Summary: Our aim is to generalize Lemma 2 and Lemma 3 of N. Nakajima
’s article “Generalized fyzzy sets” [Fuzzy Sets Syst. 32, 307–314 (1989; Zbl 0676.06017
)]. In this article, a construction of fuzzy sets without depending on a membership function, algebraic properties of a family of fuzzy sets, a ring of generalized fuzzy sets
, a complete Heyting algebra (cHa) which contains the power set
, an extension lattice
, and the set of
-fuzzy sets where
were proposed and shown that they are equivalent depending on Lemma 2 and Lemma 3. In his article Lemma 2, Lemma 3 were not shown in their generalized version. In our article we show that these lemmas can be generalized.