# 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 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:
http://www.math.wisc.edu/ kunen/moufang.psMoufang Quasigroups.)

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

Title | Moufang loop |
---|---|

Canonical name | MoufangLoop |

Date of creation | 2013-03-22 13:50:29 |

Last modified on | 2013-03-22 13:50:29 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 12 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20N05 |