A First Course in FUZZY and NEURAL CONTROL

94 CHAPTER 3. FUZZY LOGIC FOR CONTROL

The following theorem has its roots in the work of the Norwegian mathe-
matician Niels Abel in the early 1800ís, long before the idea of fuzzy sets arose.
A functionf:[0,1]→[0,1]is anorder isomorphismif it is continuous and
strictly increasing withf(0) = 0andf(1) = 1.

Theorem 3.1An Archimedean t-norm ◦is strict if and only if there is an
order isomorphismf:[0,1]→[0,1]satisfying the identity

f(x◦y)=f(x)f(y)

or equivalently, such that

x◦y=f−^1 (f(x)f(y))

Another such order isomorphismgsatisfies this condition if and only iff(x)=
g(x)rfor somer> 0.

A similar theorem is true for nilpotent t-norms.

Theorem 3.2An Archimedean t-norm◦is nilpotent if and only if there is an
order isomorphismf:[0,1]→[0,1]satisfying the identity

f(x◦y)=(f(x)+f(y)−1)∨ 0

or equivalently, such that

x◦y=f−^1 ((f(x)+f(y)−1)∨0)

Another such order isomorphismgsatisfies this condition if and only iff=g.

Another way to express the situation is: Every strict t-norm isisomorphicto
the product t-norm and every nilpotent t-norm isisomorphicto theŁukasiewicz
t-norm. The order isomorphismsfin the theorems above are calledgenerators
of the resulting t-norms.
Astrictt-normT(x, y)=x◦yis obtained from the algebraic product and a
generatorfby
x◦y=f−^1 (f(x)f(y))

The ìadditiveî generatorF:[0,1]→[0,∞]for strict t-norms that is commonly
referred to in the literature is related tofbyF(x)=−lnf(x).Thus,thereis
a decreasing functionF:[0,1]→[0,∞]with

x◦y=F−^1 (F(x)+F(y))

A nilpotent t-normT(x,y)=xïycan be obtained from the bounded product
and a generatorfby

xïy=f−^1 ((f(x)+f(y)−1)∨0)