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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 power-alternativity, that is (xx)x = x(xx) for every x in the algebra, but weaker than alternativity or associativity.Every associative algebra is 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 positive integer can be defined consistently whenever multiplication is power-associative. For example, there is no need to distinguish 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 is useful in power-associative contexts.A 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 wikiPageLength "3284".
- Power_associativity wikiPageOutDegree "21".
- Power_associativity wikiPageRevisionID "649046384".
- Power_associativity wikiPageWikiLink Abstract_algebra.
- Power_associativity wikiPageWikiLink Algebra_over_a_field.
- Power_associativity wikiPageWikiLink Alternative_algebra.
- Power_associativity wikiPageWikiLink Alternativity.
- Power_associativity wikiPageWikiLink American_Mathematical_Society.
- Power_associativity wikiPageWikiLink Associative_algebra.
- Power_associativity wikiPageWikiLink Associative_property.
- Power_associativity wikiPageWikiLink Associativity.
- Power_associativity wikiPageWikiLink Binary_operation.
- Power_associativity wikiPageWikiLink Cambridge_University_Press.
- Power_associativity wikiPageWikiLink Category:Binary_operations.
- Power_associativity wikiPageWikiLink Category:Non-associative_algebra.
- Power_associativity wikiPageWikiLink Exponentiation.
- Power_associativity wikiPageWikiLink Idempotence.
- Power_associativity wikiPageWikiLink Idempotent.
- Power_associativity wikiPageWikiLink Identity_element.
- Power_associativity wikiPageWikiLink Magma_(algebra).
- Power_associativity wikiPageWikiLink Natural_number.
- Power_associativity wikiPageWikiLink Octonion.
- Power_associativity wikiPageWikiLink Okubo_algebra.
- Power_associativity wikiPageWikiLink Positive_integer.
- Power_associativity wikiPageWikiLink Sedenion.
- Power_associativity wikiPageWikiLink Subalgebra.
- Power_associativity wikiPageWikiLink Transactions_of_the_American_Mathematical_Society.
- Power_associativity wikiPageWikiLinkText "'''power associative'''".
- Power_associativity wikiPageWikiLinkText "Power associativity".
- Power_associativity wikiPageWikiLinkText "Power_associativity".
- Power_associativity wikiPageWikiLinkText "power associative".
- Power_associativity wikiPageWikiLinkText "power associativity".
- Power_associativity hasPhotoCollection Power_associativity.
- Power_associativity wikiPageUsesTemplate Template:Cite_book.
- Power_associativity wikiPageUsesTemplate Template:Cite_journal.
- Power_associativity subject Category:Binary_operations.
- Power_associativity subject Category:Non-associative_algebra.
- Power_associativity hypernym Property.
- Power_associativity type Building.
- 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).".
- Power_associativity label "Power associativity".
- Power_associativity sameAs Potenz-assoziative_Algebra.
- Power_associativity sameAs Associativité_des_puissances.
- Power_associativity sameAs אלגברה_עם_חזקה_אסוציאטיבית.
- Power_associativity sameAs Associatività_della_potenza.
- Power_associativity sameAs Machtassociativiteit.
- Power_associativity sameAs m.0688y.
- Power_associativity sameAs Степенная_ассоциативность.
- Power_associativity sameAs Q1822837.
- Power_associativity sameAs Q1822837.
- Power_associativity sameAs 冪結合性.
- Power_associativity wasDerivedFrom Power_associativity?oldid=649046384.
- Power_associativity isPrimaryTopicOf Power_associativity.