Intuitionistic Fuzzy IMs - Index Matrices: Towards an Augmented Matrix Calculus - page 46

Its H ∗ -component has the following form (where, obviously, the information
about the IFS V ∗ is included in it):

1
2 ,

1
3 ,

1
3 ,

1
3 ,

1
3 ,

1
2 ,

2
3 ,

1
8 ,

3
4 ,

1
4 ,

1
10 ,

7
8 ,

2
5 ,

3
5 ,

H ∗ = [{ a , b , c , d , e , f , g ,

1
5 ,

1
5 } , {

1
2 ,

1
3 ,

1
3 ,

1
3 ,

1
3 ,

1
2 ,

2
3 ,

1
8 ,

3
4 ,

1
4 ,

1
10 ,

7
8 ,

h , a , b , c , d , e , f ,

2
5 ,

3
5 ,

1
5 ,

1
5 } , { μ x , y ,ν x , y }] .

g , h ,

Now, having in mind the discussion in Sect. 2.2 , we can modify the IM to the
form

1
2 ,

1
3 , b ,

1
3 ,

1
3 , c ,

1
3 ,

1
2 , d ,

2
3 ,

1
8 , e ,

3
4 ,

1
4 , f ,

1
10 ,

7
8 , g ,

2
5 ,

3
5 } ,

H ∗ =[{ a ,

2
3 ,

1
8 ,

3
4 ,

1
4 ,

1
10 ,

7
8 ,

2
5 ,

3
5 ,

1
5 ,

1
5 } , { μ x , y ,ν x , y }] .

{ d , e , f , g , h ,

The form of the new IM is

2 1 3 1 1 7 2 3 1 1

d ,

3 , 8

e ,

4 , 4

f ,

10 , 8

g ,

5 , 5

h ,

5 , 5

1 1 3 1 1 1

a ,

2 , 3 4 , 5 2 , 4

0 , 1 0 , 1 0 , 1

1 1 2

,

3 , 3 3 ,

, , , ,

b 0 0 1 0 1 0 1 0 1

1 1 1 2

c ,

3 , 2

0 , 1

5 , 5

0 , 1 0 , 1 0 , 1

2 1 1 2 3 1

d ,

3 , 8

0 , 1 0 , 1

5 , 5 4 , 5

0 , 1

3 1 2 1

e ,

4 , 4

0 , 1 0 , 1 0 , 1

3 , 6

0 , 1

1 7 1 1

f ,

10 , 8

0 , 1 0 , 1 0 , 1 0 , 1

3 , 4

2 3 3 1

g ,

5 , 5

0 , 1 0 , 1 0 , 1 0 , 1

5 , 5

Now, we can apply one or more of the level-operators N 1
ρ

N 2
ρ

N 3
ρ

3 and
as a result, the form of the graph will be changed. It is important to mention that in
the present case (when the two index sets coincide), the first two level operators must
have equal parameters and, therefore, if some vertex has to be omitted from one of
both index sets, it will be omitted from the other index set, too. For example, if we
apply operator N 5 ,

, ,

,σ ,σ ,σ 1 1 2 2 3

over G ∗ , we obtain

1
4