A duality between fuzzy domains and strongly completely distributive $L$-ordered sets

The aim of this paper is to establish a fuzzy version of the dualitybetween domains and completely distributive lattices. All values aretaken in a fixed frame $L$. A definition of (strongly) completelydistributive $L$-ordered sets is introduced. The main result inthis paper is that the category of fuzzy domains is dually equivalentto the category of strongly completely distributive $L$-orderedsets. The results in this paper establish close connections amongfuzzy-set approach of quantitative domains and fuzzy topology withmodified $L$-sober spaces and spatial $L$-frames as links. Inaddition, some mistakes in [K.R. Wagner, Liminf convergence in$Omega$-categories, Theoretical Computer Science 184 (1997)61--104] are pointed out.