Information Technology Reference
In-Depth Information
⎫
⎬
h
f
l
1
...
l
j
...
l
n
.
k
1
a
k
1
,
l
1
,
h
f
a
k
1
,
l
j
,
h
f
...
a
k
1
,
l
n
,
h
f
...,
,
.
.
.
.
⎭
. . .
. . .
k
m
a
k
m
,
l
1
,
h
f
...
a
k
m
,
l
j
,
h
f
...
a
k
m
,
l
n
,
h
f
where
K
={
k
1
,
k
2
,...,
k
m
}
,
L
={
l
1
,
l
2
,...,
l
n
}
,
H
={
h
1
,
h
2
,...,
h
f
}
, and for
1
≤
i
≤
m
,
1
≤
j
≤
n
,1
≤
g
≤
f
:
a
k
i
,
l
j
,
h
g
∈
X
.
6.2 Operations Over 3D-IMs
First, we start with operation “transposition".
EIM and its transposed EIM. Now, for 3D-IMs, there are 6 (=3!) cases: the standard
3D-IM and five different transposed 3D-IMs. The geometrical and analytical forms
of the separate transposed 3D-IMs are the following.
[1,2,3]
-transposition (identity)
⎛
⎞
[1
,
2
,
3]
H
H
⎝
⎠
=
L
L
K
K
a
k
i
,
l
j
,
h
g
}]
[
1
,
2
,
3
]
=[
[
K
,
L
,
H
,
{
K
,
L
,
H
,
{
a
k
i
,
l
j
,
h
g
}];
[1,3,2]
-transposition
⎛
⎞
[1
,
3
,
2]
H
L
⎝
⎠
=
L
H
K
K
a
k
i
,
l
j
,
h
g
}]
[
1
,
3
,
2
]
=[
[
,
,
,
{
,
,
,
{
a
k
i
,
h
g
,
l
j
}];
K
L
H
K
H
L
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