Composition of Left Regular Representation with Right

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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 $\forall x, y \in S$:

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


Proof

Let $z \in S$.

\(\displaystyle \map {\paren {\lambda_x \circ \rho_y} } z\) \(=\) \(\displaystyle \map {\lambda_x} {\map {\rho_y} z}\) Definition of Composition of Mappings
\(\displaystyle \) \(=\) \(\displaystyle \map {\lambda_x} {z * y}\) Definition of Right Regular Representation
\(\displaystyle \) \(=\) \(\displaystyle x * \paren {z * y}\) Definition of Left Regular Representation
\(\displaystyle \) \(=\) \(\displaystyle \paren {x * z} * y\) Semigroup Axiom $S1$: Associativity
\(\displaystyle \) \(=\) \(\displaystyle \map {\rho_y} {x * z}\) Definition of Right Regular Representation
\(\displaystyle \) \(=\) \(\displaystyle \map {\rho_y} {\map {\lambda_x} z}\) Definition of Left Regular Representation
\(\displaystyle \) \(=\) \(\displaystyle \map {\paren {\rho_y \circ \lambda_x} } z\) Definition of Composition of Mappings

$\blacksquare$


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