# What to call this binary operation property?

In one of my projects I came across a property that all the binary operations need to have to lead to correct results. It looked like a general property so I was wondering if it might have been described and named somewhere.

Given a binary operation $\lozenge$ with $\lozenge:X\times Y\rightarrow X$.

The property is defined by $\left(x\:\lozenge\:y_1\right)\:\lozenge\:y_2\:=\:\left(x\:\lozenge \:y_2\right)\:\lozenge\: y_1$ with $x\in X$ and $y_{1,2}\in Y$.

The operation does not have to be symmetric for this property to hold. If the operator is associative and commutative this property will always hold (first use assoc. then comm. and assoc. again). But the property can also hold when both are not the case for instance with operation $x\:\square\: y\:=\: x+1$. The operation is associative nor commutative but the does have the property given above.

How should I call this?

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Note that most people would probably not call this a binary operation unless $X = Y$ (rather, it is an action of $Y$ on $X$ from the right). – Tobias Kildetoft Apr 16 '13 at 13:56
Yes, fair point. I made it a bit more general just to be sure but in most cases where I encountered it $X=Y$ was the case (or could be converted to) so I'm also fine with the property for true binary operations. – FTPlus Apr 26 '13 at 14:16

I'd simply call it a commutative right-action of $Y$ on $X$, but I know of no standard terminology.