N-Hypersoft Sets: An Innovative Extension of Hypersoft Sets and Their Applications

Abstract

This paper introduces N-hypersoft (N-HS) sets—an enriched and versatile extension of hypersoft (HS) sets—designed to handle evaluations involving both binary and non-binary data while embodying an inherent sense of structural symmetry. The paper presents several algebraic definitions, including incomplete N-HS sets, efficient N-HS sets, normalized N-HS sets, equivalence under normalization, N-HS complements, and HS sets derived from a threshold. These definitions are accompanied by illustrative examples. Additionally, the paper delves into various set-theoretic operations within the framework of N-HS sets, such as relative null/whole N-HS sets, N-HS subsets, and N-HS extended/restricted union and intersection, presented in two different ways. Finally, the paper presents and compares decision-making methodologies regarding N-HS sets.

A progressive approach to multi-criteria group decision-making: N-bipolar hypersoft topology perspective

“…Real-world challenges often entail non-binary and discrete data structures. Addressing this, Musa et al [ 48 ] introduced N -HS sets, presenting a more comprehensive framework than conventional HS sets. N -HS sets integrate a parametrized object characterization based on ordered grades, providing a versatile representation for intricate real-world scenarios.…”

“…Definition 3 . [ 48 ] Let ¥ denote a collection of alternatives , ξ be a set representing attributes, and . Let R be a set of ordered grades, specifically R = {0, 1, …, N − 1}, where N ∈ {2, 3, …}.…”

Binary Bipolar Soft Points and Topology on Binary Bipolar Soft Sets with Their Symmetric Properties

“…In forthcoming works, we will add some other concepts of binary bipolar soft topological properties such as binary bipolar soft continuous mappings, binary bipolar soft compactness, binary bipolar soft connectedness, and binary bipolar soft separation axioms in terms of binary bipolar soft open sets and binary bipolar soft points. We extend this work to define some types of symmetrical soft sets such as N-soft sets [27][28][29], N-hypersoft sets [30], N-bipolar soft sets [31], and fuzzy N-soft sets [17,[32][33][34]. Furthermore, we can introduce the concepts as defined in [35][36][37][38] by using the binary bipolar soft sets.…”

N-bipolar hypersoft sets: Enhancing decision-making algorithms

“…However, practical problems often involve non-binary and discrete data structures. In response to this, the concept of N -hypersoft ( N -HS) sets was introduced by Musa and Asaad [ 40 ] as a broader framework than HS sets. N -HS sets incorporate the notion of parameterized characterization of objects in the universe based on a finite number of ordered grades.…”