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

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

• 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