Associator

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In abstract algebra, the term associator is used in different ways as a measure of the nonassociativity of an algebraic structure. Associators are commonly studied as triple systems.

Ring theoryEdit

For a nonassociative ring or algebra  , the associator is the multilinear map   given by

 

Just as the commutator

 

measures the degree of noncommutativity, the associator measures the degree of nonassociativity of  . 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   is an alternative ring.

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

 

The nucleus is an associative subring of R.

Quasigroup theoryEdit

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.

Higher-dimensional algebraEdit

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

 

Category theoryEdit

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

See alsoEdit

ReferencesEdit

  • 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) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. CS1 maint: discouraged parameter (link)