Information Technology Reference
In-Depth Information
Chapter 1
Index Matrices: Definitions, Operations,
Relations
The concept of Index Matrix (IM) was introduced in 1984 in [2, 3]. During the
following 25 years some of its properties were studied, but in general it was used
only as an auxiliary tool for describing the transitions of the generalized nets (see
[1, 4, 9]), the intuitionistic fuzzy relations and graphs with finite sets of vertices
(see [7, 13]) and in some decision making algorithms based on intuitionistic fuzzy
estimations (see e.g., [13]).
In the present topic, we include the author's results on IMs, published during
the last years. They contain the definitions of different types of IMs, as well as the
definitions of the operations, relations and operators over IMs.
We start with the basic definition of the concept of an IM with real number
elements, following [3, 10]. For brevity, we denote this IM by
R
-IM.
1.1 Definitions of an Index Matrix with Real Number Elements
and
(
0
,
1
)
-index Matrix
Let
I
be a fixed set of indices and
R
be the set of real numbers. Let opera-
tions
◦
,
∗
:
R
×
R
→
R
be fixed. For example, they can be the pairs,
◦
,
∗ ∈
{×
,
+
,
, or others.
Let the standard sets
K
and
L
satisfy the condition:
K
max
,
min
,
min
,
max
}
. Let over these
sets, the standard set-theoretical operations be defined. We call “IMwith real number
elements” (
,
L
⊂
I
R
-IM) the object:
l
1
l
2
...
l
n
k
1
a
k
1
,
l
1
a
k
1
,
l
2
...
a
k
1
,
l
n
k
2
a
k
2
,
l
1
a
k
2
,
l
2
...
a
k
2
,
l
n
[
K
,
L
,
{
a
k
i
,
l
j
}] ≡
,
k
m
a
k
m
,
l
1
a
k
m
,
l
2
...
a
k
m
,
l
n
