Total ordering for all convex normal fuzzy subsets of the real line
Recently, a new decomposition theorem that expresses a fuzzy set in terms of countably many α-cuts is established. Based on the new decomposition theorem, total orderings on the set consisting of all convex normal fuzzy subsets of the real line can be defined by using the lexicography. Such total orderings are generalizations of the natural ordering of all real numbers. Restricting them on the set of all fuzzy numbers yields total orderings for fuzzy numbers. With any given equivalent relation, a ranking can be obtained from a total ordering.
 D. Dubois and H. Prade, Ranking fuzzy numbers in the setting of possibility theory, Information Sciences 30, 183-224, 1983.
 G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Thery and Applications, Prentice Hall, 1995.
 Z. Wang, R. Yang, and K. S. Leung, Nonlinear integrals and Their Applications in Data Mining, World Scientific, Singapore, 2010.
 Z. Wang and L. Zhang-Westman, The cardinality of the set of all fuzzy numbers, Proc. IFSA2013. 1045-1049.
By using the knowledge on fuzzy sets and fuzzy numbers, develop total orderings for all convex fuzzy sets and, therefore, for all fuzzy numbers. Furthermore, for any given total ordering, assigning an equivalence relation yields a ranking. Some examples should be used to illustrate these results. A research paper on this topic should be completed and be submitted to some international academic conference in time.