Intuitionistic Fuzzy IMs - Index Matrices: Towards an Augmented Matrix Calculus - page 43

2.7 Extended Modal Operators Defined Over EIFIMs

Let, as above, x . Some of the extended

modal operators defined over x have the following forms (see [13, 26]):

= a , b be an IFP and let α, β ∈[ 0 , 1 ]

F

− α (

x ) = a + α.( 1 − a − b ), b + β.( 1 − a − b ) , where α + β ≤ 1

α, 1

G

α,β (

x ) = α. a ,β. b

H

α,β (

x ) = α. a , b + β.( 1 − a − b )

H α,β (

x ) = α. a , b + β.( 1 − α. a − b )

J α,β ( x ) = a + α.( 1 − a − b ), β. b

J α,β ( x ) = a + α.( 1 − a − β. b ), β. b

and let the level operators have the forms:

P α,β x = max (α, a ), min (β, b )

Q x = min (α, a ), max (β, b ) ,

α,β

for 1.

Now we define operators over EIFIMs. Let O 1
α

α, β ∈[ 0 , 1 ] and α + β ≤

O 2
α

O 3
α 1 , 2 , 3 be three opera-

,β ,β ,β 1 2 3

tors and their arguments α 1 ,β 1 ,α 2 ,β 2 ,α 3 ,β 3 satisfy the respective conditions, given

above. The three operators affect the K -, L -indices and -elements, re-

spectively. They can be applied over an EIFIM A sequentially, or simultaneously. In
the first case, their forms are

μ k i , l j ,ν k i , l j

O 1 (

α 1 ,β 1 , ⊥ , ⊥ )(

A )

l l

l n ,β n

l 1 , α

1 ,β 1

... l n , α

O 1

k k

k 1 ,

α 1 ,β 1 ( α 1 ,β 1 ) μ k 1 , l 1 ,ν k 1 , l 1 ... μ k 1 , l n ,ν k 1 , l n

= ,

. . .

. . .

O 1

k k

k m ,

α 1 ,β 1 ( α m ,β m ) μ k m , l 1 ,ν k m , l 1 ... μ k m , l n ,ν k m , l n

O 2

( ⊥ ,

α 2 ,β 2 , ⊥ )(

A )

O 2
α

l l

O 2
α

l n n

l 1 , ( α

1 ,β 1 )...

l n , ( α ,β )

,β ,β 2 2 2 2

k k

k 1 , α

1 ,β 1

μ k 1 , l 1 ,ν k 1 , l 1 ... μ k 1 , l n ,ν k 1 , l n

= ,

. . .

. . .

m ,β m

k m , α μ k m , l 1 ,ν k m , l 1 ... μ k m , l n ,ν k m , l n

O 3
α

( ⊥ , ⊥ , )( A )

,β 3 3