Associativeness of the powers

In Algebra, the associativeness of the powers is a weaker form of the Associativité and Alternativité.

A magma is known as associative powers if the under-magma generated by any element is associative. Concretely, that means that if element X is multiplied by itself several times, the order in which are carried out these multiplications does not have importance; thus, for example, x (X (xx)) = (X (xx))X = (xx) (xx)

Any associative magma is obviously associative powers , as well as the alternate magmas, as the Octonion S. Certains not-alternate magmas are it also, like the Sédénion S.

The exponentiation with a power of natural entirety different from zero can be defined in a coherent way if the multiplication is associative powers . For example, there is no ambiguity that x3 is defined like (xx) X or X (xx), because both are equal. The exponentiation with a power of zero can also be defined if the operation has a neutral element: the existence of such elements is thus particularly useful in the contexts where the associativeness of the powers is checked.

A remarkable law of substitution is valid in the associative algebras of the powers, with neutral element. She affirms that the Multiplication of the polynomials functions as waited. Are F and G two Polynôme S realities in X. For all has, we have (fg) (A) = F (A) G (a).

See too

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