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.

 

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