Representation by cuts in the framework of relational valued fuzzy sets

Generalized or relational-valued fuzzy sets are mappings from a set X to a relational system S = (S, p). Representation of collections of subsets by relational-valued fuzzy sets in the cutworthy framework is presented. It is proved that for every collection F of subsets of a set X there is a relational system S = (S, p) and a fuzzy set μ : X → S, such that the collection of cuts of μ coincides with F.