Information Technology Reference
In-Depth Information
Chapter 3
Extended Index Matrices
The discussed above IMs are, in some sense, extensions of ordinary matrices. Now,
we introduce an IM, that includes all of them as particular cases, i.e., it is an extension
of the above four types of IMs.
3.1 Definition of an Extended Index Matrix
Let
I
be again a fixed set of indices,
n
I
={
i
1
,
i
2
,...,
i
n
|
(
∀
j
:
1
≤
j
≤
n
)(
i
j
∈
I)
}
and
I
∗
=∪
1
n
≤∞
I
.
≤
n
be a fixed set of some objects. In the particular cases, they can be either real
numbers, or only the numbers 0 or 1, or logical variables, propositions or predicates,
etc.
Let operations
Let
X
◦
,
∗:
X
×
X
→
X
be fixed.
An Extended IM (EIM) with index sets
K
and
L
⊂
I
∗
)
(
,
K
L
and elements from
set
X
is called the object (see, [18]):
l
1
...
l
j
...
l
n
k
1
a
k
1
,
l
1
...
a
k
1
,
l
j
...
a
k
1
,
l
n
.
.
.
.
. . .
. . .
[
K
,
L
,
{
a
k
i
,
l
j
}] ≡
,
k
i
a
k
i
,
l
1
...
a
k
i
,
l
j
...
a
k
i
,
l
n
.
.
.
.
. . .
. . .
k
m
a
k
m
,
l
1
...
a
k
m
,
l
j
...
a
k
m
,
l
n
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