Power associativity
In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity.
Definition
An algebra is said to be power-associative if the subalgebra generated by any element is associative. Concretely, this means that if an element is performed an operation by itself several times, it doesn't matter in which order the operations are carried out, so for instance.Examples and properties
Every associative algebra is power-associative, but so are all other alternative algebras and even non-alternative flexible algebras like the sedenions, trigintaduonions, and Okubo algebras. Any Jordan algebra is power-associative. Any algebra whose elements are idempotent is also power-associative.Exponentiation to the power of any positive integer can be defined consistently whenever multiplication is power-associative. For example, there is no need to distinguish whether x3 should be defined as x or as x, 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 is useful in power-associative contexts.
Over a field of characteristic 0, an algebra is power-associative if and only if it satisfies and, where is the associator.
Over an infinite field of prime characteristic there is no finite set of identities that characterizes power-associativity, but there are infinite independent sets, as described by Gainov :
- For : and for to be the element of the algebra resulting from the obvious substitution of a into f. Then for any two such polynomials f and g, we have that.