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 $ M $-actions. 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 $ M $-actions. 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.