# 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 $\text S 1$: Associativity $\displaystyle$ $=$ $\displaystyle \map {\lambda_{x * y} } z$ Definition of Left Regular Representation

$\blacksquare$