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