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 [ ⋅ , ⋅ , ⋅ ] : R × R × R → R {\displaystyle [\cdot ,\cdot ,\cdot ]:R\times R\times R\to R} given by
[ x , y , z ] = ( x y ) z − x ( y z ) . {\displaystyle [x,y,z]=(xy)z-x(yz).}
Just as the commutator
[ x , y ] = x y − y x {\displaystyle [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 = [ w x , y , z ] − [ w , x y , z ] + [ w , x , y z ] . {\displaystyle 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 } . {\displaystyle [n,R,R]=[R,n,R]=[R,R,n]=\{0\}\ .}
The nucleus is an associative subring of R.
Quasigroup theory A quasigroup Q is a set with a binary operation ⋅ : Q × Q → Q {\displaystyle \cdot :Q\times Q\to Q} such that for each a, b in Q, the equations a ⋅ x = b {\displaystyle a\cdot x=b} and y ⋅ a = b {\displaystyle y\cdot a=b} have unique solutions x, y in Q. In a quasigroup Q, the associator is the map ( ⋅ , ⋅ , ⋅ ) : Q × Q × Q → Q {\displaystyle (\cdot ,\cdot ,\cdot ):Q\times Q\times Q\to Q} defined by the equation
( a ⋅ b ) ⋅ c = ( a ⋅ ( b ⋅ c ) ) ⋅ ( a , b , c ) {\displaystyle (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 : ( x y ) z ↦ x ( y z ) . {\displaystyle 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.
See also Commutator Non-associative algebra Quasi-bialgebra – discusses the Drinfeld associator
References 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. {{cite journal}}: Cite uses deprecated parameter |citeseerx= (help) Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5.