Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 148

Another parametric family of this class is comprised of the Frank functions,
defined by [FRA 79]:

1+ s x

1 s y 1

,

− −

[0 , 1] 2 ,t ( x, y ) = log s ∀ ( x, y ) ∈ [8.65]

s − 1

where s is a strictly positive parameter. These t-norms and their dual t-conorms satisfy
the following notable relation (and t-norms and t-conorms alone satisfy this relation):

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

∀ ∈

( x, y ) [8.66]

Examples of Frank t-norms are shown in Figure 8.6. If s is small and tends to 0, the
t-norm tends to the minimum. If s tends to + , the t-norm tends to the Lukasiewicz

t-norm. If s =1, we end up with the product again.

∞

Figure 8.6. Four examples of Frank t-norms. First line: s =0 . 1 and s =2 .

Second line: s =10 and s =1 , 000