RU and (U,N)-implications satisfying Modus Ponens

•Characterization of all residual implications for uninorms satisfying Modus Ponens.•Study of Modus Ponens for (U,N)(U,N)-implications.•Modus Ponens (T-conditionality) for implications generated from uninorms.

In this paper it is investigated when some kinds of fuzzy implication functions derived from uninorms satisfy the Modus Ponens with respect to a continuous t-norm T, or equivalently, when they are T-conditionals. The study is done for RU  -implications and (U,N)(U,N)-implications with N a continuous fuzzy negation leading to a lot of solutions in both cases. For RU-implications T  -conditionality only depends on the underlying t-norm of the uninorm used to derive the residual implication. On the contrary, for (U,N)(U,N)-implications the underlying t-norm is never relevant and only the region out of the t-norm is so. Even the t-conorm can be not relevant also in some cases.