Encyclopedia > Power associativity

  Article Content

Power associativity

In abstract algebra, power associativity is a weak form of associativity.

An algebra (or more generally a magma) is said to be power-associative if the subalgebra generated by any element is associative. Concretely, this means that if an element x is multiplied by itself several times, it doesn't matter in which order the multiplications are carried out, so for instance x(x(xx)) = (x(xx))x = (xx)(xx). This is stronger than merely saying that (xx)x = x(xx) for every x in the algebra.

Every associative algebra is obviously power-associative, but so too are alternative algebras like the octonions and even some non-alternative algebras like the sedenions.

Exponentiation to the power of any natural number other than zero can be defined consistently whenever multiplication is power-associative. For example, there is no ambiguity as to whether x3 should be defined as (xx)x or as x(xx), since these are equal. Exponentiation to the power of zero can also be defined if the operation has an identity element, so the existence of identity elements becomes especially useful in power-associative contexts.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Flapper

... derived from a fashion of wearing galoshes unbuttoned so that they flapped as the wearer walks, the term was already doccumented as in use in the United Kingdom as ...

 
 
 
This page was created in 35.4 ms