# Definition:Regular Representations

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## 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**.

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 35$: Elementary consequences of the group axioms