Associator

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. == See also ==