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}
X
{\ displaystyle X}
a
,
b
,
c
∈
X
{\ displaystyle a, b, c \ in X}
(M1)
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{\ displaystyle {\ Big (} a \ cdot (b \ cdot a) {\ Big)} \ cdot c = a \ cdot {\ Big (} b \ cdot (a \ cdot c) {\ Big)}}
and
(M2)
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{\ 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 ')
(
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b
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⋅
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=
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{\ displaystyle {\ Big (} (a \ cdot b) \ cdot c {\ Big)} \ cdot b = a \ cdot {\ Big (} b \ cdot (c \ cdot b) {\ Big)}}
and
(M2 ')
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=
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a
{\ 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 .
(
M.
,
⋅
)
{\ 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
(
M.
,
⋅
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{\ displaystyle (M, \ cdot)}
1
{\ displaystyle 1}
⋅
{\ displaystyle \ cdot}
the left alternative (because of (M1) with ):
b
=
1
{\ displaystyle b = 1}
(
a
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a
)
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c
=
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a
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(
1
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a
)
)
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c
=
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1
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=
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{\ 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
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b
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a
=
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1
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=
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=
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{\ 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
(
M.
,
⋅
)
{\ displaystyle (M, \ cdot)}
1
{\ displaystyle 1}
⋅
{\ displaystyle \ cdot}
(
a
⋅
b
)
⋅
b
=
(
(
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1
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=
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b
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1
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=
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{\ 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 ):
b
=
1
{\ displaystyle b = 1}
a
⋅
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=
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1
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)
=
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=
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⋅
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 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
(
M.
,
⋅
)
{\ displaystyle (M, \ cdot)}
⋅
{\ displaystyle \ cdot}
(M3, follows from M1)
(
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=
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=
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{\ 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')
(
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=
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{\ 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
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