Medial quasigroups

Abbreviation: MQgrp

Definition

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

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

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.

Morphisms

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: $h(x \cdot y)=h(x) \cdot h(y)$, $h(x \backslash y)=h(x) \backslash h(y)$, $h(x / y)=h(x) / h(y)$,

Definition

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

$...$ is …: $axiom$

$...$ is …: $axiom$

Example 1:

Properties

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.

Classtype variety

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

Subclasses

[[...]] subvariety
[[...]] expansion

Superclasses

[[quasigroups]] supervariety