Associator (mathematics)

In abstract algebra , the term associator is used in different ways as a measure of the deviation of an algebraic structure or a two-digit link defined there from the associative law.

Ring theory

In a non-associative ring or algebra , the associator is the multilinear map ${\ displaystyle R}$

{\ displaystyle [\ cdot, \ cdot, \ cdot]: R \ times R \ times R \ to R: [x, y, z] = (x \ circ y) \ circ zx \ circ (y \ circ z) ; \,}

where the spellings or are also common. For an (associative) ring it is always zero. ${\ displaystyle (x, y, z)}$ ${\ displaystyle A (x, y, z)}$

Identity applies to the associate

{\ displaystyle w [x, y, z] + [w, x, y] z = [wx, y, z] - [w, xy, z] + [w, x, yz]. \,}

It alternates exactly when is alternative . ${\ displaystyle R}$

It is symmetric in its rightmost arguments when is a pre-Lie algebra . ${\ displaystyle R}$

Quasi-group theory

The associator in quasi-groups also measures the deviation of the link defined there from the associative law, but otherwise its definition differs fundamentally from that in ring theory.

A quasi-group Q is a set with the two-digit link for which the equations and have the unique solutions . In a quasi-group Q the associator is through ${\ displaystyle \ cdot: Q \ times Q \ to Q}$ ${\ displaystyle \ forall a, b \ in Q}$ ${\ displaystyle a \ cdot x = b}$ ${\ displaystyle y \ cdot a = b}$ ${\ displaystyle x, y \ in Q}$