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:


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


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