Interval representations, Łukasiewicz implicators and Smets–Magrez axioms

The Smets–Magrez axiomatic is usually used to define the class of fuzzy continuous implications which are both S and R-implications (Łukasiewicz implications). Another approach is the construction of such class starting from a basic implication and applying automorphisms. Literature has shown that there is a harmony between those approaches, however in this paper we show that the extension of the Łukasiewicz implication defined on [0, 1] for interval values cannot be applied in a direct way. We show that the harmony between the Smets–Magrez axiomatic approach and the one that comes from the generation by automorphisms is not preserved when such extension is done. One of the main consequences lies on the fact that the automorphism approach induces the loss of R-implications from the resulting class of implicators. More precisely, we show that the interval version of such approaches produce two disjunct classes of implicators, meaning that, unlike the usual case, the choice of the respective approach is an important step.