mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, specifically in

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, power associativity is a property of a

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation ...

that is a weak form of

associativity In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...

Definition

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

(or more generally a

magma Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma (sometimes colloquially but incorrectly referred to as ''lava'') is found beneath the surface of the Earth, and evidence of magmatism has also ...

) is said to be power-associative if the

subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear opera ...

generated by any element is associative. Concretely, this means that if an element

x

is performed an operation

*

by itself several times, it doesn't matter in which order the operations are carried out, so for instance

x*(x*(x*x)) = (x*(x*x))*x = (x*x)*(x*x)

Examples and properties

Every

associative algebra In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...

is power-associative, but so are all other

alternative algebra In abstract algebra, an alternative algebra is an algebra over a field, algebra in which multiplication need not be associative, only alternativity, alternative. That is, one must have *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the a ...

s (like the

octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of Hypercomplex number, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or ...

s, which are non-associative) and even non-alternative

flexible algebra In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity: : a \bullet \left(b \bullet a\right) = \left(a \bullet b\right) \bullet a for any two elements ''a'' and ''b'' of the ...

s like the

sedenion In abstract algebra, the sedenions form a 16-dimension of a vector space, dimensional commutative property, noncommutative and associative property, nonassociative algebra over a field, algebra over the real numbers, usually represented by the cap ...

trigintaduonion In abstract algebra, the trigintaduonions, also known as the , , form a commutative property, noncommutative and associative property, nonassociative algebra over a field, algebra over the real numbers. Names The word ''trigintaduonion'' is d ...

s, and Okubo algebras. Any algebra whose elements are

idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

is also power-associative.

Exponentiation In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...

to the power of any

positive integer In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...

can be defined consistently whenever multiplication is power-associative. For example, there is no need to distinguish whether ''x''³ 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 In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

, so the existence of identity elements is useful in power-associative contexts. Over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

of characteristic 0, an algebra is power-associative if and only if it satisfies

,x,x 0

and

^2,x,x 0

, where

,y,z =(xy)z-x(yz)

is the

associator In abstract algebra, the term associator is used in different ways as a measure of the associativity, non-associativity of an algebraic structure. Associators are commonly studied as triple systems. Ring theory For a non-associative ring or non ...

(Albert 1948). Over an infinite field of

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

characteristic

p>0

there is no finite set of identities that characterizes power-associativity, but there are infinite independent sets, as described by Gainov (1970): * For

p=2

,x^2,x 0

and

^,x,x 0

for

n=3,2^k

(

k=2,3...)

* For

p=3

^,x,x 0

for

n=4,5,3^k

(

k=1,2...)

* For

p=5

^,x,x 0

for

n=3,4,6,5^k

(

k=1,2...)

* For

p>5

^,x,x 0

for

n=3,4,p^k

(

k=1,2...)

A substitution law holds for real power-associative algebras with unit, which basically asserts that multiplication of

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

s 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 .

References

* * * * * {{cite book , first=R. D. , last=Schafer , title=An introduction to non-associative algebras , publisher=Dover , year=1995 , orig-year=1966 , isbn=0-486-68813-5 , page
128–148
, url=https://archive.org/details/introductiontono0000scha/page/128 Non-associative algebra Properties of binary operations

Definition

Examples and properties

See also

References