General form of lattice-valued fuzzy sets under the cutworthy approach

In this note a new solution of problem of Synthesis of fuzzy sets is presented. In other words, necessary and sufficient conditions are formulated, under which for a given family of subsets F of a set X and a fixed complete lattice L there is a fuzzy set μ:X→L, such that the collection of cuts of μ coincides with F. Moreover, it is proved that the general form of lattice-valued fuzzy sets (considering families of cuts) are the type of fuzzy sets having the codomain {0,1}c for a suitable chosen cardinal c.