##  [Alternativity](/alternativity-0) 

 Definition

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

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

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

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

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

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

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

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

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.