Multidimensional Fuzzy Sets
Extensions of fuzzy sets; Partial orders; Admissible orders; Multidimensional aggregation functions.
Since arising fuzzy set theory many extensions proposals have been given, one of them is the n-dimensional fuzzy set in which its elements are tuples of size n whose components are values in [0; 1], ordered in increasing form, called n-dimensional intervals. Generally, these sets are used to develop tools that aid in modeling situations involving decision-making where given a problem, each interval represents the opinions of a specialist on the criteria of some attribute for this problem. However, these sets are not able to deal with situations in which a particular expert can, for example, refrain from any decision-making criteria, and therefore, we would have in the same problem coexisting n-dimensional intervals with different values of n. Thus, we need a new fuzzy set extension in which its elements (intervals) can have any dimensions. In this work, we present the concept of multidimensional fuzzy sets as a generalization of the n-dimensional fuzzy sets in which the elements can have different dimensions. We also present a way to generate comparisons (ordinations) of these elements of different dimensions, discuss conditions under which these sets have lattice structure and
introduce the concepts of admissible orders and multidimensional aggregations on multidimensional
fuzzy sets.