The property of an algebra that powers of any single element are unambiguously defined by the rule a^m a^n = a^{m+n} for all nonnegative integers m,n, equivalently all parenthesizations of repeated multiplication of the same element agree.

Principle

Ensure that the subalgebra generated by one element is associative so that polynomial expressions in a single generator and exponentiation are well defined despite the larger algebra possibly being nonassociative.

Demonstration

Jordan algebras and alternative algebras are power-associative: for a fixed a, the products a(a(a…)) and ((…(aa)a)…) coincide, hence one can form a^k unambiguously and evaluate polynomials in a.

Misapplication

Assuming power-associativity implies associativity for products of different elements; power-associativity guarantees only that repeated self-products associate, not mixed triples of distinct elements.

Consequence

Allows a consistent calculus of powers and polynomial functional calculus for single elements, supports definitions of exponentials or minimal polynomials when combined with other structure, and simplifies spectral considerations.

Reversal

Without power-associativity, different parenthesizations of a^n may yield different results, making the notion of a^n or of polynomials in a ill-defined; conversely full associativity guarantees power-associativity but is stronger.

Boundary

Refers specifically to monogenic subalgebras and does not constrain associators with distinct entries; it can be stated with or without a unit and may require characteristic restrictions for certain conclusions.

Semantic Tension

Competes with alternativity: alternativity implies power-associativity and extra two-generator associativity (Artin's theorem), but power-associativity alone leaves open many nonassociative behaviors.

Synthesis

Power-associativity isolates the minimal associativity needed to treat powers and polynomials in one element reliably: it enforces agreement of all parenthesizations of repeated self-product while permitting broader nonassociative phenomena.