The law of importation for discrete implications

In this paper the law of importation is studied for implications defined on a finite chain L. We give a whole characterization of those S, R, QL and D-implications, derived from smooth discrete t-norms and t-conorms, that satisfy the law of importation with a discrete t-norm image. QL and D-implications derived from a special family of non-smooth t-conorms are also studied. Moreover, the study of the law of importation for these two last types of implications, in the framework of the unit interval [0,1], is completed from already known results on the topic.