From fuzzy sets to the decompositions of non-rigid sets

The concept of identifying a fuzzy subset of a set R with its collection of level sets and its collection of membership values plays an important role in many real life applications. Such identification is possible only if the fuzzy set can be recaptured. This implies that uniqueness of the fuzzy set plays an important role. It has been recently shown that in order to have uniqueness it is necessary and sufficient that the collection S of membership values of the fuzzy set be a rigid set. With this in mind, the authors went on to show that if S has the min-max-property then S is a rigid set and left the converse as an open problem. We start this paper by constructing a counterexample that shows that rigid sets do not have to have the min-max-property. We then take a closer look at non-rigid sets by decomposing them into isomorphism-invariant components and into S-connected components in the hope of getting a characterization of non-rigid sets, which will lead to a characterization of rigid sets and hence of uniqueness of the fuzzy set. We also investigate, in the case of nonuniqueness, the number of fuzzy sets corresponding to a collection of level sets and a collection of membership values.