A First Course in FUZZY and NEURAL CONTROL

3.3. COMBINING FUZZY SETS 95

The ìadditiveî generatorF:[0,1]→[0,∞]for nilpotent t-norms that is com-
monly referred to in the literature is related tofbyF(x)/F(0) = (1−f(x)).
Thus, there is a decreasing functionF:[0,1]→[0,∞]with

xïy=F−^1 ((F(x)+F(y))∧F(0))

A ìmultiplicativeî generatorG:[0,1]→[0,1]is obtained fromfbyG(x)=
ef(x)−^1 , giving a third representation for nilpotent t-norms

xïy=G−^1 ((G(x)G(y))∨G(0))

Here are two well-known examples of strict t-norms. For other strict and
nilpotent t-norms, see Tables 3.1 (a) and 3.2 (a) starting on page 97.

ïHamacher one-parameter family of t-norms:

x◦Hy=

xy

x+y−xy

generator:f(x)=e

x−x 1

ïThe Frank one-parameter family of t-norms includes strict, nilpotent, and

idempotent t-norms:

x◦Fay=loga

∑

1+

(ax−1) (ay−1)

a− 1

∏

, 0 <a<∞,a 6 =1

x◦F 1 y= lima→ 1 loga

h

1+(a

x−1)(ay−1)

a− 1

i

=xy

x◦F∞y=lima→∞loga

h

1+(a

x−1)(ay−1)

a− 1

i

=(x+y−1)∨ 0

x◦F 0 y= lima→ 0 +loga

h

1+(a

x−1)(ay−1)

a− 1

i

=x∧y

generators:Fa(x)=

ax− 1

a− 1

, 0 <a<∞,a 6 =1;F 1 (x)=x

Triangular conorms The corresponding generalization for OR is atriangu-
lar conormort-conorm.

Definition 3.3At-conormis a binary operation∗:[0,1]◊[0,1]→[0,1]
satisfying for allx,y,z∈[0,1]

1.x∗y=y∗x(∗is commutative)

2.x∗(y∗z)=(x∗y)∗z(∗is associative)

3.x∗0=0∗x=x( 0 is an identity)

4.y≤zimpliesx∗y≤x∗z(∗is increasing in each variable)