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.
Ring theory
For a nonassociative ring or algebra R, the associator is the multilinear map [\cdot,\cdot,\cdot] : R \times R \times R \to R given by
[x,y,z] = (xy)z - x(yz).
Just as the commutator
[x, y] = xy - yx
measures the degree of noncommutativity, the associator measures the degree of nonassociativity of R.
For an associative ring or algebra the associator is identically zero.
The associator in any ring obeys the identity
w[x,y,z] + [w,x,y]z = [wx,y,z] - [w,xy,z] + [w,x,yz].
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
[n,R,R] = [R,n,R] = [R,R,n] = \{0\} \ .
The nucleus is an associative subring of R. Quasigroup theory
A quasigroup Q is a set with a binary operation \cdot : Q\times Q\to Q such that for each a,b in Q, the equations a\cdot x = b and y\cdot a = b have unique solutions x,y in Q.
In a quasigroup Q, the  associator is the map (\cdot,\cdot,\cdot) : Q\times Q\times Q\to Q defined by the equation
(a\cdot b)\cdot c=(a\cdot (b\cdot c))\cdot (a,b,c)
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
a_{x,y,z} : (xy)z \mapsto x(yz).
Category theory
In category theory, the associator expresses the associative properties of the internal product functor in monoidal categories.
See also
Commutator
Non-associative algebra
Quasi-bialgebra – discusses the Drinfeld associator
References
