Does commutativity imply Associativity?
Consider the operation $(x,y) \mapsto xy+1$ on the integers.
A basic example is the "midpoint" binary operation: $a*b = \frac{a+b}{2}$
In general, if $P(u,v)$ is any polynomial in two variables with rational coefficients, then $x*y = P(x+y,xy)$ is rarely associative - I'd be curious under what conditions on $P$ this operation would be associative.
My example is $P(u,v)=\frac{u}{2}$ and Marlu's example is $P(u,v)=1+v$.