Fuzzy and Intuitionistic Fuzzy Operators with Application in Decision-Making - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 43

max operators. t -norms and t -conorms that belong to the conditional class
are as follows:

p p p
(,) in([( )( )] ,)

(/) 1

1. Yager's-norm [18]:

t Y xy =− −+− 1 1 x 1 y 1

p

p

wit h decreasing generator fx x () ( =− 1 )

p

− 1 1 / p

with f y y () =− 1 , p > 0

(3.3)

* (,)min([

p p (/) 1 p

T - conorm: Yxy x y ] , ) 1

= + p

with increasing genera tor

p

() /

1 p

− 1

gx x () = and g yy =

p p

2. Zadeh's t operators:

Txy (,)min(,) = x y

(3.4)

* (,)max(,)

Txy = x y

3. Weber's operator [15]:

xy xy +−+

+

1 λ

⎛
⎜

⎞
⎟

T -norm: W xy (,)max = , 0 , λ >− 1

1 λ

x

=− +
+

ln( 1

1

λ
λ

)

with decreasing generator fx () 1

ln( )

1

− 1 1 − y

with f y () [( = 1 + λ ) − 1 ], y ≤ 1

λ

(3.5)

* (,)min(

T -conorm: W xy = xy xy + + λ , ) 1

() ln(

1
1

+
+

λ
λ

x )

where the increasing ge nerator is gx =

ln( )

1

y

gy

− 1

y

with () [( = 1 + λ ) − 1 ], ≤ 1

λ

4. Schweizer and Sklar's [13] t -norm and t -conorm:

r r 1 / r

Sxy (,) max( , = 0 x + y − 1 )]

(3.6)

* (,)

r r 1 r

Sxy =− −+−− 1 [max( ,( )( ) 0 1 x 1 y 1 )] , r > 0