closed non associative binary operation

Question

I'm Trying to show that a binary operation does not have to be associative in order to maintain closure, identity element and inverse element - on a 5 elements set. Any ideas for a set + binary operation that shows that? I can't find a way to lose associativity without throwing out closure as well.

Answer 1

Consider the set $\{0,1,2,3,4\}$ and the opertion $+$ thus defined:

1. $n+n=n$;
2. $0+n=n+0=n$;
3. otherwise, $m+n=0$.

It has an identity element ($0$) and each element has an inverse (several, indeed, except for $0$). But $(2+1)+1=0+1=1$, whereas $2+(1+1)=2+1=0$.