Proposition: Let $Q$ be a nonempty quasigroup.

I) The following conditions are equivalent.

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.