satisfying
the following condition:
X
i
X
i
(DU
≤
)
UA
A
(
)
=
UA
C
(
)
for all
i
∈
V
d
.
An L
-reduct in DRSM is a minimal condition attribute subset
A
•
ↆ
C
satisfying
the following condition:
X
i
X
i
X
i
X
i
(DL
)
LA
A
(
)
=
LA
C
(
)
and LA
A
(
)
=
LA
C
(
)
for all
i
∈
V
d
.
AU
-reduct in DRSM is a minimal condition attribute subset
A
•
ↆ
C
satisfying
the following condition:
X
i
X
i
X
i
X
i
(DU
)
UA
A
(
)
=
UA
C
(
)
and UA
A
(
)
=
UA
C
(
)
for all
i
∈
V
d
.
AB
-reduct in DRSM is a minimal condition attribute subset
A
•
ↆ
C
satisfying
the following condition:
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