Algebras of Fuzzy Sets in Logics based on Continuous Triangular Norms

Associated with any [0,1]-valued propositional logic with a complete algebraic semantics, one can consider algebras of families of fuzzy sets over a classical universe, endowed with the appropriate operations. For the three most important schematic extensions of Hájek’s Basic (Fuzzy) Logic, we investigate the existence and the structure of such algebras of fuzzy sets in the corresponding algebraic varieties. In the general case of Basic Logic itself, and in sharp contrast to the three aforementioned extensions, we show that there actually exist different, incomparable notions of algebras of fuzzy sets.