n-Dimensional Fuzzy Negations

$n$ -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 $L_{n}([0,1])$ . This paper investigates a special extension from $[0,1]$ — $n$ -representable fuzzy negations on $L_{n}([0,1])$ , 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 $L_{n}([0,1])$ , 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 $L_{n}([0,1])$ are preserved. Finally, we provide a method for multiexpert decision-making problems based on ${\mathcal{N}}$ -reciprocal $n$ -dimensional fuzzy preference relations over a set of alternatives preserving the ${\mathcal{N}}$ -reciprocity for an $n$ -dimensional fuzzy negation ${\mathcal{N}}$ .