# Definition:Regular Representations/Left Regular Representation

## Definition

Let $\struct {S, \circ}$ be a magma.

The mapping $\lambda_a: S \to S$ is defined as:

$\forall x \in S: \map {\lambda_a} x = a \circ x$

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

## Also known as

Some sources use a hyphen: left-regular representation.

## Also defined as

Some treatments of abstract algebra and group theory define this construct for semigroups.

Some define it only for groups.

## Also see

• Results about regular representations can be found here.