Composition of Regular Representations

Theorem

Let $\left({S, *}\right)$ be a semigroup.

Let $\lambda_x, \rho_x$ be the left and right regular representations of $\left({S, *}\right)$ with respect to $x$.

Let $\lambda_x \circ \lambda_y$, $\rho_x \circ \rho_y$ etc. be defined as the composition of the mappings $\lambda_x$ and $\lambda_y$ etc.

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

$(1): \quad \lambda_x \circ \lambda_y = \lambda_{x * y}$

$(2): \quad \rho_x \circ \rho_y = \rho_{y * x}$

$(3): \quad \lambda_x \circ \rho_y = \rho_y \circ \lambda_x$.

Proof

Let $z \in S$.

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

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({\lambda_x \circ \lambda_y}\right) \left({z}\right)\)	\(=\)	\(\displaystyle \lambda_x \left({\lambda_y \left({z}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Composition of Mappings
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \lambda_x \left({y * z}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Left Regular Representation
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle x * \left({y * z}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Left Regular Representation
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({x * y}\right) * z\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Associativity
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \lambda_{x * y} \left({z}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Left Regular Representation

$\blacksquare$

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

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({\rho_x \circ \rho_y}\right) \left({z}\right)\)	\(=\)	\(\displaystyle \rho_x \left({\rho_y \left({z}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Composition of Mappings
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \rho_x \left({z * y}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Right Regular Representation
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({z * y}\right) * x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Right Regular Representation
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle z * \left({y * x}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Associativity
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \rho_{y * x} \left({z}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Right Regular Representation

$\blacksquare$

• $\lambda_x \circ \rho_y = \rho_y \circ \lambda_x$:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({\lambda_x \circ \rho_y}\right) \left({z}\right)\)	\(=\)	\(\displaystyle \lambda_x \left({\rho_y \left({z}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Composition of Mappings
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \lambda_x \left({z * y}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Right Regular Representation
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle x * \left({z * y}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Left Regular Representation
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({x * z}\right) * y\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Associativity
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \rho_y \left({x * z}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Right Regular Representation
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \rho_y \left({\lambda_x \left({z}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Left Regular Representation
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({\rho_y \circ \lambda_x}\right) \left({z}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Composition of Mappings

$\blacksquare$

Sources

• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): Exercise $6.2$