Associators are commonly studied as triple systems.
For a non-associative ring or algebra R, the associator is the multilinear map
The associator is symmetric in its two rightmost arguments when R is a pre-Lie algebra.
The nucleus is the set of elements that associate with all others: that is, the n in R such that The nucleus is an associative subring of R. A quasigroup Q is a set with a binary operation
As with its ring theory analog, the quasigroup associator is a measure of nonassociativity of Q.
In higher-dimensional algebra, where there may be non-identity morphisms between algebraic expressions, an associator is an isomorphism In category theory, the associator expresses the associative properties of the internal product functor in monoidal categories.