Power associativity

related topics
{math, number, function}

In abstract algebra, power associativity is a property of a binary operation which 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, but weaker than associativity, which requires that (xy)z = x(yz) for every x, y, and z in the algebra.

Every associative algebra is obviously power-associative, but so are all other alternative algebras (like the octonions, which are non-associative) 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.

A nice substitution law holds for real power-associative algebras with unit, which basically asserts that multiplication of polynomials works as expected. For f a real polynomial in x, and for any a in such an algebra define f(a) 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 (fg) (a) = f(a)g(a).


Full article ▸

related documents
Almost all
Church–Rosser theorem
Leaf node
Measurable function
Probability axioms
Integer sequence
Equivalence class
Center (group theory)
Gauss–Legendre algorithm
Krull dimension
Digital Signature Algorithm
Gabriel Lamé
Sieve of Eratosthenes
Ring homomorphism
Mary (programming language)
Catalan's constant
Timeline of programming languages
Laurent polynomial
Endomorphism ring
Two-out-of-five code
Euclidean distance
HTML scripting
Doubling the cube
Markov process