POWERSET MONAD, FILTER MONAD AND PRIMEFILTER MONAD IN THE CATEGORY OF SETS WITH MONOID ACTIONS

We construct a powerset monad, a filter monad and a primefilter monad in the category M-Set of sets with Mactions. To investigate the categories of algebras of these monads in M-Set, we consider the category of complete semilattices (resp. continuous lattices, compact Hausd orff spaces) with Mactions. We show that if M is a group, this category is isomorphic to the category of algebras of the powerset monad (resp. filter monad, primefilter monad) in M-Set.