We define the Cartesian product, composition, union and join on interval-valued fuzzy graphs and investigate some of their properties. We also introduce the notion of interval-valued fuzzy complete graphs and present some properties of self complementary and self weak complementary interval-valued fuzzy complete graphs

Group decision‐making is a process wherein multiple individuals interact simultaneously, analyze problems, evaluate the possible available alternatives, characterized by multiple conflicting criteria, and choose suitable alternative solution to the problem. Technique for establishing order preference by similarity to the ideal solution (TOPSIS) is a well‐known method for multiple‐criteria decision‐making. The purpose of this study is to extend the TOPSIS method to solve multicriteria group decision‐making problems equipped with Pythagorean fuzzy data, in which the assessment information on feasible alternatives, provided by the experts, is presented as Pythagorean fuzzy decision matrices having each entry characterized by Pythagorean fuzzy numbers. A revised closeness index is utilized to obtain the ranking of alternatives and to identify the optimal alternative. The developed Pythagorean fuzzy TOPSIS (PF‐TOPSIS) is illustrated by a flow chart. At length, practical examples interpreting the applicability of our proposed PF‐TOPSIS are solved.

A characterization of an h-hemiregular hemiring in terms of a fuzzy h-ideal is provided. Some properties of prime fuzzy h-ideals of h-hemiregular hemirings are investigated. It is proved that a fuzzy subset ζ of a hemiring S is a prime fuzzy left (right) h-ideal of S if and only if ζ is two-valued, ζ(0) = 1, and the set of all x in S such that ζ(x) = 1 is a prime (left) right h-ideal of S. Finally, the similar properties for maximal fuzzy left (right) h-ideals of hemirings are considered.