Toyoda–Bruck theorem

Remarks

The converse direction is a direct computation. With $x \cdot y = \varphi(x) + \psi(y) + c$, the product $(x \cdot y) \cdot (u \cdot v)$ equals $\varphi^2(x) + \varphi\psi(y) + \psi\varphi(u) + \psi^2(v) + \varphi(c) + \psi(c) + c$. Swapping $y$ and $u$ gives the same value because $\varphi\psi = \psi\varphi$, and the operation is a quasigroup because $\varphi$ and $\psi$ are bijective. For the main direction, one fixes an element of $Q$ and defines a new addition on $Q$ from the quasigroup operation by an isotopy. Mediality forces this addition to be associative and commutative, and the quasigroup operation then becomes affine in each argument.

Example: on $\Z/5\Z$ the operation $x \cdot y = 2x + 3y$ is a medial quasigroup, with $\varphi(x) = 2x$, $\psi(y) = 3y$ and $c = 0$. Example: on $\Z/3\Z$ the operation $x \cdot y = -x - y$ is a medial quasigroup, with $\varphi = \psi = -\id$ and $c = 0$.

The quasigroup hypothesis is needed. The commutative semigroup $(\N, \cdot)$ is medial, but it is not of this form. In an operation $\varphi(x) + \psi(y) + c$ each map $y \mapsto x \cdot y$ is a bijection, while in $(\N, \cdot)$ the map $y \mapsto 0 \cdot y$ is constant.

Going deeper

For a student

The theorem is the main structure result for medial quasigroups. It reduces questions about medial quasigroups to questions about abelian groups with two commuting automorphisms. It is also called the Bruck–Murdoch–Toyoda theorem. If the medial quasigroup is moreover idempotent, so that $x \cdot x = x$, then putting $x = 0$ gives $c = 0$, and then $\varphi + \psi = \id$. Such quasigroups are the affine combinations $x \cdot y = \varphi(x) + (\id - \varphi)(y)$, as in the midpoint operation $(x + y)/2$ on $\Q$.