Convex combinations of fuzzy logical operations

One may ask when (non-trivial) convex combinations of fuzzy logical operations result in operations of the same type. This question has been intensively studied for triangular norms and it still remains open for continuous ones. An equivalent problem is obtained for triangular conorms. We show equivalence with the analogous problem for S-implications. Negative answers are obtained for strong fuzzy negations and for R-implications corresponding to continuous triangular norms. The situation for Q-implications and related problems are discussed.