# Category:Regular Representations

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This category contains results about **Regular Representations**.

Definitions specific to this category can be found in Definitions/Regular Representations.

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

### Left Regular Representation

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 $a$**.

### Right Regular Representation

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 $a$**.

## Subcategories

This category has only the following subcategory.

## Pages in category "Regular Representations"

The following 23 pages are in this category, out of 23 total.

### C

### L

### R

- Regular Representation of Invertible Element is Permutation
- Regular Representation on Subgroup is Bijection to Coset
- Regular Representation wrt Cancellable Element on Finite Semigroup is Bijection
- Regular Representations in Group are Permutations
- Regular Representations in Semigroup are Permutations then Structure is Group
- Regular Representations of Subset Product
- Regular Representations wrt Element are Permutations then Element is Invertible
- Right and Left Regular Representations in Topological Group are Homeomorphisms
- Right Cancellable iff Right Regular Representation Injective
- Right Regular Representation by Inverse is Transitive Group Action
- Right Regular Representation of 0 is Bijection in B-Algebra
- Right Regular Representation of Subset Product
- Right Regular Representation wrt Right Cancellable Element on Finite Semigroup is Bijection