Moufang identities

A two-digit link on a set fulfills the Moufang identities (named after the German mathematician Ruth Moufang ), if the equations for all ${\displaystyle \cdot }$ ${\displaystyle X}$ ${\displaystyle a,b,c\in X}$

(M1) ${\displaystyle {\Big (}a\cdot (b\cdot a){\Big )}\cdot c=a\cdot {\Big (}b\cdot (a\cdot c){\Big )}}$

and

(M2) ${\displaystyle (a\cdot b)\cdot (c\cdot a)=a\cdot {\Big (}(b\cdot c)\cdot a{\Big )}}$

be valid.

The following equations are also referred to as Moufang identities:

(M1 ') ${\displaystyle {\Big (}(a\cdot b)\cdot c{\Big )}\cdot b=a\cdot {\Big (}b\cdot (c\cdot b){\Big )}}$

and

(M2 ') ${\displaystyle (a\cdot b)\cdot (c\cdot a)={\Big (}a\cdot (b\cdot c){\Big )}\cdot a}$

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 . ${\displaystyle (M,\cdot )}$

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 ${\displaystyle (M,\cdot )}$ ${\displaystyle 1}$ ${\displaystyle \cdot }$

• the left alternative (because of (M1) with ): ${\displaystyle b=1}$

${\displaystyle (a\cdot a)\cdot c={\Big (}a\cdot (1\cdot a){\Big )}\cdot c=a\cdot {\Big (}1\cdot (a\cdot c){\Big )}=a\cdot (a\cdot c)}$

• the flexibility law (because of (M2) with ): ${\displaystyle c=1}$

${\displaystyle (a\cdot b)\cdot a=(a\cdot b)\cdot (1\cdot a)=a\cdot {\Big (}(b\cdot 1)\cdot a{\Big )}=a\cdot (b\cdot a)}$

Apply in a neutral element , however, 'and (M2 Moufang identities (M1)'), then for the combination ${\displaystyle (M,\cdot )}$ ${\displaystyle 1}$ ${\displaystyle \cdot }$

• the legal alternativity (because of (M1 ') with ): ${\displaystyle c=1}$

${\displaystyle (a\cdot b)\cdot b={\Big (}(a\cdot b)\cdot 1{\Big )}\cdot b=a\cdot {\Big (}b\cdot (1\cdot b){\Big )}=a\cdot (b\cdot b)}$

• the flexibility law (because of (M2 ') with ): ${\displaystyle b=1}$

${\displaystyle a\cdot (a\cdot b)=(a\cdot 1)\cdot (c\cdot a)={\Big (}a\cdot (1\cdot c){\Big )}\cdot a=(a\cdot b)\cdot a}$

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 ${\displaystyle (M,\cdot )}$ ${\displaystyle \cdot }$

(M3, follows from M1) ${\displaystyle {\Big (}(a\cdot b)\cdot a{\Big )}\cdot c={\Big (}a\cdot (b\cdot a){\Big )}\cdot c=a\cdot {\Big (}b\cdot (a\cdot c){\Big )}}$

(M3 ', follows from M1') ${\displaystyle {\Big (}(a\cdot b)\cdot c{\Big )}\cdot b=a\cdot {\Big (}b\cdot (c\cdot b){\Big )}=a\cdot {\Big (}(b\cdot c)\cdot b{\Big )}}$

literature

• 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