In abstract algebra, power associativity is a weak form of associativity. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, associativity is a property that a Binary operation can have
An algebra (or more generally a magma) is said to be power-associative if the subalgebra generated by any element is associative. In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure. In Algebra (mathematics, the word "algebra" usually means a Vector space or module equipped with an additional bilinear operation 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 are all other alternative algebras (like the octonions, which are non-associative) and even some non-alternative algebras like the sedenions. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive In Abstract algebra, an alternative algebra is an algebra in which multiplication need not be Associative, only alternative. In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real In Abstract algebra, sedenions form a 16- dimensional algebra over the reals.
Exponentiation to the power of any natural number other than zero can be defined consistently whenever multiplication is power-associative. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an 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. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that
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).
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