Fuzzy power structures

Power structures are obtained by lifting some mathematical structure (operations, relations, etc.) from an universe X to its power set P(X). A similar construction provides fuzzy power structures: operations and fuzzy relations on X are extended to operations and fuzzy relations on the set F(X) of fuzzy subsets of X. In this paper we study how this construction preserves some properties of fuzzy sets and fuzzy relations (similarity, congruence, etc.). We define the notions of good, very good, Hoare good and Smith good fuzzy relation and establish some connections between them, generalizing some results of Brink, Bosnjak and Madarasz on power structures.