Information Technology Reference
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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 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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