View source for Alternative identity

A condition on a [[binary operation]] $\cdot$ on a set $X$: that for all $x, y \in X$
$$
x \cdot (x \cdot y) = (x \cdot x) \cdot y 
$$
and
$$
x \cdot (y \cdot y) = (x \cdot y) \cdot y \ .
$$

[[Alternative rings and algebras]] are those whose multiplication satisfies the alternative identity, which in this case is equivalent to the [[associator]] $(x,y,z)$ being an alternating function of three variables.

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