Equivalent finite fuzzy sets and Stirling numbers

In this paper we study the number of k-level equivalence classes of fuzzy subsets of a finite set of n elements under a natural equivalence. This number is related to Stirling numbers. Viewing fuzzy subsets as functions from a set into the unit interval, we also associate a kernel partition with every equivalence class of fuzzy subsets. After some elementary properties of the equivalence, we provide a recurrence relation and a generating function concerning the number of k-level fuzzy subsets using Stirling numbers.