A Bounded Lattice of Normal and Non-Normal Secondary Membership Functions and an Application in CWW

To the best of our knowledge, almost all general type-2 (GT2) fuzzy inference systems (FISs) currently used in practice rely on discretizing the domains of the primary and secondary membership functions (MFs) and have type-2 (T2) inputs, antecedents, and - if applicable - consequents whose secondary MFs are convex and normal. Only a few recent studies are concerned with non-convex, normal secondary MFs and their applications in T2 FISs. Other authors have measured the similarity between GT2 fuzzy sets (FSs) with non-normal secondary MFs but have not used them in T2 FISs. In this paper, we clarify that the class of convex upper semicontinous FSs on [0], [1] yields a bounded lattice in the sense of Garrett Birkhoff. This result implies that the class of GT2 FSs whose secondary MFs are convex and upper semicontinuous, but not necessarily normal, constitutes a bounded lattice as well. We employ the latter in order to propose GT2 FISs having inputs, antecedents, and consequents that are GT2 FSs with not necessarily normal secondary MFs. We finish by presenting a simple, but novel application in computing with words (CWW).