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Sylow's theorems are fundamental theorems in classical group theory that are of paramount importance. The extension of these theorems into diverse fuzzy contexts emerges as a compelling area of exploration. This study introduces the novel concept of the conjunctive complex fuzzy conjugate element within the conjunctive complex fuzzy subgroup of a group, elucidating numerous crucial properties of this concept. Additionally, it propounds the notion of the conjunctive complex fuzzy p- subgroup within the conjunctive complex fuzzy subgroup (CCFSG) and delineates various indispensable characteristics associated with this construct. Additionally, the paper formulates the conjunctive complex fuzzy version of the Cauchy theorem for finite groups. Lastly, it defines the concept of the conjunctive complex fuzzy Sylow p -subgroup for a finite group and conducts a generalization of Sylow's theorems within a conjunctive complex fuzzy environment. [ABSTRACT FROM AUTHOR]

The application of a complex fuzzy logic system based on a linear conjunctive operator represents a significant advancement in the field of data analysis and modeling, particularly for studying physical scenarios with multiple options. This approach is highly effective in situations where the data involved is complex, imprecise and uncertain. The linear conjunctive operator is a key component of the fuzzy logic system used in this method. This operator allows for the combination of multiple input variables in a systematic way, generating a rule base that captures the behavior of the system being studied. The effectiveness of this method is particularly notable in the study of phenomena in the actual world that exhibit periodic behavior. The foremost aim of this paper is to contribute to the field of fuzzy algebra by introducing and exploring new concepts and their properties in the context of conjunctive complex fuzzy environment. In this paper, the conjunctive complex fuzzy order of an element belonging to a conjunctive complex fuzzy subgroup of a finite group is introduced. Several algebraic properties of this concept are established and a formula is developed to calculate the conjunctive complex fuzzy order of any of its powers in this study. Moreover, an important condition is investigated that determines the relationship between the membership values of any two elements and the membership value of the identity element in the conjunctive complex fuzzy subgroup of a group. In addition, the concepts of the conjunctive complex fuzzy order and index of a conjunctive complex fuzzy subgroup of a group are also presented in this article and their various fundamental algebraic attributes are explored structural. Finally, the conjunctive complex fuzzification of Lagrange's theorem for conjunctive complex fuzzy subgroups of a group is demonstrated. [ABSTRACT FROM AUTHOR]

Symmetry is a key part of the study of basic forces and particles, as well as the creation of mathematical models that help scientists in various scientific disciplines understand complex events. Scientists can figure out what a system is made of and how it works by looking at its symmetry. They can then use this information to make predictions and create new materials and technologies. Humanity has conquered many once-fatal diseases due to medical research and technological advancements. Although this progress is encouraging, there are still a great many areas that require continual human efforts. An effort is made in this article to choose the best treatment strategy to completely manage the pandemic of COVID-19 under conjunctive complex fuzzy knowledge. In this paper, the concept of conjunctive complex fuzzy relations is presented and numerous set theoretical aspects of this phenomenon are established. The investigation of this ideology is further expanded to describe different sorts of essential structural conjunctive complex fuzzy relations. Matrix and graphical representations of the formation of these newly specified relations are also provided. Moreover, this concept has been successfully employed to provide a therapy strategy for a rapid recovery from COVID-19. Furthermore, a comparative analysis is conducted to demonstrate the validity and applicability of the suggested approaches compared to existing methods. [ABSTRACT FROM AUTHOR]

Complex fuzzy sets are the novel extension of Zadeh's fuzzy sets. In this paper, we comprise the introduction to the concept of ξ -complex fuzzy sets and proofs of their various set theoretical properties. We define the notion of α , δ -cut sets of ξ -complex fuzzy sets and justify the representation of an ξ -complex fuzzy set as a union of nested intervals of these cut sets. We also apply this newly defined concept to a physical situation in which one may judge the performance of the participants in a given task. In addition, we innovate the phenomena of ξ -complex fuzzy subgroups and investigate some of their fundamental algebraic attributes. Moreover, we utilize this notion to define level subgroups of these groups and prove the necessary and sufficient condition under which an ξ -complex fuzzy set is ξ -complex fuzzy subgroup. Furthermore, we extend the idea of ξ -complex fuzzy normal subgroup to define the quotient group of a group G by this particular ξ -complex fuzzy normal subgroup and establish an isomorphism between this quotient group and a quotient group of G by a specific normal subgroup G A ξ . [ABSTRACT FROM AUTHOR]