Associator

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

Ring theory
For a non-associative 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 non-commutativity, the associator measures the degree of non-associativity 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.