More on picture fuzzy sets and their properties

Abstract

In this paper, some basic properties of the set of a picture fuzzy set truth values D* are studied. Also, using an adequate order ≤ of D*, some picture fuzzy sets operations are introduced by meaning a punctual order (point by point). As well as the order of D* is used to show some characteristic sets of a picture fuzzy set, such as support, kernel, α-cut, strong α-cut and picture fuzzy line of degree α of a picture fuzzy set, where α ∈ D*, have been defined, some properties of them have been established and some decomposition theorems of picture fuzzy sets have been proved. Finally, some of Atanassov’s modal operators are extended to the picture fuzzy case.