{{inline |date=May 2024}} 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 <math>[\cdot,\cdot,\cdot] : R \times R \times R \to R</math> given by : <math>[x,y,z] = (xy)z - x(yz).</math>

Just as the commutator : <math>[x, y] = xy - yx</math> 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 : <math>w[x,y,z] + [w,x,y]z = [wx,y,z] - [w,xy,z] + [w,x,yz].</math>

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 : <math>[n,R,R] = [R,n,R] = [R,R,n] = \{0\} \ .</math>

The nucleus is an associative subring of ''R''.

== Quasigroup theory ==

A quasigroup ''Q'' is a set with a binary operation <math>\cdot : Q \times Q \to Q</math> such that for each ''a'', ''b'' in ''Q'', the equations <math>a \cdot x = b</math> and <math>y \cdot a = b</math> have unique solutions ''x'', ''y'' in ''Q''. In a quasigroup ''Q'', the associator is the map <math>(\cdot,\cdot,\cdot) : Q \times Q \times Q \to Q</math> defined by the equation : <math>(a\cdot b)\cdot c = (a\cdot (b\cdot c))\cdot (a,b,c)</math> 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 : <math> a_{x,y,z} : (xy)z \mapsto x(yz).</math>

== 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&nbsp;– discusses the ''Drinfeld associator''

== References ==

* {{cite journal |title=Identities for the Associator in Alternative Algebras |first1=M. |last1=Bremner |first2=I. |last2=Hentzel |journal=Journal of Symbolic Computation |volume=33 |issue=3 |date=March 2002 |pages=255–273 |doi=10.1006/jsco.2001.0510 |citeseerx=10.1.1.85.1905 }} * {{cite book |first=Richard D. |last=Schafer |title=An Introduction to Nonassociative Algebras |url=https://archive.org/details/introductiontono0000scha |url-access=registration |year=1995 |orig-date=1966 |publisher=Dover |isbn=0-486-68813-5 }}

Category:Non-associative algebra

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