Implications Satisfying the Law of Importation with a Given Uninorm

Abstract

In this paper a characterization of all fuzzy implications with continuous e-natural negation that satisfy the law of importation with a given uninorm U is provided. The cases when the considered uninorm U is representable or a uninorm in umin are studied separately and detailed descriptions of those implications with continuous natural negation with respect to e that satisfy the law of importation with a uninorm in these classes are done. In the process some important examples are included. © Springer International Publishing Switzerland 2014.