Residual implications revisited. Notes on the Smets–Magrez Theorem

It is well known that continuous residual implications are conjugate with the Imageukasiewicz implication. This fact was first obtained by Smets and Magrez (Internat. J. Approx. Reason. 1 (1987) 327). In this paper we show that the assumption of the monotonicity in this theorem can be omitted. We are also interested in the characterization of the other classes of fuzzy implications. As a result, the characterization of fuzzy implications, which are conjugate with the Goguen implication, is obtained. Besides these main results we present some new facts concerning the properties of fuzzy implications.