Abbreviation: Qgrp

A \emph{quasigroup} is a structure $\mathbf{A}=\langle A,\cdot ,\backslash,/\rangle$ of type $\langle 2,2,2\rangle $ such that

$(y/x)x = y$, $x(x\backslash y) = y$

$(xy)/y = x$, $x\backslash(xy) = y$

Remark:

Let $\mathbf{A}$ and $\mathbf{B}$ be quasigroups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

$h(xy)=h(x)h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$.

Example 1:

$\begin{array}{lr} f(1)= &1
f(2)= &1
f(3)= &1
f(4)= &5
f(5)= &35
f(6)= &1411
f(7)= &1130531
\end{array}$

Loops

Medial quasigroups