Powerset theory, monads, Kleisli category, fuzzy type relations, monadic relations

Powerset operators in categories with relations defined by monads as morphisms are investigated. Such categories are, in fact, Kleisli categories of monads in clone form. Examples of such categories used in lattice-valued F-transform theory are presented. It is proven that for arbitrary powerset operator P in a category K with a monad T there exists a powerset operator eP of the Kleisli category KT, which extends the original powerset operator of the category K.