Multidimensional fuzzy sets: Negations and an algorithm for multi-attribute group decision making

Multidimensional fuzzy sets (MFS) is a new extension of fuzzy sets on which the membership values of an element in the universe of discourse are increasingly ordered vectors on the set of real numbers in the interval $[0,1]$. This paper aims to investigate fuzzy negations on the set of increasingly ordered vectors on $[0,1]$, i.e. on $L_{\infty }\left(\right[0,1\left]\right)$, MFN in short, with respect to some partial order. In this paper we study partial orders, giving special attention to admissible orders on $L_n\left(\right[0,1\left]\right)$ and $L_{\infty }\left(\right[0,1\left]\right)$. In addition, we study the possibility of existence of strong multidimensional fuzzy negations and some properties and methods to construct such operators. In particular, we define the ordinal sums of n-dimensional negations and ordinal sums of multidimensional fuzzy negations on a multidimensional product order. A multi-attribute group decision making algorithm is presented.