Interpolativity of at-least and at-most models of monotone single-input single-output fuzzy rule bases

Interpolativity is one of the most important properties of a fuzzy inference system. It is well known that normal antecedent fuzzy sets forming a Ruspini partition constitute a practical setting ensuring interpolativity. In case of a fuzzy rule base expressing a monotone relationship, another desirable property is the monotonicity of the resulting function (after defuzzification). Unfortunately, this goal may often only be reached through the application of the at-least and/or at-most modifiers to the antecedent and consequent fuzzy sets. However, this approach does not seem compatible with the practical setting of a Ruspini partition. This paper shows that the situation is less conflicting than it seems, and that interpolativity can still be guaranteed, in the same practical setting, and, interestingly, from two different modeling points of view. This paper addresses the case of single-input single-output fuzzy rules.