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 duality between domains and completely distributive lattices. All values are taken in a fixed frame $L$. A definition of (strongly) completely distributive $L$-ordered sets is introduced. The main result in this paper is that the category of fuzzy domains is dually equivalent to the category of strongly completely distributive $L$-ordered sets. The results in this paper establish close connections among fuzzy-set approach of quantitative domains and fuzzy topology with modified $L$-sober spaces and spatial $L$-frames as links. In addition, some mistakes in [K.R. Wagner, Liminf convergence in $\Omega$-categories, Theoretical Computer Science 184 (1997) 61--104] are pointed out.