Power associativity: Difference between revisions
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In [[abstract algebra]], '''power associativity''' is a weak form of [[associativity]]. |
In [[abstract algebra]], '''power associativity''' is a weak form of [[associativity]]. |
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A [[magma]] (that is, a [[set]] with a [[binary operation]] on it) is said to be power-associative if the [[subalgebra|submagma]] generated by any element is a [[semigroup]] (where the operation must be associative). |
A [[magma|Magma_(algebra)]] (that is, a [[set]] with a [[binary operation]] on it) is said to be power-associative if the [[subalgebra|submagma]] generated by any element is a [[semigroup]] (where the operation must be associative). |
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This is equivalent to saying that (<i>x</i><i>x</i>)<i>x</i> = <i>x</i>(<i>x</i><i>x</i>) for every <i>x</i> in the magma. |
This is equivalent to saying that (<i>x</i><i>x</i>)<i>x</i> = <i>x</i>(<i>x</i><i>x</i>) for every <i>x</i> in the magma. |
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Revision as of 09:08, 1 December 2002
In abstract algebra, power associativity is a weak form of associativity.
A Magma_(algebra) (that is, a set with a binary operation on it) is said to be power-associative if the submagma generated by any element is a semigroup (where the operation must be associative). This is equivalent to saying that (xx)x = x(xx) for every x in the magma.
Similarly, and more commonly, an algebra A is said to be power-associative if the subalgebra generated by any element of A is associative. This is equivalent to saying that the multiplicative magma of A is power-associative. Every associative algebra is obviously power-associative, but so too are alternative algebras like the octonions and even some non-alternative algebras like the sedenions.
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.