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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 $\struct {A_R, \oplus}$.

Also see

  • Results about associators can be found here.