Associator Explained

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

[ ⋅ , ⋅ , ⋅ ]:R x R x R\toR

given by

[x,y,z]=(xy)z-x(yz).

[x,y]=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

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\} .

The nucleus is an associative subring of R.

Quasigroup theory

⋅ :Q x Q\toQ

such that for each a, b in Q,the equations

a ⋅ x=b

and

y ⋅ a=b

have unique solutions x, y in Q. In a quasigroup Q, the associator is the map

( ⋅ , ⋅ , ⋅ ):Q x Q x Q\toQ

defined by the equation

(a ⋅ b) ⋅ c=(a ⋅ (b ⋅ c)) ⋅ (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\mapstox(yz).

Category theory

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

References

Identities for the Associator in Alternative Algebras . M. . Bremner . I. . Hentzel . Journal of Symbolic Computation . 33 . 3 . March 2002 . 255–273 . 10.1006/jsco.2001.0510 . 10.1.1.85.1905 .
Book: Schafer, Richard D. . An Introduction to Nonassociative Algebras . registration . 1995 . 1966 . Dover . 0-486-68813-5 .