A Complementary Study on General Interval Type-2 Fuzzy Sets

Liang et al. in 2000 defined interval type-2 fuzzy sets (IT2FSs), which constitute a subset of type-2 fuzzy sets. While the membership degrees in the former are functions from [0, 1] to [0, 1] (fuzzy truth values), the membership degrees in IT2FSs only take their values in \lbrace {\text{0}},{\text{1}}\rbrace. Although all the initial work on IT2FSs involved convex membership degrees only, in 2015, Bustince et al. began the study on IT2FSs in general, including certain sets with nonconvex membership degrees. However, these are obviously early stages, with a lot of open problems regarding the theoretical structure of IT2FSs. For example, as far as we know, no negation operator has been obtained in this context. Therefore, it seems appropriate to continue with the study started in previous papers, delving deeper into the properties and operations of IT2FSs. Consequently, this work studies the structure of the set of functions from [0, 1] to \lbrace {\text{0}},{\text{1}}\rbrace (expanding the set considered by Bustince et al. ), from which we have removed the constant function \mathbf{0}, to offer a different study to the one carried out by Walker and Walker. More specifically, we consider join and meet operations, partial order derived from each one, and the negation operators in that set. Among other results, we provide new characterizations of join and meet operations and of partial orders on the set of functions from [0, 1] to \lbrace {\text{0}},{\text{1}} \rbrace ; we also present the first negation operators on this set.