Type-2 implications on non-interactive fuzzy truth values

In this paper we consider algebras of fuzzy truth values equipped with an implication operation. Generalizing the definition of type-1 fuzzy implications, we discuss extended fuzzy S-implications and coimplications and also extended residual implications and coimplications, in particular to the extended residual of minimum. We study the algebraic properties of these operations and their relationships to extended t-norms and t-conorms in particular to meet and join. These investigations are intended to provide a theoretical background for type-2 approximate reasoning applications.