Does there exist an anti-associative structure on a set with three elements?

Question

A friend and I discussed whether there exist operations that are never associative in the sense that $$x(yz)\neq (xy)z$$ for all x, y, z and after pondering I found a simple example on a set with 2 elements. An example of an operation on a set with 4 elements was also readily found. But 3 elements eludes me. I have found these three restrictions that may all serve to disprove this existence.

1. At least two pairs of elements do not commute
2. No element can be idempotent
3. The following identity can never hold $$xy=yz=y$$ for a triple of x,y,z.

An example or a proof of non-existence would be much appreciated.

Answer 1

Here's a simple example of an anti-associative structure on any set $A$ with more than one element. Let $f:A\to A$ be a mapping with no fixed points and define $xy=f(y)$. Then for any $x,y,z\in A$, $$(xy)z=f(z)\ne f(f(z))=x(yz).$$