Abbreviation: MQgrp

A \emph{medial quasigroup} is a quasigroup $\mathbf{A}=\langle A,\cdot,\backslash,/\rangle$ such that

$\cdot$ is \emph{medial}: $(xy)(zw)=(xz)(yw)$

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It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

Let $\mathbf{A}$ and $\mathbf{B}$ be medial quasigroups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \cdot y)=h(x) \cdot h(y)$, $h(x \backslash y)=h(x) \backslash h(y)$, $h(x / y)=h(x) / h(y)$,

An \emph{…} is a structure $\mathbf{A}=\langle A,\ldots\rangle$ of type $\langle …\rangle$ such that

$\ldots$ is …: $axiom$

$\ldots$ is …: $axiom$

Example 1:

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

$\begin{array}{lr}

f(1)= &1\\
f(2)= &\\
f(3)= &\\
f(4)= &\\
f(5)= &\\

\end{array}$ $\begin{array}{lr}

f(6)= &\\
f(7)= &\\
f(8)= &\\
f(9)= &\\
f(10)= &\\

\end{array}$

[[...]] subvariety

[[...]] expansion

[[quasigroups]] supervariety