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- Power_associativity abstract "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.Every associative algebra is obviously power-associative, but so are all other alternative algebras (like the octonions, which are non-associative) and even some non-alternative algebras like the sedenions and Okubo algebras. 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).".
- Power_associativity wikiPageID "25054".
- Power_associativity wikiPageRevisionID "593783190".
- Power_associativity hasPhotoCollection Power_associativity.
- Power_associativity subject Category:Binary_operations.
- Power_associativity subject Category:Non-associative_algebra.
- Power_associativity type BinaryOperations.
- Power_associativity type BooleanOperation113440935.
- Power_associativity type DataProcessing113455487.
- Power_associativity type Operation113524925.
- Power_associativity type PhysicalEntity100001930.
- Power_associativity type Process100029677.
- Power_associativity type Processing113541167.
- Power_associativity comment "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.Every associative algebra is obviously power-associative, but so are all other alternative algebras (like the octonions, which are non-associative) and even some non-alternative algebras like the sedenions and Okubo algebras. ".
- Power_associativity label "Associatività della potenza".
- Power_associativity label "Associativité des puissances".
- Power_associativity label "Machtassociativiteit".
- Power_associativity label "Potenz-assoziative Algebra".
- Power_associativity label "Power associativity".
- Power_associativity label "Степенная ассоциативность".
- Power_associativity label "冪結合性".
- Power_associativity sameAs Potenz-assoziative_Algebra.
- Power_associativity sameAs Associativité_des_puissances.
- Power_associativity sameAs Associatività_della_potenza.
- Power_associativity sameAs Machtassociativiteit.
- Power_associativity sameAs m.0688y.
- Power_associativity sameAs Q1822837.
- Power_associativity sameAs Q1822837.
- Power_associativity sameAs Power_associativity.
- Power_associativity wasDerivedFrom Power_associativity?oldid=593783190.
- Power_associativity isPrimaryTopicOf Power_associativity.