On the continuity of residuals of triangular norms (Q1049061)

R-implications are based on the functional \(I_T(x,y)=\sup\{t\in [0,1] \;| \;T(x,t) \leq y\}\), where \(T\) is a t-norm. The author of this paper presents a complete characterization of the class of continuous R-implications obtained from any arbitrary t-norm. In particular, it is shown that an R-implication \(I_T\) is continuous if and only if \(T\) is a nilpotent t-norm. Using this result, the exact intersection between two classes of implication functions, i.e., the continuous subsets of R-implications and (S, N)-implications, are determined by showing that the only continuous (S, N)-implication that is also an R-implication obtained from any t-norm, not necessarily left-continuous, is the Łukasiewicz implication, up to isomorphism.