A First Course in FUZZY and NEURAL CONTROL

3.3. COMBINING FUZZY SETS 93

The most widely used t-norm, minimum, is also the largest t-norm ñ that
is,
x◦y≤x∧y

for any t-norm◦. The smallest t-norm, called thedrastic product,isnot
continuous.

Drastic product

x◦y=

Ω

x∧y if x∨y=1

0 if x∨y< 1

The most common examples of a t-norm, other than minimum, are the
product t-norm,oralgebraic product

x◦y=xy

and theŁukasiewicz^1 t-normorbounded product

x◦y=max{x+y− 1 , 0 }=(x+y−1)∨ 0

These three t-norms (minimum, algebraic product, and bounded product) are
depicted in the following plots:

0 0.2

x 0.8^1

0

0.5y

0

0.2

0.4

0.6

0.8

1

z

x∧y

0 0.2

x 0.8^1

0.2 0

y0.4

0

0.2

0.4

0.6

0.8

1

z

xy

0 0.2

0.8^1

0.2 0

0.4

0

0.2

0.4

0.6

0.8

1

max{x+y− 1 , 0 }

A t-norm that satisfiesx◦x=xfor allxis calledidempotent.Themini-
mum t-norm is idempotent, and it is the only idempotent t-norm. A continuous
t-norm that satisfiesx◦x<xfor allx 6 =0, 1 is calledArchimedean.Allcon-
tinuous t-norms we will consider, other than minimum, are Archimedean. The
three examples depicted above are basic, in the sense that all other continuous
t-norms can be obtained from these three in a straightforward way.
An Archimedean t-norm for whichx◦x=0only whenx=0is calledstrict.
The product t-norm is the prototype for a strict t-norm. Other Archimedean
t-norms arenilpotent.TheŁukasiewicz t-norm is the prototype for a nilpotent
t-norm.

(^1) Łukasiewicz (1878ó1956) was a Polish logician and philosopher.