On the uniqueness of $L$-fuzzy sets in the representation of families of   sets
  This paper deals with the uniqueness of $L$-fuzzy sets in the representation of a given family of subsets of nonempty set. It first shows a formula of the number of $L$-fuzzy sets whose collection of cuts coincides with a given family of subsets of a nonempty set, and then provides a necessary and sufficient condition under which such $L$-fuzzy sets are unique. 
Peng He, Xue-ping Wang
math.LO cs.LO