Definition:Regular Representations/Left Regular Representation

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

Some sources use a hyphen: left-regular representation.

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:Right Regular Representation

• Left 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

• 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