A property of a (not necessarily associative) algebra stating that the associator (a,b,c)=(ab)c−a(bc) vanishes whenever two adjacent arguments coincide, i.e. (a,a,b)=(b,a,a)=0 for all a,b.

Principle

Weaken full associativity by demanding associativity only on shells where two neighboring slots agree; this guarantees associativity of subalgebras generated by two elements (Artin's theorem) and yields useful identities.

Demonstration

The octonion algebra is alternative but nonassociative: any product where two adjacent factors coincide associates, and every subalgebra generated by two octonions is associative, illustrating alternativity concretely.

Misapplication

Inferring full associativity from alternativity or assuming identities valid for associative rings hold generally; alternativity does not imply triple associativity for arbitrary distinct elements.

Consequence

Alternativity implies power-associativity and that monogenic and two-generated subalgebras are associative, enabling well-defined powers and many classical manipulations within those subalgebras.

Reversal

If alternativity is removed one may obtain algebras where two-generated subalgebras are nonassociative and powers can be ambiguous; conversely full associativity is the stronger condition that forces all associators to vanish.

Boundary

This property concerns the ternary associator and applies to algebras over a ring or field; it neither requires a unit nor excludes other pathologies but does not control associators with three distinct entries.

Semantic Tension

Tension with power-associativity: alternativity implies power-associativity and two-generated associativity, but power-associativity alone does not force alternativity; they are related but distinct weakenings of associativity.

Synthesis

Alternativity is the middle ground between associativity and weaker identities: it forces associators with repeated adjacent entries to vanish, yielding associative behavior for one- and two-generated subalgebras while allowing genuine nonassociativity in larger tensorings.