# Composition of Right Regular Representations

## Theorem

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

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

Let $\rho_x \circ \rho_y$ be defined as the composition of the mappings $\rho_x$ and $\rho_y$.

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

$\rho_x \circ \rho_y = \rho_{y * x}$

## Proof

Let $z \in S$.

 $\displaystyle \map {\paren {\rho_x \circ \rho_y} } z$ $=$ $\displaystyle \map {\rho_x} {\map {\rho_y} z}$ Definition of Composition of Mappings $\displaystyle$ $=$ $\displaystyle \map {\rho_x} {z * y}$ Definition of Right Regular Representation $\displaystyle$ $=$ $\displaystyle \paren {z * y} * x$ Definition of Right Regular Representation $\displaystyle$ $=$ $\displaystyle z * \paren {y * x}$ Semigroup Axiom $S1$: Associativity $\displaystyle$ $=$ $\displaystyle \map {\rho_{y * x} } z$ Definition of Right Regular Representation

$\blacksquare$