Definition:Regular Representations

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

Left Regular Representation
The mapping $\lambda_a: S \to S$ is defined as:


 * $\forall a \in S: \lambda_a \left({x}\right) = a \circ x$

This is known as the left regular representation of $\left ({S, \circ}\right)$ with respect to $a$.

Right Regular Representation
The mapping $\rho_a: S \to S$ is defined as:


 * $\forall a \in S: \rho_a \left({x}\right) = x \circ a$

This is known as the right regular representation of $\left ({S, \circ}\right)$ with respect to $a$.

Regular Representations as Subset Product
It can be seen that the left and right regular representations of a semigroup are examples of the subset product where one of the subsets is a singleton.

That is, for any semigroup $\left ({S, \circ}\right)$, we have:


 * $\lambda_a \left({S}\right) = \left \{{a}\right\} \circ S = a \circ S$


 * $\rho_a \left({S}\right) = S \circ \left \{{a}\right\} = S \circ a$