The associator in any ring obeys the identity
The associator is symmetric in its two rightmost arguments when 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
It turns out that any two of being implies that the third is also the zero set.
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.
In higher-dimensional algebra, where there may be non-identity morphisms between algebraic expressions, an associator is an isomorphism
- Bremner and Hentzel. "Identities for the Associator in Alternative Algebras" (PDF).
- Schafer, Richard D. (1995) . An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5.
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