Definition:Regular Representations/Right Regular Representation

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

• Definition:Left Regular Representation

• Right Regular Representation of Subset Product

• 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