I'm looking at proving that the Łukasiewicz T-norm operator, or bounded product T-norm, is a T-norm, but I'm stuck on associativity.
The operator is defined as:
$xTy = max[0,x+y-1]$
Trying to show associativity:
$(aTb)Tc =$
$max[0,max[0,a+b-1]+c-1] =$
$max[0,max[c-1, a+b+c-2]] =$
$max[0,max[c-a,b+c-1]+a-1] =$
$max[0,a+max[c-a,b+c-1]-1] =$
At this point, I need to show that this is equivalent to...
$=max[0,a+max[0,b+c-1]-1]=aT(bTc)/$
But I can't figure out how to resolve $c-a$ to zero.
Hopefully someone can help!
Picking up from where you have $max[0,max[0,a+b−1]+c−1]$, there are two possibilities for the inner $max$. It either evaluates to $a+b−1$ or $0$.
In the first case we will have:
$$ max[0,max[0,a+b−1]+c−1] = max[0, a+b+c-2] $$
And in the second case we have: $$ max[0,max[0,a+b−1]+c−1] = max[0, c-1] $$
But $c-1 \in [-1, 0]$ since $c \in [0, 1]$. So we have: $$ max[0, c-1] = 0 $$
If you do the same for your other equation:
$$ aT(bTc) = \begin{cases} max[0, a+b+c-2], \text{ when }(bTc) \neq 0 \\ max[0, a-1] = 0, \text{ otherwise} \end{cases} $$