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 given byJust as the commutator
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
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
The nucleus is an associative subring of R.
Quasigroup theory
A quasigroup Q is a set with a binary operation such that for each a, b in Q,the equations and have unique solutions x, y in Q. In a quasigroup Q, the associator is the map defined by the equation
for all a, b, c in Q. As with its ring theory analog, the quasigroup associator is a measure of nonassociativity of Q.