Definition:Regular Representations/Left Regular Representation

Definition
Let $\left ({S, \circ}\right)$ be an algebraic structure. The mapping $\lambda_a: S \to S$ is defined as:


 * $\forall a \in S: \lambda_a \left({x}\right) = a \circ x$

This is known as the left regular representation of $\left ({S, \circ}\right)$ with respect to $a$.

Also known as
Some sources use a hyphen: left-regular representation.

Also defined as
Although the left regular representation is defined here in the context of the general algebraic structure, many treatments of abstract algebra define this construct only for semigroups.

Also see

 * Right Regular Representation


 * Regular Representations of Invertible Elements are Permutations
 * Regular Representations in Group are Permutations