# Composition of Regular Representations

## Theorem

Let $\left({S, *}\right)$ be a semigroup.

Let $\lambda_x, \rho_x$ be the left and right regular representations of $\left({S, *}\right)$ with respect to $x$.

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

Then $\forall x, y \in S$, the following results hold:

### Composition of Left Regular Representations

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

### Composition of Right Regular Representations

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

### Composition of Left Regular Representation with Right

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