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 $z \in S$.
\(\ds \map {\lambda_x \circ \lambda_y} z\) | \(=\) | \(\ds \map {\lambda_x} {\map {\lambda_y} z}\) | Definition of Composition of Mappings | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\lambda_x} {y * z}\) | Definition of Left Regular Representation | |||||||||||
\(\ds \) | \(=\) | \(\ds x * \paren {y * z}\) | Definition of Left Regular Representation | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x * y} * z\) | Semigroup Axiom $\text S 1$: Associativity | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\lambda_{x * y} } z\) | Definition of Left Regular Representation |
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: An Introduction to Groups: Exercise $2$