Medial magma

Summary

A medial magma is a magma that satisfies the medial law $(x y)(u v) = (x u)(y v)$: the two middle elements of a product of four may be swapped. Every commutative semigroup is medial, and so is every operation of the form $x \cdot y = \varphi(x) + \psi(y) + c$ on an abelian group with commuting endomorphisms $\varphi, \psi$. For quasigroups, the Toyoda–Bruck theorem says that these affine operations are the only medial ones. Mediality is the binary case of the entropic property of general algebras.