Definition:Regular Representations

Definition
Let $\left ({S, \circ}\right)$ be an algebraic structure.

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

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


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


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

Also defined as
Although the left and right regular representations are defined here in the context of the general algebraic structure, many treatments of abstract algebra define this construct only for semigroups.