Composition of Left Regular Representation with Right

Theorem

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

Let $\lambda_x, \rho_y$ be the left and right regular representations of $\struct {S, *}$ with respect to $x$ and $y$ respectively.

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

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

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

Let $z \in S$.

$\ds \map {\paren {\lambda_x \circ \rho_y} } z$

$=$

$\ds \map {\lambda_x} {\map {\rho_y} z}$

$\ds $

$=$

$\ds \map {\lambda_x} {z * y}$

$\ds $

$=$

$\ds x * \paren {z * y}$

$\ds $

$=$

$\ds \paren {x * z} * y$

$\ds $

$=$

$\ds \map {\rho_y} {x * z}$

$\ds $

$=$

$\ds \map {\rho_y} {\map {\lambda_x} z}$

$\ds $

$=$

$\ds \map {\paren {\rho_y \circ \lambda_x} } z$

$\blacksquare$

1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: An Introduction to Groups: Exercise $2$