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

Document Type : Research Paper


1 Department of Mathematics, Hebei University of Science and Technology, Shijiazhuang 050018, P.R. China

2 Department of Mathematics, Shaanxi Normal University, Xi'an 710062, P.R. China


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.


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