Moufang loop

Proposition: Let $Q$ be a nonempty quasigroup.

I) The following conditions are equivalent.

$(x(yz))x$$=$$(xy)(zx)\mathit{\hspace{1em}\hspace{1em}}\text{for all }x,y,z\in Q$(1)

$((xy)z)y$$=$$x(y(zy))\mathit{\hspace{1em}\hspace{1em}}\text{for all }x,y,z\in Q$(2)

$(xz)(yx)$$=$$x((zy)x)\mathit{\hspace{1em}\hspace{1em}}\text{for all }x,y,z\in Q$(3)

$((yz)y)x$$=$$y(z(yx))\mathit{\hspace{1em}\hspace{1em}}\text{for all }x,y,z\in Q$(4)

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 theexistence of an identity element. And Bol and Bruck [2] show thatthe four conditions are equivalent for loops.

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

The 16-element set of unit octonions over $\mathbb{Z}$ is an exampleof 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:http://www.math.wisc.edu/ kunen/moufang.psMoufang Quasigroups.)

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