Uniqueness results in the representation of families of sets by fuzzy sets

The important concept of identifying a fuzzy subset of a set R with its corresponding chain of level sets is widely used in the literature. We investigate in this paper the problem of characterization of all fuzzy subsets of R that can be identified with a given arbitrary family C of subsets of R together with a given arbitrary subset S of [0,1]. Although our main focus is on the problem of uniqueness, we start by giving necessary and sufficient conditions for both existence and uniqueness of such fuzzy subsets. We continue by obtaining several algebraic and topological properties under which uniqueness is guaranteed. To further support our results, we present some examples which we hope will shed more light on the above mentioned problem of identification. We finish by raising a few natural questions that are left as open problems for further investigation.