A note on Nguyen–Fullér–Keresztfalvi theorem and Zadeh's extension principle

This paper is devoted to the analysis of the generalised form of the Nguyen–Fullér–Keresztfalvi theorem (NFK theorem). The classical NFK theorem expresses the Zadeh extension principle in terms of α-cuts of fuzzy sets, but it is subjected to some constraining assumptions. These assumptions concern all data in the problem: shape of fuzzy sets, topology of underlying spaces, and regularity of functions and t-norms. In this paper we analyse consequences of dropping these assumptions. In order to prove the generalised version of the NFK theorem, we introduce a notion of level sets as a generalisation of the collection of α-cuts. We discuss properties of the level sets and then we formulate the general NFK theorem, which does not require assumptions on the shape of fuzzy sets, t-norms, nor topology of underlying spaces. Finally, we return to the classical formulation of the NFK theorem and we show that it can be extended to the class of fuzzy sets with unbounded supports.