Total orderings defined on the set of all fuzzy numbers

In this work, a new concept of upper dense sequence in interval (0,1](0,1] is introduced. There are infinitely many upper dense sequences in interval (0,1](0,1]. Using any upper dense sequence, a new decomposition theorem for fuzzy sets is established and proved. Then, using a chosen upper dense sequence as one of the necessary reference systems, infinitely many total orderings on the set of all fuzzy numbers can be well defined. Among them, a common upper dense sequence based on the binary numbers is suggested as a natural default option. Another upper dense sequence based on the rational numbers is also suggested. Regarding real numbers as special fuzzy numbers, all of these total orderings defined by using the suggested upper dense sequences are consistent with the natural ordering of real numbers. Building total ordering on the set of all fuzzy numbers in such a way is significant for fuzzy data analysis and, therefore, may be used in decision making with fuzzy information.