n-Dimensional Fuzzy Negations

TL;DR

This paper extends fuzzy negations to n-dimensional fuzzy sets, analyzing their properties, and introduces methods to generate new negations while preserving key characteristics like continuity and monotonicity.

Contribution

It characterizes n-representable fuzzy negations on n-dimensional intervals, preserving core properties and providing a method to generate new negations via automorphisms.

Findings

Characterization of continuous, monotone n-representable fuzzy negations.

Preservation of properties like continuity and monotonicity in n-dimensional extensions.

Method for generating new n-dimensional fuzzy negations through automorphisms.

Abstract

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 n-dimensional fuzzy negation, in which properties such as representability, continuity and monotonicity on Ln([0,1]) are preserved.