# Composition of Left Regular Representation with Right

## Theorem

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

Let $\lambda_x, \rho_y$ be the left and right regular representations of $\struct {S, *}$ with respect to $x$ and $y$ respectively.

Let $\lambda_x \circ \rho_y$, $\rho_y \circ \lambda_x$ etc. be defined as the composition of the mappings $\lambda_x$ and $\rho_y$.

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

$\lambda_x \circ \rho_y = \rho_y \circ \lambda_x$

## Proof

Let $z \in S$.

 $\displaystyle \map {\paren {\lambda_x \circ \rho_y} } z$ $=$ $\displaystyle \map {\lambda_x} {\map {\rho_y} z}$ Definition of Composition of Mappings $\displaystyle$ $=$ $\displaystyle \map {\lambda_x} {z * y}$ Definition of Right Regular Representation $\displaystyle$ $=$ $\displaystyle x * \paren {z * y}$ Definition of Left Regular Representation $\displaystyle$ $=$ $\displaystyle \paren {x * z} * y$ Semigroup Axiom $\text S 1$: Associativity $\displaystyle$ $=$ $\displaystyle \map {\rho_y} {x * z}$ Definition of Right Regular Representation $\displaystyle$ $=$ $\displaystyle \map {\rho_y} {\map {\lambda_x} z}$ Definition of Left Regular Representation $\displaystyle$ $=$ $\displaystyle \map {\paren {\rho_y \circ \lambda_x} } z$ Definition of Composition of Mappings

$\blacksquare$