Monotonicity of SISO Fuzzy Relational Inference With an Implicative Rule Base

A fuzzy relational inference (FRI) mechanism is appraised based on the different desirable properties it possesses. Among these properties, monotonicity of an FRI has not received much attention. In this work, we investigate the monotonicity of a single-input single-output FRI with an implicative form of the rule base. In all the previous works that deal with monotonicity of an FRI with the implicative form of rule base, the employed fuzzy implications come from a residuated lattice. It can be noticed that this rich underlying structure plays a major role in proving the results. Further, we also modify the given monotone rule base. This work differs from the previous works in that: 1) the fuzzy implications employed in it do not come from any known residuated structure on [0, 1]; and 2) the original rule base is employed without any alteration. We determine conditions under which monotonicity of an FRI, where the rule base is modeled by a strict fuzzy implication, can be ensured without transforming the original rule base. Thus, the results in this work further augment the case for considering fuzzy implications, other than those from the residuated setting, to be used in applications.