$n$ -Dimensional Fuzzy Negations
Benjamín Bedregal, Ivan Mezzomo, Renata Reiser · IEEE Transactions on Fuzzy Systems · 2018
n-dimensional fuzzy sets theory is a broad research area extending fuzzy set theory where the membership values are n-tuples of real numbers in the unit interval [0, 1] orderly increased, called n-dimensional intervals. The set of n-dimensional intervals is denoted by Ln([0, 1]). This paper investigates a special extension from [0, 1]-n-representable fuzzy negations on Ln([0, 1]), summarizing the class of such functions that are continuous and monotone by part. The main properties of (strong) fuzzy negations on [0, 1] are preserved by representable (strong) fuzzy negation on Ln([0, 1]), mainly related to the analysis of degenerate elements and equilibrium points. The conjugate obtained by the action of an n-dimensional automorphism on an n-dimensional fuzzy negation provides a method to obtain another n-dimensional fuzzy negation, in which properties such as representability, continuity, and monotonicity on Ln([0, 1]) are preserved. Finally, we provide a method for multiexpert decisionmaking problems based on N-reciprocal n-dimensional fuzzy preference relations over a set of alternatives preserving the N reciprocity for an n-dimensional fuzzy negation N.