The Scott open V-set monad and its categories of algebras

For a completely distributive lattice V, a novel class of lattice-valued Scott open sets, referred to as Scott open V-sets, is introduced on the powerset. These sets are utilized to construct a monad over the category of sets, termed the Scott open V-set monad. It is demonstrated that the category of Eilenberg-Moore algebras for the Scott open V-set monad is isomorphic to that of algebraic V-modules, and the category of Kleisli monoids with respect to this monad is isomorphic not only to the category of algebraic V-closure spaces but also to that of lax algebras for the finite powerset monad.