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 , the associator is the multilinear map given by
Just as the commutator measures the degree of noncommutativity, the associator measures the degree of nonassociativity of . It is identically zero for an associative ring or algebra.
The associator in any ring obeys the identity
The associator is alternating precisely when is an alternative ring.
The associator is symmetric in its two rightmost arguments when 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
It turns out that any two of being implies that the third is also the zero set.
A quasigroup Q is a set with a binary operation such that for each a,b in Q, the equations and have unique solutions x,y in Q. In a quasigroup Q, the associator is the map defined by the equation
for all a,b,c in Q. 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.