# Definition:Regular Representations/Left Regular Representation

< Definition:Regular Representations(Redirected from Definition:Left Regular Representation)

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## 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

For the **left regular representation**, some sources use a hyphen: **left-regular representation**.

Some sources refer to the **left regular representation** as **left multiplication**.

## Also defined as

Some treatments of abstract algebra and group theory define the **regular representations** for semigroups.

Some define it only for groups.

## Also see

- Regular Representation of Invertible Element is Permutation
- Regular Representations in Group are Permutations

- Results about
**regular representations**can be found**here**.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{II}$: Groups: Problem $\text{EE}$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 35$: Elementary consequences of the group axioms