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: