Associator

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 theory

For a nonassociative 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 noncommutativity, the associator measures the degree of nonassociativity 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\} \ .

It turns out that any two of $([n,R,R],[R,n,R] , [R,R,n])$ being $\{0\}$ implies that the third is also the zero set.

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.

References

Bremner, M.; Hentzel, I. (March 2002). "Identities for the Associator in Alternative Algebras" (PDF). Journal of Symbolic Computation. 33 (3): 255–273. doi:10.1006/jsco.2001.0510.
Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5.

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