Moufang loops

Definition

A loop satisfing the following identities

	((xy)z)x = x[y(zx)]   
	y(x(yz)) = [(yx)y]z,
	(yx)(zy) = (y(xz))y.
	
	

Examples

• Any group
• Here is a Moufang loop M(S_3, 2) which isn't a group.

  	 * |  e  1  2  3  a  b  eu 1u 2u 3u au bu 
  	 --+--------------------------------------
  	 e |  e  1  2  3  a  b  eu 1u 2u 3u au bu 
  	 1 |  1  e  a  b  2  3  1u eu bu au 3u 2u 
  	 2 |  2  b  e  a  3  1  2u au eu bu 1u 3u 
  	 3 |  3  a  b  e  1  2  3u bu au eu 2u 1u 
  	 a |  a  3  1  2  b  e  au 2u 3u 1u bu eu 
  	 b |  b  2  3  1  e  a  bu 3u 1u 2u eu au
  	eu | eu 1u 2u 3u bu au   e  1  2  3  b  a
  	1u | 1u eu au bu 3u 2u   1  e  b  a  2  3
  	2u | 2u bu eu au 1u 3u   2  a  e  b  3  1
  	3u | 3u au bu eu 2u 1u   3  b  a  e  1  2
  	au | au 3u 1u 2u eu bu   a  2  3  1  e  b
  	bu | bu 2u 3u 1u au eu   b  3  1  2  a  e
  	

Structure

Representation

Decision problems

Identity problem:

Word problem:

Spectra and growth

Finite spectrum:

Free spectrum:

Growth series:

History/Importance

References

V. D. Belousov, Fundamentals of the theory of quasigroup and loop (in Russian)

Subsystems