In
mathematics, specifically in
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, power associativity is a property of a
binary operation that is a weak form of
associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
.
Definition
An
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
(or more generally a
magma
Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natural sa ...
) 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 operat ...
generated by any element is associative. Concretely, this means that if an element
is performed an operation
by itself several times, it doesn't matter in which order the operations are carried out, so for instance
.
Examples and properties
Every
associative algebra is power-associative, but so are all other
alternative algebra In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
*x(xy) = (xx)y
*(yx)x = y(xx)
for all ''x'' and ''y'' in the algebra.
Every associative algebra is o ...
s (like the
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s, which are non-associative) and even some non-alternative algebras like the
sedenion
In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers; they are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic to ...
s and
Okubo algebra In abstract algebra, algebra, an Okubo algebra or pseudo-octonion algebra is an 8-dimensional algebra over a field, non-associative algebra similar to the one studied by Susumu Okubo. Okubo algebras are composition algebras, flexible algebras (''A'' ...
s. 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
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to r ...
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 ''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 operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
, 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
and
, where
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-as ...
(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
there is no finite set of identities that characterizes power-associativity, but there are infinite independent sets, as described by Gainov (1970):
* For
:
and
for
(
* For
:
for
(
* For
:
for
(
* For
:
for
(
A substitution law holds for
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
power-associative algebras with unit, which basically asserts that multiplication of
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
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 .
See also
*
Alternativity
In abstract algebra, alternativity is a property of a binary operation. A magma ''G'' is said to be if (xx)y = x(xy) for all x, y \in G and if y(xx) = (yx)x for all x, y \in G. A magma that is both left and right alternative is said to be () ...
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