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.

Ring theory[edit]

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 measures the degree of noncommutativity, the associator measures the degree of nonassociativity of R. It is identically zero for an associative ring or algebra.

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\} \ .

Quasigroup theory[edit]

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[edit]

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[edit]

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

See also[edit]

References[edit]