module over a ring is a general mathematical concept for many examples of mathematicalobjects that can be added to each other and multiplied by scalar numbers.in this paper, we consider a module over a ring as a universe and by using the notion of reference points, we provide local approximations for  subsets of the universe.

In this dissertation, we study the picture fuzzy graph (PFG) in decision-making problem based on relation and some types of graphs such as complete graphs, strong complement are described also Pythagorean decision making environment. Further, present an application Fuzzy Graphs Picture

Picture fuzzy set (PFS) is more comprehensive tool than intuitionistic (IFS) for modeling the uncertain decision-making problems. In this paper, a new picture entropy measure proposed and proved that satisfies axiomatic definition of measures sets. Besides this, useful mathematical properties are also investigated. The justification established by discussing its particular cases compares it wit...

A set S of vertices in a graph G=(V,E) is a dominating set ofG if every vertex of V-S is adjacent to some vertex of S.For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from somevertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongestdominating set as well a...

recently, gasimov and yenilmez proposed an approach for solving two kinds of fuzzy linear programming (flp) problems. through the approach, each flp problem is first defuzzified into an equivalent crisp problem which is non-linear and even non-convex. then, the crisp problem is solved by the use of the modified subgradient method. in this paper we will have another look at the earlier defuzzifi...

In this paper, we focus on combining the theories of picture fuzzy sets rings and establishing a new framework for commutative rings. The aim manuscript is to apply set dealing with several kinds in Moreover, introduce notions ideals some properties them are obtained. Finally, give suitable definitions operations over ring, as composition, product intersection.