Residual implications revisited. Notes on the Smets-Magrez theorem. (Q1885725)

This paper elaborates on the fuzzy extension of the binary implication. More specifically it refines the well-known result of \textit{P. Smets} and \textit{P. Magrez} [Int. J. Approx. Reasoning 1, 327--347 (1987; Zbl 0643.03018)] that states: a continuous \([0,1]^2\)-\([0,1]\) mapping \(I\), satisfying the exchange principle, the monotonicity with respect to the second variable and the equivalency \(x\leq y\Leftrightarrow I(x,g)= 1\), is nothing else than the \(\varphi\)-transform (with (\(\varphi\) an increasing permutation of \([0,1]\)) of the Łukasiewicz fuzzy implication. Already \textit{J. Fodor} and \textit{M. Roubens} proved [Fuzzy preference modelling and multicriteria decision support (Theory and Decision Library, Series D: Systems Theory, Knowledge Engineering and Problem Solving 14, Dordrecht: Kluwer) (1994; Zbl 0827.90002)] that the conditions required by Smets and Magrez [loc. cit.] are too strong. In this paper the author proves that the assumption of monotonicity can be reduced, more precisely he proves that a \([0,1]^2\)-\([0,1]\) operator \(I\) is decreasing with respect to its first variable as soon as \(I\) satisfies the exchange principle and the above equivalency. Finally, the author proves an analogue of the Smets-Magrez caracterization for the well-known Goguen fuzzy implication, i.e. the residual implication of the product triangular norm.