Correct models of fuzzy IF–THEN rules are continuous

We show that a system of fuzzy IF–THEN rules being modeled correctly works as a partially given (fuzzy) function. Its behavior is determined by a chosen structure for IF–THEN rules which assigns meaning to fuzzy sets in the IF and THEN parts as well as to basic connectives. This partial function can be extended to a whole domain of fuzzy sets with the help of the Compositional Rule of Inference which plays a role of a mechanism for computing the dependent functional values. We investigate the question whether this function is continuous. Two new notions are introduced: a correct model of fuzzy IF–THEN rules in a structure and a continuous model of fuzzy IF–THEN rules with respect to given data. We prove that correctness of models of fuzzy IF–THEN rules is connected with the problem of solvability of the respective system of fuzzy relation equations. In this and only this case a model of fuzzy IF–THEN rules is continuous. As a side result we have established a new criterion of a solvability of a system of fuzzy relation equations with sup-*-composition.