Power associativity
Encyclopedia
In abstract algebra
, power associativity is a property of a binary operation
which 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.
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, but weaker than alternativity
or associativity, which requires that (xy)z = x(yz) for every x, y, and z in the algebra.
Every associative algebra
is obviously power-associative, but so are all other alternative algebra
s (like the octonion
s, which are non-associative) and even some non-alternative algebras like the sedenion
s. Any algebra whose elements are idempotent is also power-associative.
Exponentiation
to the power of any natural number
other than zero
can be defined consistently whenever multiplication is power-associative.
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.
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).
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, power associativity is a property of a binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
which is a weak form of associativity
Associativity
In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
.
An algebra
Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
(or more generally a magma
Magma (algebra)
In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M \times M \rightarrow M....
) is said to be power-associative if the subalgebra
Subalgebra
In mathematics, the word "algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures...
generated by any element is associative.
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, but weaker than alternativity
Alternativity
In abstract algebra, alternativity is a property of a binary operation. A magma G is said to be left alternative if y = x for all x and y in G and right alternative if y = x for all x and y in G...
or associativity, which requires that (xy)z = x(yz) for every x, y, and z in the algebra.
Every associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
is obviously power-associative, but so are all other alternative algebra
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 = y*x = y...
s (like the octonion
Octonion
In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...
s, which are non-associative) and even some non-alternative algebras like the sedenion
Sedenion
In abstract algebra, sedenions form a 16-dimensional non-associative algebra over the reals obtained by applying the Cayley–Dickson construction to the octonions...
s. Any algebra whose elements are idempotent is also power-associative.
Exponentiation
Exponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
to the power of any natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
other than zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
can be defined consistently whenever multiplication is power-associative.
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
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
, so the existence of identity elements becomes especially useful in power-associative contexts.
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).