Picture Fuzzy Ordering and D∗-Based Lattices

In this paper, some fundamental concepts related to fuzzy relations, fuzzy lattices, intuitionistic fuzzy relations, and intuitionistic fuzzy lattices are extended to the picture fuzzy setting. Also, the structure of the set ????∗ of membership values of the picture fuzzy set that plays the role of a prototype for the picture fuzzy set was studied, and some of its basic properties were discussed. Furthermore, we have introduced the concepts of picture fuzzy filters in a crisp lattice, crisp filters in a picture fuzzy lattice, and picture fuzzy filters in a picture fuzzy lattice, and some of their properties, subtle differences, and extensions in terms of picture fuzzy sets are proved. As well as giving many characterizations of picture fuzzy filters in a picture fuzzy lattice. Finally, we present the necessary and sufficient requirements for a picture fuzzy subset to be a picture fuzzy prime filter.