Associator

In abstract algebra, the term associator is used in different ways as a measure of the nonassociativity of an algebraic structure. Associators are commonly studied as triple systems.

For a nonassociative ring or algebra ${\displaystyle R}$, the associator is the multilinear map ${\displaystyle [\cdot ,\cdot ,\cdot ]:R\times R\times R\to R}$ given by

Just as the commutator measures the degree of noncommutativity, the associator measures the degree of nonassociativity of ${\displaystyle R}$. It is identically zero for an associative ring or algebra.

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