Does commutativity imply Associativity?

Question

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.

Answer 1

Consider the operation $(x,y) \mapsto xy+1$ on the integers.

Answer 2

The easiest Jordan algebra is symmetric square matrices with the operation $$ A \ast B = (AB + BA)/2, $$ similar to a Lie algebra but with a plus sign.

http://en.wikipedia.org/wiki/Jordan_algebra

Answer 3

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 $P(u,v)=1+v$.

Answer 4

Let $A = \{e,x,y\}$. Define $\cdot$ on $A$ to be $a\cdot e=a$ for all $a$, $e\cdot a= a$ for all a, and $a\cdot b=e$ for all $a$ and $b$ such that $a\neq e$ and $b\neq e$, (i.e. $a,b \in \{x,y\}$).

This operation is commutative, $e$ is the identity, (everything even has an inverse), but is not associative since $(x \cdot y) \cdot y = e \cdot y = y$ and $x \cdot (y \cdot y) = x \cdot e = x$.