The property $(xy)x = xy$ is one of the equations satisified by a *directoid*.  Various properties have names ($xy = yx$ is commutativity, $xx=x$ is idempotency, etc).  The wikipedia page for <a href="http://en.wikipedia.org/wiki/Magma_%28algebra%29">Magma</a> has a long list of named properties, but not this one.

Directoids also satisfy $y(xy)=xy$ which is similar to the one above (although which should be called a 'left' property and which 'right', I am unsure).  Last, there is $x((xy)z)=(xy)z$ - for which I have also not been able to find a name.  Have these been named?