Associator

Ring theory

For a nonassociative ring or algebra R , the associator is the multilinear map [ ⋅ , ⋅ , ⋅ ] : R × R × R → R given by

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

The associator in any ring obeys the identity

The associator is alternating precisely when R is an alternative ring.

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

It turns out that any two of ( [ n , R , R ] , [ R , n , R ] , [ R , R , n ] ) being { 0 } implies that the third is also the zero set.

Quasigroup theory

A quasigroup Q is a set with a binary operation ⋅ : Q × Q → Q such that for each a,b in Q, the equations a ⋅ x = b and y ⋅ a = b have unique solutions x,y in Q. In a quasigroup Q, the associator is the map ( ⋅ , ⋅ , ⋅ ) : Q × Q × Q → Q 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.

Higher-dimensional algebra

In higher-dimensional algebra, where there may be non-identity morphisms between algebraic expressions, an associator is an isomorphism

Category theory

In category theory, the associator expresses the associative properties of the internal product functor in monoidal categories.