The n-hyperSpherical Neutrosophic sets defined on some  algebraic structures

Marimuthu Dharmalingam; G. S. Mahapatra; I. Silambarasan; R. Vijaykrishnaraj; P. Rajakumari; R.  Selvakumar

Authors

Marimuthu Dharmalingam NIT,Puducherry,Karaikal-609609;
G. S. Mahapatra NIT,Puducherry,Karaikal-609609;
I. Silambarasan Department of Mathematics, Vel Tech Multi Tech Dr.Rangarajan Dr.Sakunthala Engineering College, Avadi-600062;
R. Vijaykrishnaraj Department of Mathematics, Bharath Institute of Higher Education and Research, Chennai, India;
P. Rajakumari Department of Mathematics, Jeppiaar Institute of Technology, Sunguvachatram, Chennai;
R. Selvakumar Department of Mathematics, Vel Tech Multi Tech Dr.Rangarajan Dr.Sakunthala Engineering College, Avadi-600062;

Keywords:

Intuitionistic fuzzy set; Pythagorean fuzzy set; Picture fuzzy set; Spherical fuzzy set; Spherical Neutrosophic set; Algebraic sum; Algebraic product; Scalar multiplication; Exponentiation operations.

Abstract

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.

DOI: 10.5281/zenodo.16887587