n-Dimensional Fuzzy Negations

n-Dimensional fuzzy sets is a fuzzy set extension 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 aims to investigate a special extension from [0, 1] – n-representable fuzzy negations on Ln([0, 1]), summarizing the class of such functions which 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 action of an n-dimensional automorphism on an n-dimensional fuzzy negation provides a method to obtain other ndimensional fuzzy negation, in which properties such as representability, continuity and monotonicity on Ln([0, 1]) are preserved.