Proposition: Let $Q$ be a nonempty quasigroup.

I) The following conditions are equivalent. \begin{eqnarray} (x(yz))x &=& (xy)(zx) \qquad\forall x,y,z\in Q \\ ((xy)z)x &=& x(y(zx)) \qquad\forall x,y,z\in Q \\ (yx)(zy) &=& (y(xz))y \qquad\forall x,y,z\in Q \\ y(x(yz)) &=& ((yx)y)z \qquad\forall x,y,z\in Q \end{eqnarray} II) If $Q$ satisfies those conditions, then $Q$ has an identity element (i.e., $Q$ is a loop).

For a proof, we refer the reader to the two references. Kunen in [1] shows that that any of the four conditions implies the existence of an identity element. And Bol and Bruck [2] show that the four conditions are equivalent for loops.

Definition:A nonempty quasigroup satisfying the conditions (1)-(4) is called a Moufang quasigroup or, equivalently, a Moufang loop (after Ruth Moufang, 1905-1977).

The 16-element set of unit octonions over $\mathbb{Z}$ is an example of a nonassociative Moufang loop. Other examples appear in projective geometry, coding theory, and elsewhere.

References

[1] Kenneth Kunen, Moufang Quasigroups, J. Algebra 83 (1996) 231-234. (A preprint in PostScript format is available from Kunen's website: Moufang Quasigroups.)

[2] R. H. Bruck, A Survey of Binary Systems, Springer-Verlag, 1958.