The law of importation versus the exchange principle on fuzzy implications

Some open problems on fuzzy implications dealing with the so-called importation law are studied and totally or partially solved in this work. A weaker version of the law of importation, called the weak law of importation, is introduced. The relationships of these two properties and the exchange principle are studied. In particular, it is proved that the law of importation is stronger than the exchange principle. On the other hand, the three properties are equivalent for some kind of fuzzy implications, those that satisfy a boundary property. Along this study, new characterizations of (S,N)-implications, R-implications and their counterparts for uninorms based on the weak law of importation are showed.