Uninorm based residual implications satisfying the Modus Ponens property with respect to a uninorm.

In any fuzzy rules based system the inference management is usually carried out by the so-called fuzzy implication functions. In this framework, the Modus Ponens property becomes essential to make forward inferences and it is well known that this inference rule is guaranteed when the conjunction and the implication function used in the process satisfy the corresponding functional inequality. Such inequality has been extensively studied for many kinds of implication functions when the conjunction is modelled by a t-norm. However, the use of conjunctive uninorms to model conjunctions is an increasingly widespread option in fuzzy systems and for this reason the study of the Modus Ponens with respect to a conjunctive uninorm U instead of a t-norm T, which we call here U-Modus Ponens or U-conditionality, becomes very important. In this paper this new property is deeply analyzed and it is shown that usual implications derived from t-norms and t-conorms do not satisfy it, but many solutions appear among those implications derived from uninorms. In particular, the U-Modus Ponens for the case of residual implications derived from uninorms, or RU-implications, is investigated in detail when the uninorm U lies in any of the four most usual classes of uninorms.