New Negations on the Type-2 Membership Degrees

Hernandez et al. [9] established the axioms that an operation must fulfill in order to be a negation on a bounded poset (partially ordered set), and they also established in [14] the conditions that an operation must satisfy to be an aggregation operator on a bounded poset. In this work, we focus on the set of the membership degrees of the type-2 fuzzy sets, and therefore, the set M of functions from [0, 1] to [0, 1]. In this sense, the negations on M respect to each of the two partial orders defined in this set are presented for the first time. In addition, the authors show new negations on L (set of the normal and convex functions of M) that are different from the negations presented in [9] applying the Zadeh’s Extension Principle. In particular, negations on M and on L are obtained from aggregation operators and negations. As results to highlight, a characterization of the strong negations that leave the constant function 1 fixed is given, and a new family of strong negations on L is presented.