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 $\forall x, y \in S$:


 * $(1): \quad \lambda_x \circ \lambda_y = \lambda_{x * y}$
 * $(2): \quad \rho_x \circ \rho_y = \rho_{y * x}$
 * $(3): \quad \lambda_x \circ \rho_y = \rho_y \circ \lambda_x$.

Proof
Let $z \in S$.


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


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


 * $\lambda_x \circ \rho_y = \rho_y \circ \lambda_x$: