Fuzzy Groups: some group-theoretic analogs
Information Sciences: an International Journal
Fuzzy Sets and Systems
Measurement of membership functions and their acquisition
Fuzzy Sets and Systems - Special memorial volume on foundations of fuzzy reasoning
Fuzzy Sets and Systems
Certain fuzzy ideals of rings redefined
Fuzzy Sets and Systems
A generalization of the representation theorem
Fuzzy Sets and Systems
On a representation of posets by fuzzy sets
Fuzzy Sets and Systems
On an equivalence of fuzzy subgroups I
Fuzzy Sets and Systems
Simulation, Knowledge-Based Computing, and Fuzzy Statistics
Simulation, Knowledge-Based Computing, and Fuzzy Statistics
Completion of ordered structures by cuts of fuzzy sets: an overview
Fuzzy Sets and Systems - Logic and algebra
Representing ordered structures by fuzzy sets: an overview
Fuzzy Sets and Systems - Logic and algebra
On an equivalence of fuzzy subgroups II
Fuzzy Sets and Systems - Logic and algebra
Uniqueness results in the representation of families of sets by fuzzy sets
Fuzzy Sets and Systems
Some notes on equivalent fuzzy sets and fuzzy subgroups
Fuzzy Sets and Systems
Uniqueness in the generalized representation by fuzzy sets
Fuzzy Sets and Systems
On a non-nested level-based representation of fuzziness
Fuzzy Sets and Systems
International Journal of Approximate Reasoning
Fuzzy identities with application to fuzzy semigroups
Information Sciences: an International Journal
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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.