Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 147

These two equations are usually not verified by strictly monotonic, archimedean
t-norms and t-conorms.

Any strictly monotonic, archimedean t-norm (or t-conorm) can be defined based
on multiplicative generators, by equivalence with additive generators [CHE 89]:

[0 , 1] 2 ,t ( x, y )= h − 1 h ( x ) h ( y ) ,

∀ ∈

( x, y ) [8.61]

where h is a strictly increasing function of [0 , 1] into [0 , 1] such that h (0) = 0 and
h (1) = 1. The equivalence with the additive form is obtained simply by defining:

h = e − f [8.62]

where f is an additive generating function.

The most common t-norms and t-conorms in this class are the product and the
algebraic sum:

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

∀ ( x, y ) ∈ T ( x, y )= x + y − xy. [8.63]

The only rational t-norms of this class are the Hamacher t-norms defined by
[HAM 78]:

xy

[0 , 1] 2 , ∀ ( x, y ) ∈

xy ) ,

[8.64]

γ +(1 − γ )( x + y −

where γ is a positive parameter (for γ =1we get the product again). They are illus-
trated in Figure 8.5.

Figure 8.5. Two examples of Hamacher t-norms, for γ =0 (left)

and γ =0 . 4 (right)