What's medial law? While i'm reading about AG-groupoids, there is noted that identity $$(xy)z=(zy)x$$ implies "medial law": $$(xy)(uv)=(xu)(yv)$$ Can anyone tell me what is medial law? I cannot translate it to my language, but I'm highly interested what the "medial law" means.

Definition: An LA-semigroup (also called Abel-Grassmann's groupoid) satisfies the medial law if $(ab)(cd) = (ac)(bd)$ for all $a,b,c,d$.

Reference: Left almost semigroups defined by a free algebra .

Comment: For the meaning of "medial" see here. The elements in the "middle" are interchanged.