Abbreviation: MLoop

A \emph{Moufang loop} is a loops $\mathbf{A}=\langle A,\cdot ,\backslash,/,e\rangle$ such that

$((xy)z)x = x(y(zx))$, $y(x(yz)) = ((yx)y)z$, $(yx)(zy) = (y(xz))y$

Remark:

Let $\mathbf{A}$ and $\mathbf{B}$ be Moufang 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:

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

Groups

Loops