On new contrapositivisation techniques for (interval-valued) fuzzy (co)implications and their generalizations - Oliveira, Fernando Neres de (2023)
In this work, we introduce several contrapositivisation operators for fuzzy implications, we present a wide study of each of these operators with respect to the main properties routinely required by fuzzy implications, we prove that the classes of these contrapositivisators are invariant by automorphisms and present some conditions for the -compatibility of the respective contrapositivisations, we propose some construction methods of classes of triangular norms (quasi-overlaps), triangular conorms (quasi-groupings) and aggregation functions from these contrapositivisators; we introduce the Min-Max contrapositivisation technique for fuzzy implications and some of its generalizations; we introduce four contrapositivisation operators for fuzzy coimplications so-called co-upper, co-lower, co-medium and co-(,)- contrapositivisators, we characterize these operators from the point of view of the properties that are usually attributed to the fuzzy coimplications, we present sufficient conditions for the -compatibility of the co-upper, co-lower, co-medium and co-(,)- contrapositivisations, we show that the classes of co-upper, co-lower, co-medium and co-(,)- contrapositivisators are closed under the action of automorphisms and we propose a construction method of triangular conorms from co-(,)- contrapositivisators of (,)-coimplications and fuzzy negations; and finally, we propose the classes of interval-valued upper, lower and medium contrapositivisators and broadly characterize each one of them, we show that both classes are invariant by interval-valued automorphisms, we introduce the notion of -compatibility for the respective interval-valued contrapositivisations and we prove that the best interval representations of real-valued upper, lower and medium contrapositivisators are, respectively, interval-valued upper, lower and medium contrapositivisators.