Abbreviation: Loop

A \emph{loop} is a structure $\mathbf{A}=\langle A,\cdot ,\backslash,/,e\rangle $ of type $\langle 2,2,2,0\rangle $ such that

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

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

$e$ is an identity for $\cdot$: $xe = x$, $ex = x$

Remark:

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

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

Example 1:

$\begin{array}{lr} f(1)= &1
f(2)= &1
f(3)= &1
f(4)= &2
f(5)= &6
f(6)= &109
f(7)= &23746
f(8)= &106228849
f(9)= &9365022303540
f(10)= &20890436195945769617
f(11)= &1478157455158044452849321016
\end{array}$

Groups

Quasigroups