# Definition:Regular Representations/Right Regular Representation

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## Definition

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

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 known as

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

However, this can be confusing: when the term **right** appears hyphenated in this manner, it usually has the meaning of **perpendicular** or **orthogonal**.

Some sources refer to the **right regular representation** as **right 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
**the right regular representation**can be found**here**.

## Sources

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