Definition:Regular Representations

Definition
Let $\struct {S, \circ}$ be a magma.

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 $\struct {S, \circ}$, we have:


 * $\map {\lambda_a} S = \set a \circ S = a \circ S$


 * $\map {\rho_a} S = S \circ \set a = S \circ a$

Also defined as
Some treatments of abstract algebra define this construct only for semigroups.

Some define it only for groups.