Double hierarchy linguistic term set (DHLTS) is a powerful tool when expressing the real thoughts of experts and handling complex linguistic information considering that it divides complex linguistic information into two simple linguistic hierarchies in which the first hierarchy linguistic term set (LTS) is the main linguistic hierarchy and the second hierarchy LTS is the linguistic feature or detailed supplementary of each linguistic term in the first hierarchy LTS. Some extensions of DHLTS have been developed, such as the double hierarchy hesitant fuzzy LTS, the unbalanced DHLTS, the linguistic preference ordering, the double hierarchy linguistic preference relation, and the double hierarchy hesitant fuzzy linguistic preference relation. In recent years, DHLTS and its extensions have been researched by scholars in lots of fields, including the extended concepts, the operational laws, the comparative methods, the measure methods, the consistency and consensus methods, the decision-making methods, the applications, and so forth. Therefore, the purpose of this paper is to review all the researches of DHLTS and its extensions, as well as proposing some challenges in the future.

In practical decision-making processes, we can utilize various types of fuzzy sets to express the uncertain and ambiguous information. However, we may encounter such the situations: the sum of the support (membership) degree and the against (nonmembership) degree to which an alternative satisfies a criterion provided by the decision maker may be bigger than 1 but their square sum is equal to or less than 1. The Pythagorean fuzzy sets (PFS), as the generalization of the fuzzy sets, can be used to effectively deal with this issue. Therefore, to enrich the theory of PFS, it is very necessary to investigate the fundamental properties of Pythagorean fuzzy information. In this paper, we first describe the change values of Pythagorean fuzzy numbers (PFNs), which are the basic components of PFSs, when considering them as variables. Then we divide all the change values into the eight regions by using the basic operations of PFNs. Finally, we develop several Pythagorean fuzzy functions and study their fundamental properties such as continuity, derivability, and differentiability in detail.