On special fuzzy implications

Special implications were introduced by Hájek and Kohout [Fuzzy implications and generalized quantifiers, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 4 (1996) 225–233] in their investigations on some statistics on marginals. They have either suggested or only partially answered three important questions, especially related to special implications and residuals of t-norms. In this work we investigate these posers in-depth and give complete answers. Toward this end, firstly we show that many of the properties considered as part of the definition of special implications are redundant. Then, a geometric interpretation of the specialty property is given, using which many results and bounds for such implications are obtained. We have obtained a characterization of general binary operations whose residuals become special. Finally, some constructive procedures to obtain special fuzzy implications are proposed and methods of obtaining special implications from existing ones are given, showing that there are infinitely many fuzzy implications that are special but cannot be obtained as residuals of t-norms.