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0, an algebra is power-associative if and only if it satisfies and , where is the associator (Albert 1948).

Over an infinite field of prime characteristic there is no finite set of identities that characterizes power-associativity, but there are infinite independent sets, as described by Gainov (1970):

* For : and for (
* For : for (
* For : for (
* For : for (

A substitution law holds for real

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