Misplaced Pages

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (May 2024) (Learn how and when to remove this message)

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 ${\displaystyle :R\times R\times R\to R}$ given by

${\displaystyle =(xy)z-x(yz).}$

Just as the commutator

${\displaystyle =xy-yx}$

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

${\displaystyle w+z=-+.}$

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

${\displaystyle ===\{0\}\ .}$

The nucleus is an associative subring of R.

Quasigroup theory

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.

Higher-dimensional algebra

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

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

Category theory

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

See also

• Commutator
• Non-associative algebra
• Quasi-bialgebra – discusses the Drinfeld associator

References

• Bremner, M.; Hentzel, I. (March 2002). "Identities for the Associator in Alternative Algebras". Journal of Symbolic Computation. 33 (3): 255–273. CiteSeerX 10.1.1.85.1905. doi:10.1006/jsco.2001.0510.
• Schafer, Richard D. (1995) . An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5.

This algebra-related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Non-associative algebra
• Algebra stubs