# Composition of Left Regular Representations

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## Theorem

Let $\struct {S, *}$ be a semigroup.

Let $\lambda_x$ be the left regular representation of $\struct {S, *}$ with respect to $x$.

Let $\lambda_x \circ \lambda_y$ be defined as the composition of the mappings $\lambda_x$ and $\lambda_y$.

Then $\forall x, y \in S$:

- $\lambda_x \circ \lambda_y = \lambda_{x * y}$

## Proof

Let $z \in S$.

\(\displaystyle \map {\lambda_x \circ \lambda_y} z\) | \(=\) | \(\displaystyle \map {\lambda_x} {\map {\lambda_y} z}\) | Definition of Composition of Mappings | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map {\lambda_x} {y * z}\) | Definition of Left Regular Representation | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle x * \paren {y * z}\) | Definition of Left Regular Representation | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {x * y} * z\) | Semigroup Axiom $S1$: Associativity | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map {\lambda_{x * y} } z\) | Definition of Left Regular Representation |

$\blacksquare$

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $6$: An Introduction to Groups: Exercise $2$