Stratification structures on a kind of completely distributive lattices and their applications in the theory of topological molecular lattices (Q1125370)

Based on the stratifications of the family \(L^X\) of all \(L\)-fuzzy sets on \(X\), the authors introduce the concept of a stratification structure in a completely distributive lattice. The authors prove that the category of stratified completely distributive lattices and stratification-preserving homomorphisms is equivalent to a category whose objects are the families of all fuzzy \(L\)-fuzzy sets on \(X\), where \(L\) is an irreducible completely distributive lattice and \(X\) a non-empty set, and whose morphisms are bi-induced maps. As an application, the authors introduce a type of compactness having the character of stratifications, similar to Wang's fuzzy compactness [\textit{Wang Guojun}, J. Math. Anal. Appl. 94, 1-23 (1983; Zbl 0512.54006)], on topological molecular lattices [\textit{Wang Guojun}, Fuzzy Sets Syst. 47, 351-376 (1992; Zbl 0783.54032)].