A characterization of discrete uninorms having smooth underlying operators

In this paper discrete uninorms U such that their underlying t-norm T and t-conorm S are smooth are characterized. The different cases combining when T is the minimum or the Łukasiewicz t-norm and S is the maximum or the Łukasiewicz t-conorm, are given separately and the number of discrete uninorms with these underlying operators is given. The same is done for the general case when T and/or S are smooth, but ordinal sums.