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) $$\lambda_x \circ \lambda_y = \lambda_{x * y}$$;
 * 2) $$\rho_x \circ \rho_y = \rho_{y * x}$$;
 * 3) $$\lambda_x \circ \rho_y = \rho_y \circ \lambda_x$$.

Proof
Let $$z \in S$$.


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

$$ $$ $$ $$ $$


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

$$ $$ $$ $$ $$


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

$$ $$ $$ $$ $$ $$ $$