# Alternative body

an alternative body is a body (in the mathematical sense), into neither the commutative law still the associative law to apply must. Instead it is demanded that the multiplication the characteristic of the Alternativität has.

## definition

a quantity of M with two linkages + and * is a Alternativörper, if applies:

• (M, +) is a Abel group, whose neutral element is called 0;
• (M \ {0}, *) is a quasi-'s group with neutral element, which is called 1;
• To the linkage * the Alternativität applies: o · (o · p) = (o · o) · p and o · (p · p) = (o · p) · p.
• it applies the distributive law a* (b+c) = a*b + a*c.

## example

the most well-known example of an alternative body are the Oktonionen.

## characteristics

from the Alternativität follows further the flexibility law

• o · (p · o) = (o · p) · o.

Furthermore in an alternative body the Moufang identities apply to the linkage *

• [A · (b · A)] · C = A · [b · (A · C)]

and

• (A · b) · (C · A) = A · [(b · C) · A]

Ruth Moufang showed 1934 that three arbitrary elements A, b, C from one Alternative bodies, which meet the relation (A * b) * C = A * (b * C), an inclined body produce. This is an aggravation of the sentence of Artin. The sentence of Artindevelops for the special case C = 1.