Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 149

The second useful family of t-norms and t-conorms is comprised of the nilpotent
operators, which have the following general form:

[0 , 1] 2 ,t ( x, y )= f ∗ f ( x )+ f ( y ) ,

∀ ( x, y ) ∈ [8.67]

where f is a decreasing bijection of [0 , 1] into [0 , 1], such that f (0) = 1, f (1) = 0 and
f ∗ ( x )= f − 1 ( x ) if x [0 , 1], with f ∗ ( x )=0if x

∈ ≥ 1. The general form of nilpotent

t-conorms is inferred from it by duality.

These operators satisfy the excluded middle law and the non-contradiction law.

The most common t-norms and t-conorms in this class are the Lukasiewicz opera-
tors:

[0 , 1] 2 ,t ( x, y ) = max(0 ,x + y

∀ ∈ −

( x, y ) 1) ,T ( x, y )=min(1 ,x + y ) .

Examples of generating functions f have been suggested by Schweizer and Sklar
[SCH 63], as well as by Yager [YAG 80].

Operators combining t-norms and t-conorms have been suggested, for example, by
[ZIM 80]:

[0 , 1] 2 ,C γ ( x, y )= t ( x, y ) 1 − γ T ( x, y ) γ ,

∀ ( x, y ) ∈ [8.68]

where γ is a parameter in [0 , 1].

8.5.3. Mean operators

A mean operator is a function m :[0 , 1] [0 , 1] such that:

- the result of the combination is always included between the min and the max:

× [0 , 1] →

[0 , 1] 2 , min( x, y )

∀ ( x, y ) ∈ ≤ m ( x, y ) ≤ max( x, y ),but m =minand m =max;

- m is commutative;
- m is increasing with respect to the two variables:

( x, x ,y,y ) [0 , 1] 4 , ( x x and y y )= m ( x ,y ) . ∀ ∈ ≤ ≤ ⇒ m ( x, y ) ≤

The first property entails that m is always an idempotent operation:

∀ x ∈ [0 , 1] ,m ( x, x )= x.