Definition:Associator

Definition
Let $\struct {A_R, \oplus}$ be an algebra over a ring.

Consider the trilinear mapping $\sqbrk {\cdot, \cdot, \cdot}: A_R^3 \to A_R$ defined as:
 * $\forall a, b, c \in A_R: \sqbrk {a, b, c} := \paren {a \oplus b} \oplus c - a \oplus \paren {b \oplus c}$

Then $\sqbrk {\cdot, \cdot, \cdot}$ is known as the associator of $\struct {A_R, \oplus}$.

It can be considered a measure of how much associativity of $\oplus$ fails in $\struct {A_R, \oplus}$.

Note that trivially if $\struct {A_R, \oplus}$ is an associative algebra, then:
 * $\forall a, b, c \in A_R: \sqbrk {a, b, c} = \mathbf 0_R$

Also see

 * Definition:Commutator