Multidimensional fuzzy implications

Abstract

Multidimensional fuzzy sets (MFS) is a new extension of fuzzy sets on which the membership values of an element in the discourse universe are increasingly ordered vectors on the set of real numbers in the interval [0,1]. This paper aims to investigate fuzzy negations and, mainly, fuzzy implications on the set of increasingly ordered vectors on [0,1], i.e. on L∞([0,1]), MFN and MFI in short, with respect to some partial order. In this paper we study partial orders, giving special attention to admissible orders on L∞([0,1]). In addition, some properties and methods to construct such operators from fuzzy negations and fuzzy implications, respectively, are provided and we demonstrate that the action of the group of automorphisms on fuzzy implications on L∞([0,1]) preserves several original properties of the implication. Finally, through a specific type of representable MFI, we are able to generate a class of MFN called natural m-negations.