Medial magma

Definition
A magma $$(S,*)$$ is termed a medial magma if it satisfies the following identity:

$$(a * b) * (c * d) = (a * c) * (b * d) \ \forall \ a,b,c,d \in S$$.

This identity is termed the medial identity.

If a medial magma has a two-sided neutral element (i.e., identity element) then it must be an abelian monoid. This is a special case of the general result known as Eckmann-Hilton duality, which says that if two binary operations commute and have a common neutral element, they must coincide and both be commutative.