Associator
# 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.

