Limit structures over completely distributive lattices

摘要

The notion of a limit molecular lattice―a completely distributive lattice with some kinds of convergence of ideals on it―is introduced. It is proved that the category of limit molecular lattices is a cartesian closed category containing the category of topological molecular lattices as a bireflective full subcategory and the category of limit spaces as a reflective and coreflective subcategory, respectively. Noteworthy is the distinctive order property of the convergence of ideals on the limit molecular lattice compared to that of limit spaces. The L-fuzzy version of fuzzy limit space is introduced, and it is proved that the category of L-fuzzy limit spaces is a nonfull coreflective subcategory of the category of limit molecular lattices.