medial quasigroup

A medial quasigroup is a quasigroup such that, for any choice of four elements $a,b,c,d$, one has

$$(a\cdot b)\cdot (c\cdot d)=(a\cdot c)\cdot (b\cdot d).$$

Any commutative quasigroup is trivially a medial quasigroup. A nontrivial class of examples may be constructed as follows. Take a commutative group $(G,+)$ and two automorphisms $f,g:G\to G$ which commute with each other, and an element $c$ of $G$. Then, if we define an operation $\cdot :G\times G\to G$ as

$$x\cdot y=f(a)+g(b)+c,$$

$(G,\cdot )$ is a medial quasigroup.

Reference:

V. D. Belousov, Fundamentals of the theory of quasigroups and loops (in Russian)