The $n$-dimensional fuzzy sets and Zadeh fuzzy sets based on the finite valued fuzzy sets

Abstract

The connections among the $n$-dimensional fuzzy set, Zadeh fuzzy set and the finite-valued fuzzy set are established in this paper. The $n$-dimensional fuzzy set, a special $L$-fuzzy set, is first defined. It is pointed out that the $n$-dimensional fuzzy set is a generalization of the Zadeh fuzzy set, the interval-valued fuzzy set, the intuitionistic fuzzy set, the interval-valued intuitionistic fuzzy set and the three dimensional fuzzy set. Then, the definitions of cut set on $n$-dimensional fuzzy set and $n$-dimensional vector level cut set of Zadeh fuzzy set are presented. The cut set of the $n$-dimensional fuzzy set and $n$-dimensional vector level set of the Zadeh fuzzy set are both defined as $n+1$-valued fuzzy sets. It is shown that a cut set defined in this way has the same properties as a normal cut set of the Zadeh fuzzy set. Finally, by the use of these cut sets, decomposition and representation theorems of the $n$-dimensional fuzzy set and new decomposition and representation theorems of the Zadeh fuzzy set are constructed.