# Definition:Regular Representations

## Definition

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

### Left Regular Representation

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 $a$.

### Right Regular Representation

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

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

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

## 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.