Information Technology Reference
In-Depth Information
6.6 Operation “Substitution” Over an 3D-IM
Let the 3D-IM
A
be given.
First, local substitution over the IM is defined for the pairs of indices
=[
K
,
L
,
H
,
{
a
k
,
l
,
h
}]
(
p
,
k
)
and/or
(
q
,
l
)
and/or
(
r
,
h
)
, respectively, by
p
k
;⊥;⊥
A
=
(
a
k
i
,
l
j
,
h
g
}
,
K
−{
k
}
)
∪{
p
}
,
L
,
H
,
{
A
=
K
a
k
i
,
l
j
,
h
g
}
,
q
l
;⊥
⊥;
,(
L
−{
l
}
)
∪{
q
}
,
H
,
{
A
=
K
a
k
i
,
l
j
,
h
g
}
.
r
h
⊥; ⊥;
,
L
,(
H
−{
h
}
)
∪{
r
}
,
{
Second,
p
k
;
A
p
k
;⊥;⊥
⊥;
⊥; ⊥;
A
q
l
;
r
h
q
l
;⊥
r
h
=
=
(
a
k
i
,
l
j
,
h
g
}
.
K
−{
k
}
)
∪{
p
}
,(
L
−{
l
}
)
∪{
q
}
,(
H
−{
h
}
)
∪{
r
}
,
{
Let the sets of indices
P
={
p
1
,
p
2
,...,
p
m
}
,
Q
={
q
1
,
q
2
,...,
q
n
}
,
R
=
{
r
1
,
r
2
,...,
r
s
}
=
(
),
=
(
),
=
(
).
be given, where
m
car d
K
n
car d
L
s
car d
H
Third, for them we define sequentially:
P
K
;⊥;⊥
A
=
p
1
k
1
A
,
p
2
k
2
···
p
m
k
m
;⊥;⊥
A
=
A
,
Q
L
;⊥
q
1
l
1
q
2
l
2
···
q
n
l
n
;⊥
⊥;
⊥;
A
=
A
,
R
H
r
1
h
1
r
2
h
2
···
r
s
h
s
;⊥
⊥; ⊥;
⊥;
P
K
;
A
=
p
1
k
1
A
=
P
,
Q
,
R
,
{
a
k
,
l
,
h
}
.
Q
L
;
R
H
p
2
k
2
···
p
m
k
m
;
q
1
l
1
q
2
l
2
···
q
n
l
n
;
r
1
h
1
r
2
h
2
···
r
s
h
s
6.7 An Example with Bookshops
Let us have bookshops
B
1
,
C
c
. Obviously,
some bookshops can be in one company and in different towns. Let us interested in
the sales of the topics with titles
A
1
,
B
2
,...,
B
b
in different towns
C
1
,
C
2
,...,
A
2
,...,
A
a
.
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