Alternative magma

Definition
A magma $$(S,*)$$ is termed an alternative magma if it is both a defining ingredient::left-alternative magma and a defining ingredient::right-alternative magma, i.e., it satisfies the following two identities:


 * $$(x * x) * y = x * (x * y) \ \forall \ x,y \in S$$
 * $$x * (y * y) = (x * y) * y \ \forall \ x,y \in S$$

Property obtained by the opposite operation
Suppose $$(S,*)$$ is a magma and we define $$\cdot$$ on $$S$$ as $$a \cdot b := b * a$$. Then, $$(S,*)$$ is an alternative magma if and only if $$(S,\cdot)$$ is an alternative magma.