Associator

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

Ring theory

In a non-associative ring or algebra of the associator is the multilinear map

Whereby the spellings or are common. For an ( associative ) ring he is always equal to zero.

For the associator identity applies

He alternates iff is alternative.

It is symmetrical in its rightmost arguments, if a pre- Lie algebra.

Quasigroup theory

Also, the associator in quasigroups measures the deviation of the defined there logic from the associative law, but its definition differs otherwise fundamentally from that in the ring theory.

A quasigroup Q is a quantity with the two-digit shortcut to have the equations and the unique solutions. In a quasigroup Q is the associator by

Defined.