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.

For a non-associative ring or algebra R, the associator is the multilinear map ${\displaystyle [\cdot ,\cdot ,\cdot ]:R\times R\times R\to R}$ given by

${\displaystyle [x,y,z]=(xy)z-x(yz).}$

Just as the commutator

${\displaystyle [x,y]=xy-yx}$

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

${\displaystyle w[x,y,z]+[w,x,y]z=[wx,y,z]-[w,xy,z]+[w,x,yz].}$

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

${\displaystyle [n,R,R]=[R,n,R]=[R,R,n]=\{0\}\ .}$

The nucleus is an associative subring of R.

A quasigroup Q is a set with a binary operation ${\displaystyle \cdot :Q\times Q\to Q}$ such that for each a, b in Q, the equations ${\displaystyle a\cdot x=b}$ and ${\displaystyle y\cdot a=b}$ have unique solutions x, y in Q. In a quasigroup Q, the associator is the map ${\displaystyle (\cdot ,\cdot ,\cdot ):Q\times Q\times Q\to Q}$ defined by the equation

${\displaystyle (a\cdot b)\cdot c=(a\cdot (b\cdot c))\cdot (a,b,c)}$

for all a, b, c in Q. As with its ring theory analog, the quasigroup associator is a measure of nonassociativity of Q.

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

${\displaystyle a_{x,y,z}:(xy)z\mapsto x(yz).}$

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