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

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.

Also defined as

Some treatments of abstract algebra and group theory define this construct 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 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