Composition of Right Regular Representations

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Theorem

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

Let $\rho_x$ be the right regular representation of $\struct {S, *}$ with respect to $x$.

Let $\rho_x \circ \rho_y$ be defined as the composition of the mappings $\rho_x$ and $\rho_y$.


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

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


Proof

Let $z \in S$.

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

$\blacksquare$


Sources