The n-hyperSpherical Neutrosophic sets defined on some algebraic structures

A generalization of spherical neutrosophic sets are n-Hyper-spherical neutrosophic sets. This article explores the desirable features of n-Hyper-spherical neutron elementary sets (n-HSNS) by defining algebraic sums and products of these sets. In addition, we construct exponentiation (An) and scalar multiplication (nA) operations for n-Hyper-spherical neutrosophic sets. Lastly, for n-Hyper-spherical neutron elementary sets, we construct a new operation (@) and explore the distributive laws when used in conjunction with ,⊠,⊓, and ⊔. Significant Statement: Significant methodological issues were identified and examined in the existing theories that were reviewed. Several advanced theories—such as neuroscopy theory, Pythagorean theory, spherical set theory, image set theory, and others—were introduced to tackle the core limitations of Atanassov’s intuitionistic fuzzy sets. However, the results derived from these newly proposed theories turned out to be considerably less effective than anticipated. Therefore, in this paper, we suggest an alternative method based on extending spherical sets within the framework of n-hyperspherical neutron elementary sets. In this context, we are investigating a crucial category of complex fuzzy algebraic systems.