Composition of Left Regular Representations

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Theorem

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

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

Let $\lambda_x \circ \lambda_y$ be defined as the composition of the mappings $\lambda_x$ and $\lambda_y$.


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

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


Proof

Let $z \in S$.

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

$\blacksquare$


Sources