# Moufang loop

 $\displaystyle(x(yz))x$ $\displaystyle=$ $\displaystyle(xy)(zx)\qquad\text{for all }x,y,z\in Q$ (1) $\displaystyle((xy)z)y$ $\displaystyle=$ $\displaystyle x(y(zy))\qquad\text{for all }x,y,z\in Q$ (2) $\displaystyle(xz)(yx)$ $\displaystyle=$ $\displaystyle x((zy)x)\qquad\text{for all }x,y,z\in Q$ (3) $\displaystyle((yz)y)x$ $\displaystyle=$ $\displaystyle y(z(yx))\qquad\text{for all }x,y,z\in Q$ (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  shows that that any of the four conditions implies the existence of an identity element. And Bol and Bruck  show that the four conditions are equivalent for loops.

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

 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.)

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

Title Moufang loop MoufangLoop 2013-03-22 13:50:29 2013-03-22 13:50:29 yark (2760) yark (2760) 12 yark (2760) Definition msc 20N05