A note on cut properties of semilattices valued fuzzy sets

Harina Monim, Sri Wahyuni, Indah E. Wijayanti

Semilattice valued fuzzy sets are investigated in the framework of cuts. A theorem of synthesis for such fuzzy sets is proved. Families of cuts for meet(join)-fuzzy sets are proved to be special semi-closure systems. Conversely, for every such semi-closure system, we prove existence of a semilattice and a semilattice-valued fuzzy set whose collection of cuts is the given semi-closure system. We also show that for an arbitrary collection of subsets of a nonempty set, there is a semilattice valued fuzzy set whose collection of cuts contains these subsets. Using meet-irreducible elements in a finite meet-semilattice, we give conditions under which all the cuts of a meet-semilattice-valued fuzzy set are different and we describe a representation of the semilattice by the collection of cuts ordered dually by inclusion.