Power associativity

In mathematics, specifically in abstract algebra, power associativity is the property of a binary operation that integer powers (

x

n

{\displaystyle x^{n}}

) are well-defined; it implies that

x

m

∗

x

n

=

x

m

+

n

{\displaystyle x^{m}*x^{n}=x^{m+n}}

for all positive integers

m

,

n

{\displaystyle m,n}

. It is a weak form of associativity.

Definition

An algebra (or more generally a magma) is power-associative if the subalgebra generated by any element is associative. So any product of

n

{\displaystyle n}

instances of an element

x

{\displaystyle x}

has the same value, denoted

x

n

{\displaystyle x^{n}}

, regardless of parenthesization. For example,

x

∗

(

x

∗

(

x

∗

x

)

)

=

{\displaystyle x*(x*(x*x))=\,}

x

∗

(

(

x

∗

x

)

∗

x

)

=

{\displaystyle x*((x*x)*x)=\,}

(

x

∗

x

)

∗

(

x

∗

x

)

=

{\displaystyle (x*x)*(x*x)=\,}

(

x

∗

(

x

∗

x

)

)

∗

x

=

{\displaystyle (x*(x*x))*x=\,}

(

(

x

∗

x

)

∗

x

)

∗

x

=

{\displaystyle ((x*x)*x)*x=\,}

(

x

∗

x

)

2

=

{\displaystyle (x*x)^{2}=\,}

x

4

{\displaystyle x^{4}}

.

Examples and properties

Every associative algebra is power-associative, but so are all other alternative algebras (like the octonions, which are non-associative) 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 (xx)x or as x(xx), since these are equal.