Uniqueness in the generalized representation by fuzzy sets

The concept of representation of a given family of sets by a fuzzy set in such a way that the level sets of the fuzzy set are precisely the given sets is widely used in the literature for the classification and study of fuzzy algebraic structures on the same underlying set. The problem of existence of such fuzzy sets was investigated by various authors. As far as we know, the problem of uniqueness of such fuzzy sets was not studied before. We establish, among other things, necessary and sufficient conditions under which uniqueness is guaranteed in the cases of complete as well as partial representations of the level sets of the fuzzy set.