Information Technology Reference
In-Depth Information
⎧
⎨
⎩
a
k
i
,
l
j
,
if
β
γ
=
k
i
∈
K
−{
k
f
}
and
δ
ε
=
l
j
∈
L
−{
l
g
}
b
p
r
,
q
s
,
if
β
γ
=
p
r
∈
P
−{
m
d
}
and
δ
ε
=
q
s
∈
Q
−{
n
e
}
α
β
γ
,δ
ε
=
c
t
u
,v
w
,
if
β
γ
=
t
u
∈
R
and
δ
ε
=
v
w
∈
S
0
,
otherwise
Theorem 7
For the above EIMs A
,
a
k
f
,
l
g
and b
m
d
,
n
e
(
A
|
(
a
k
f
,
l
g
))
|
(
b
m
d
,
n
e
)
=
A
|
((
a
k
f
,
l
g
)
|
(
b
m
d
,
n
e
)).
From the first definition of a hierarchical operator it follows that
A
|
(
a
k
f
,
l
g
)
l
1
...
l
g
−
1
q
1
...
q
u
l
g
+
1
...
l
n
k
1
a
k
1
,
l
1
...
a
k
1
,
l
g
−
1
0
...
0
a
k
1
,
l
g
+
1
...
a
k
1
,
l
n
.
.
.
.
.
.
.
.
.
.
...
...
...
k
f
−
1
a
k
f
−
1
,
l
1
a
k
f
−
1
,
l
g
−
1
0
0
a
k
f
−
1
,
l
g
+
1
a
k
f
−
1
,
l
n
p
1
0
...
0
b
p
1
,
q
1
...
b
p
1
,
q
v
0
...
0
=
.
.
.
.
.
.
.
.
.
.
.
p
u
0
...
0
b
p
u
,
q
1
...
b
p
u
,
q
v
0
...
0
k
f
+
1
a
k
f
+
1
,
l
1
...
a
k
f
+
1
,
l
g
−
1
0
...
0
a
k
f
+
1
,
l
g
+
1
...
a
k
f
+
1
,
l
n
.
.
.
.
.
.
.
.
.
.
k
m
a
k
m
,
l
1
...
a
k
m
,
l
g
−
1
0
...
0
a
k
m
,
l
g
+
1
...
a
k
m
,
l
n
From this form of the IM A
|
(
a
k
f
,
l
g
)
we see that for the hierarchical operator the
following equality holds
.
Theorem 8
Let A
=[
K
,
L
,
{
a
k
i
,
l
j
}]
be an IM and let a
k
f
,
l
g
=[
P
,
Q
,
{
b
p
r
,
q
s
}]
be its
element. Then
|
(
a
k
f
,
l
g
)
=
(
[{
k
f
}
,
{
l
g
}
,
{
}]
)
⊕
a
k
f
,
l
g
.
A
A
0
a
k
f
,
l
n
in the IM
A now are replaced by
“0”.
Therefore, as a result of this operator, information is
being lost
.
Below, we modify the first hierarchical operator, so that all the information from
the IMs, participating in it, be kept. The new—second—form of this operator for the
above defined IM A and its fixed element a
k
f
,
l
g
,is
We see that the elements a
k
f
,
l
1
,
a
k
f
,
l
2
,...,
a
k
f
,
l
g
−
1
,
a
k
f
,
l
g
+
1
,...,
|
∗
(
A
a
k
f
,
l
g
)
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