Eine zweistellige Verknüpfung
⋅
{\displaystyle \cdot }
X
{\displaystyle X}
Moufang-Identitäten (benannt nach der deutschen Mathematikerin Ruth Moufang ), wenn für alle
a
,
b
,
c
∈
X
{\displaystyle a,b,c\in X}
(M1)
(
a
⋅
(
b
⋅
a
)
)
⋅
c
=
a
⋅
(
b
⋅
(
a
⋅
c
)
)
{\displaystyle {\Big (}a\cdot (b\cdot a){\Big )}\cdot c=a\cdot {\Big (}b\cdot (a\cdot c){\Big )}}
und
(M2)
(
a
⋅
b
)
⋅
(
c
⋅
a
)
=
a
⋅
(
(
b
⋅
c
)
⋅
a
)
{\displaystyle (a\cdot b)\cdot (c\cdot a)=a\cdot {\Big (}(b\cdot c)\cdot a{\Big )}}
gelten.
Dual dazu werden auch folgende Gleichungen als Moufang-Identitäten bezeichnet:
(M1')
(
(
a
⋅
b
)
⋅
c
)
⋅
b
=
a
⋅
(
b
⋅
(
c
⋅
b
)
)
{\displaystyle {\Big (}(a\cdot b)\cdot c{\Big )}\cdot b=a\cdot {\Big (}b\cdot (c\cdot b){\Big )}}
und
(M2')
(
a
⋅
b
)
⋅
(
c
⋅
a
)
=
(
a
⋅
(
b
⋅
c
)
)
⋅
a
{\displaystyle (a\cdot b)\cdot (c\cdot a)={\Big (}a\cdot (b\cdot c){\Big )}\cdot a}
In einer Quasigruppe
(
M
,
⋅
)
{\displaystyle (M,\cdot )}
neutralen Elements zu. Eine Quasigruppe, in der also (mindestens) eine der Moufang-Identitäten erfüllt ist, ist demnach eine Loop , die dann auch Moufang-Loop genannt wird.
Bei den Moufang-Identitäten handelt es sich um eine abgeschwächte Form des Assoziativgesetzes . Außer für assoziative Verknüpfungen gelten die Moufang-Identitäten auch für Alternativkörper wie zum Beispiel die Oktonionen .
Gelten in einem Magma
(
M
,
⋅
)
{\displaystyle (M,\cdot )}
neutralen Element
1
{\displaystyle 1}
⋅
{\displaystyle \cdot }
(
a
⋅
a
)
⋅
c
=
(
a
⋅
(
1
⋅
a
)
)
⋅
c
=
a
⋅
(
1
⋅
(
a
⋅
c
)
)
=
a
⋅
(
a
⋅
c
)
{\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)}
(
a
⋅
b
)
⋅
a
=
(
a
⋅
b
)
⋅
(
1
⋅
a
)
=
a
⋅
(
(
b
⋅
1
)
⋅
a
)
=
a
⋅
(
b
⋅
a
)
{\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)}
Gelten in
(
M
,
⋅
)
{\displaystyle (M,\cdot )}
neutralen Element
1
{\displaystyle 1}
⋅
{\displaystyle \cdot }
(
a
⋅
b
)
⋅
b
=
(
(
a
⋅
b
)
⋅
1
)
⋅
b
=
a
⋅
(
b
⋅
(
1
⋅
b
)
)
=
a
⋅
(
b
⋅
b
)
{\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)}
a
⋅
(
a
⋅
b
)
=
(
a
⋅
1
)
⋅
(
c
⋅
a
)
=
(
a
⋅
(
1
⋅
c
)
)
⋅
a
=
(
a
⋅
b
)
⋅
a
{\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 einem flexiblen Magma
(
M
,
⋅
)
{\displaystyle (M,\cdot )}
⋅
{\displaystyle \cdot }
(M3, folgt aus M1)
(
(
a
⋅
b
)
⋅
a
)
⋅
c
=
(
a
⋅
(
b
⋅
a
)
)
⋅
c
=
a
⋅
(
b
⋅
(
a
⋅
c
)
)
{\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', folgt aus M1')
(
(
a
⋅
b
)
⋅
c
)
⋅
b
=
a
⋅
(
b
⋅
(
c
⋅
b
)
)
=
a
⋅
(
(
b
⋅
c
)
⋅
b
)
{\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 )}}
John Horton Conway , Derek Smith: On Quaternions and Octonions Hardcover , 2003, ISBN 1568811349 , insbesondere S. 88Kenneth Kunen : Moufang quasigroups , Journal of Algebra, Vol. 183, Issue 1, 1996, Seiten 231–234Ruth Moufang: Zur Struktur von Alternativkörpern , Math. Ann., Vol. 110, 1935, Seiten 416–430 
