A 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,Equational theories of medial groupoids, Algebra Universalis, 171983,174–190MRreview

Jaroslav Jezek, Tomas Kepka,Medial groupoids, Rozpravy Ceskoslovenske Akad. Ved Rada Mat. Prirod. Ved, 931983,93MRreview

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