A \emph{medial groupoid} is a structure $\mathbf{G}=\langle G,\cdot\rangle$, where $\cdot $ is an infix binary operation such that

$\cdot$ mediates: $(x\cdot y)\cdot(z\cdot w)=(x\cdot z)\cdot (y\cdot w)$

Let $\mathbf{G}$ and $\mathbf{H}$ be medial groupoids. A morphism from $\mathbf{G}$ to $\mathbf{H}$ is a function $h:G\rightarrow H$ that is a homomorphism:

$h(xy)=h(x)h(y)$

Jaroslav Jezek, Tomas Kepka,\emph{Equational theories of medial groupoids}, Algebra Universalis, \textbf{17}1983,174–190MRreview

Jaroslav Jezek, Tomas Kepka,\emph{Medial groupoids}, Rozpravy Ceskoslovenske Akad. Ved Rada Mat. Prirod. Ved, \textbf{93}1983,93MRreview

Example 1: $\langle S,*\rangle$, where $\langle S,+,\cdot\rangle$ is any commutative semiring, $a,b\in S$, and $x*y=a\cdot x+b\cdot y$.

$\begin{array}{lr} f(1)= &1
f(2)= &
f(3)= &
f(4)= &
f(5)= &
f(6)= &
f(7)= &
\end{array}$

Medial semigroups

Commutative medial groupoids

Groupoids