Stratification structures on a kind of completely distributive lattices and their applications in the theory of topological molecular lattices

Abstract

In this paper, we shall introduce the concept of stratification structures on completely distributive lattices by direct product decompositions of completely distributive lattices, and prove that there is, up to isomorphism, a unique stratification structure on any normal completely distributive lattice. Then we shall give the concept of stratified completely distributive lattices and prove that the category of stratified completely distributive lattices and stratification-preserving homomorphisms is equivalent to the category whose objects are completely distributive lattices of the form L^X, where L is an irreducible completely distributive lattice and L^X denotes the family of all L-fuzzy sets on a non-empty set X, and whose morphisms are bi-induced maps. As an application of these results, we shall give a definition of compactness which has the character of stratifications for a kind of topological molecular lattices.