New Negations on the Membership Functions of Type-2 Fuzzy Sets

Abstract

Type-2 fuzzy sets (T2FSs) were introduced by L. A. Zadeh in 1975 as an extension of type-1 fuzzy sets (T1FSs). In this extension, the degree to which an element belongs to a set is just a label of the linguistic variable “TRUTH,” which allows to represent reality in a more appropriate way. On the other hand, negations play an essential role within fuzzy sets theory. In fact, they are necessary in order to obtain, for example, complements of fuzzy sets, dual of a t-norm or a t-conorm, entropies, implications, as well as to study the possible contradictions appearing in a fuzzy system. However, meanwhile negations on [0, 1] (set of the membership degrees of a fuzzy set) have been deeply studied throughout the literature, the same has not happened with the negations on <inline-formula><tex-math notation= LaTeX >${\\bf M}$</tex-math></inline-formula> = <inline-formula><tex-math notation= LaTeX >$\\text{Map} \\;([\\text{0}, \\text{1}],[\\text{0}, \\text{1}])$</tex-math></inline-formula>, set of functions from [0, 1] to [0, 1] (and also set of the membership degrees of a T2FS), and so many aspects of the negations on <inline-formula><tex-math notation= LaTeX >${\\bf M}$</tex-math></inline-formula> have not yet been investigated. In a previous paper, the axioms that an operation must satisfy to be considered a negation or a strong negation on a bounded partially ordered set were established. A family of strong negations on <bold>L</bold> and set of normal and convex functions of <inline-formula><tex-math notation= LaTeX >${\\bf M}$</tex-math></inline-formula> were also presented. Moreover, let us note that the main characteristic of fuzzy systems is just the flexibility in order to be able to represent knowledge according to each situation, offering different models among which the expert can choose the one that best suits his/her criteria. Thus, it seems useful to find broad sets of negations in <inline-formula><tex-math notation= LaTeX >${\\bf M}$</tex-math></inline-formula> and in <bold>L</bold>, which, as far as we know, have not been done by other researchers. According to these ideas, in this paper, the authors first present new negations and strong negations on <bold>L</bold>, and then show, for the first time, some negations on <inline-formula><tex-math notation= LaTeX >${\\bf M}$</tex-math></inline-formula> with respect to each of the two partial orders defined in this set.