This chapter gives some essential scopes to study some plithogenic algebraic structures. Here the notion of plithogenic subgroup has been introduced and explored. It has been shown that subgroups defined earlier in the crisp, fuzzy, intuitionistic fuzzy, as well as neutrosophic environments, can also be represented as plithogenic fuzzy subgroups. Furthermore, introducing function in plithogenic setting, some homomorphic characteristics of plithogenic fuzzy subgroup have been studied. Also, the notions of plithogenic intuitionistic fuzzy subgroup, plithogenic neutrosophic subgroup have been introduced and their homomorphic characteristics have been analyzed.
Top1. Introduction
Crisp set (CS) theory has certain drawbacks. It is quite insufficient in case of handling real-life problems. Fuzzy set (FS) theory (Zadeh, 1965) is more reliable in tackling such scenarios. Since the very beginning of FS theory, many researchers have carried out that perception in various realistic problems. But eventually some other set theories have emerged like intuitionistic fuzzy set (IFS) (Atanassov, 1986), neutrosophic set (NS) (Smarandache, 2005), Pythagorean FS (Yager, 2013), Plithogenic set (PS) (Smarandache, 2017), etc., which are capable of handling uncertainty better than FSs. As a result, these set theories are preferred by most of the researchers to solve different real-life problems in which uncertainty plays a crucial role. Actually, NS is a generalization of IFS, which is further a generalization of FS. Smarandache’s contributions towards the development of NS theory are remarkable. For instance, he has contributed in developing neutrosophic measure and probability (Smarandache, 2013), calculus (Smarandache & Khalid, 2015), psychology (Smarandache, 2018), etc. Also, NS theory has a vast area of applications. Furthermore, Smarandache has introduced the notion of PS (Smarandache, 2018) theory which is a generalization of CS, FS, IFS and NS theories. He has further generalized PS and developed the notions of refined PS, plithogenic multiset, plithogenic bipolar set, plithogenic tripolar set, plithogenic multipolar set, plithogenic complex set, etc. Furthermore, he has developed plithogenic logic, probability and statistics (Smarandache, 2000; Smarandache, 2017) and shown that all those notions are generalizations of crisp logic, probability and statistics. Presently, PS theory is extensively applied in various decision-making problems as well as in other applied fields. The following Table 1 consists of some important contributions in NS and PS theory.