In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that 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.
by itself several times, it doesn't matter in which order the operations are carried out, so for instance
Exponentiation to the power of any positive integer can be defined consistently whenever multiplication is power-associative.
Over a field of characteristic 0, an algebra is power-associative if and only if it satisfies
there is no finite set of identities that characterizes power-associativity, but there are infinite independent sets, as described by Gainov (1970): A 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).