Complete many-valued lattices

Let Sup be the category of complete lattices and join preserving maps. The aim of this paper is to show that the theory of complete many-valued lattices exists. For this purpose we recall the concept of many-valued preordered sets and show that the category of many-valued join-complete lattices is isomorphic to the category of right modules in Sup — a theorem which goes back to I. Stubbe in a more general context given by quantaloid enriched categories. Finally, we will explain the presented theory by some natural examples, and do not hesitate to apply the theory to a simple stochastic linear programming problem.