Moufang identities

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A two-digit link on a set fulfills the Moufang identities (named after the German mathematician Ruth Moufang ), if the equations for all




be valid.

The following equations are also referred to as Moufang identities:

(M1 ')


(M2 ')

In a quasi-group , one of these four equations implies the other three. In addition, each of these equations assures the existence of a neutral element . A quasi-group, in which (at least) one of the Moufang identities is fulfilled, is therefore a loop , which is then also called a Moufang loop .

Relation to other forms of associativity

The Moufang identities are a weakened form of the associative law . In addition to associative links , the Moufang identities also apply to alternative bodies such as octonions .

If the Moufang identities (M1) and (M2) apply in a magma with a neutral element , then the link applies

  • the left alternative (because of (M1) with ):

Apply in a neutral element , however, 'and (M2 Moufang identities (M1)'), then for the combination

  • the flexibility law (because of (M2 ') with ):

In a flexible magma , in which the law of flexibility applies to the link , M2 'follows directly from M2 (and vice versa), and the following additional identities apply

(M3, follows from M1)
(M3 ', follows from M1')


  • John Horton Conway , Derek Smith: On Quaternions and Octonions Hardcover , 2003, ISBN 1568811349 , especially p. 88
  • Kenneth Kunen: Moufang quasigroups , Journal of Algebra, Vol. 183, Issue 1, 1996, pages 231-234
  • Ruth Moufang: To the structure of alternative bodies , Math. Ann., Vol. 110, 1935, pages 416-430