Alternative

In abstract algebra , a branch of mathematics , alternativeity is a weakening of the associative law .

definition

An algebraic structure with a two-digit link is called an alternative if the two statements below apply to all : ${\ displaystyle (M, \ cdot)}$ ${\ displaystyle \ cdot}$ ${\ displaystyle o, p \ in M}$

Left alternative :

{\ displaystyle o \ cdot (o \ cdot p) = (o \ cdot o) \ cdot p}

Legal alternativity :

{\ displaystyle o \ cdot (p \ cdot p) = (o \ cdot p) \ cdot p}

meaning

If a link is associative , it applies by definition

{\ displaystyle (o \ cdot q) \ cdot p = o \ cdot (q \ cdot p)}

for everyone . The consequence of this is that brackets are superfluous in the notation; after all, the result does not depend on the sequence in which the links are executed. Often one therefore saves the brackets. If the associative law does not apply, brackets cannot be removed without reason. However, if the alternative applies, the brackets can at least be left out if or if . In particular, the same factors can be combined to form potencies. This means that it is possible products of the kind to summarize. ${\ displaystyle o, q, p}$ ${\ displaystyle q = o}$ ${\ displaystyle q = p}$ ${\ displaystyle (a \ cdot a) \ cdot ((a \ cdot a) \ cdot b)}$ ${\ displaystyle a ^ {4} \ cdot b}$

Each associative link is alternative.

Examples

For an alternative field , a generalized field, multiplication only requires the alternative instead of the associative law. In contrast to a field , multiplication does not have to be commutative either .

The real octon ions form such an alternative body. Their multiplication is alternative, but neither associative nor commutative.

literature

Günther Eisenreich: Lexicon of Algebra . Akademie-Verlag, Berlin 1989, ISBN 3-05-500231-8 .
EN Kuz'min IP and Shestakov: Algebra VI . Combinatorial and Asymptotic Methods of Algebra: Nonassociative Structures. Ed .: AI Kostrikin and IR Shafarevich. Springer, 1995.
Ruth Moufang: On the structure of alternative bodies . In: Mathematical Annals . Volume 110, Number 1, 1935, pp. 416-430 .
Max August Zorn : Theory of Alternative Rings . In: Dep. Math. Sem . Volume 8, Number 1. Hamburg 1930, p. 123-147 , doi : 10.1007 / BF02940993 .

Web links

Eric W. Weisstein : Alternative Algebra . In: MathWorld (English).

Individual evidence

^ Eisenreich (1989)

[1] Eisenreich (1989)