On fuzzy-valued operations and fuzzy-valued fuzzy sets

This paper is devoted to investigate fuzzy-valued operations and the lattice structures of the algebra of fuzzy values. The algebra of fuzzy values is proven to be a complete completely distributive lattice, and the list of all join-irreducible elements is provided. Fuzzy-valued t-norms and t-conorms are induced by arbitrary t-norms and t-conorms regardless of their continuity, respectively. Moreover, fuzzy-valued fuzzy implications are constructed from fuzzy-valued t-norms induced by left-continuous t-norms and residuated lattices are constructed on the algebra of fuzzy values. As a consequence, fuzzy-valued fuzzy rough sets and fuzzy-valued fuzzy complete lattices are proposed and investigated in fuzzy-valued fuzzy sets.