Partially ordered fuzzy power set monads on the category of L-sets and their associated categories of topological space objects

Abstract

For a GL-monoid L with a complete lattice L underneath, this paper considers the L-unbalanced power object monad on the category Set(L) of L-sets, and introduces a new monad on Set(Lop) called the L-power L-set monad. The Kleisli and Eilenberg-Moore categories of these monads are studied. Both monads are enhanced to partially ordered monads that allow us to apply the theory of topological space objects developed by Ulrich Höhle. We give the explicit descriptions of the category of topological space objects in Set(L) with respect to the partially ordered L-unbalanced power object monad and the category of topological space objects in Set(Lop) with respect to the partially ordered L-power L-set monad. Furthermore, it is shown that the L-power L-set monad on Set(Lop) determines a monad on Set(L) in a one-to-one manner, and all existing results can be directly converted into this monad.