Does commutativity imply Associativity? Does commutativity imply associativity? I'm asking this because I was trying to think of structures that are commutative but non-associative but couldn't come up with any. Are there any such examples?

NOTE: I wasn't sure how to tag this so feel free to retag it.
Consider the operation $(x,y) \mapsto xy+1$ on the integers.
For our example the carrier set will be the set ${\Bbb Z}^{\Bbb Z}$ of all functions from $\Bbb Z$ to $\Bbb Z$.

For any two function $f, g \in {\Bbb Z}^{\Bbb Z}$ we can form the pointwise sum, $f + g: n \mapsto f(n) + g(n)$.

We have the (left) bilateral shift operator $T$ on ${\Bbb Z}^{\Bbb Z}$,

$\tag 1 T: (a_k)_{k=-\infty}^\infty \mapsto (a_{k+1})_{k=-\infty}^\infty$

We define a binary operation $\oplus$ on the set ${\Bbb Z}^{\Bbb Z}$ as follows,

$\tag 2 (f,g) \mapsto f \oplus g = T \circ (f+g)$

This binary operation is commutative but not associative; it satisfies

$\tag 3 f \oplus g =  h \oplus k \; \text{ iff } \; f + g = h + k$