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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 Magma 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?

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Just to be clear, you are indicating that "multiplication" is not associative? I see, Wikipedia uses a different shorthand in order to minimize parentheses, the multiplication is an asterisk and xy means (x * y). So I imagine they would write your property xy*x = xy. One guy claims these were introduced by Jezek and Quackenbush (1990), I guess you know that ams.org/mathscinet-getitem?mr=1025835 –  Will Jagy May 14 '10 at 19:52
Correct, multiplication is not associative. And I did read that paper (and several others besides), but not one 'names' these properties! –  Jacques Carette May 14 '10 at 19:59
I started to call this "absorption", but that is not quite correct (absorption is a relationship between two operations). Still, perhaps that name will jog someone's memory. –  Austin Mohr May 12 '11 at 1:29
I have seen people dealing with Left Regular Bands call this 'left regularity'. I might be mistaken. –  Vasu vineet Dec 26 '11 at 15:28
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