A Fuzzy Lower Powerdomain

The power set of a domain D can be seen as the function space D ! 2 of all functions in the pointwise order. One computational version thereof is the lifted lower powerdomain, which we can think of as the lattice of Scott-closed subsets or, alternatively, as the order dual of the function space D ! 2]. Therefore, logic and the ordinary power set, as well as its information-theoretic version, rest upon 2. If we replace 2 by the unit interval I then the order dual of D ! I] gives rise to the fuzzy lower power domain F D]. On maps g 2 D ! E] the function F g] is the upper adjoint of g ! id I ]. In this way F ] is a locally continuous functor from DCPO to distributive complete lattices and maps preserving suprema. If D is an L-domain then F D] is a distributive co-continuous lattice. If D is an FS-domain then F D] is completely distributive; in particular, U F ] is a locally continuous endofunctor on FS where U is the obvious forgetful functor.