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.

1. Introduction

Uncertain data play a significant role in numerous practical challenges across diverse fields like physical sciences, engineering technologies, environmental sciences, and social sciences. To address this issue, various mathematical theories have been developed. These include fuzzy set theory [1], rough set theory [2], evidence theory [3], intuitionistic fuzzy sets [4], and probabilistic theories. These theories provide researchers with effective tools to tackle different forms of uncertainties encountered in decision-making scenarios.

In 1999, the concept of soft sets [5] originated as a means to overcome certain limitations in previous models and is currently playing a crucial role in various domains, including decision-making [6,7,8,9], data analysis [10], topology [11], and medical diagnosis [12]. In terms of its theoretical advancement, Maji et al. [13] made significant contributions by introducing fundamental algebraic operations for soft sets. Their work laid the foundation for understanding the mathematical structure and properties of soft sets. Building upon this, Ali et al. [14] further expanded the study by introducing additional operations and exploring their implications. The authors [15,16,17] extended soft set theory by relaxing certain conditions on parameter sets and introducing generalizations of fundamental concepts.

From an alternative standpoint, there has been a burgeoning research focus on extending soft sets to accommodate uncertain environments. Zou and Xiao [10] were pioneers in investigating soft sets within an environment of incomplete information. Building upon their work, Feng et al. [18], as well as Ali [19], integrated soft sets, fuzzy sets, and rough sets. Moreover, Feng et al. [20] merged soft sets with rough sets and employed them in the context of multi-criteria group decision-making, as examined in [21]. Maji et al. [22] developed intuitionistic fuzzy soft sets. Jiang et al. [23] introduced interval-valued intuitionistic fuzzy soft sets. Ma et al. [24] conducted surveys on decision-making approaches related to hybrid soft set models.

Based on the recent surveys on hybrid soft set models, it is evident that numerous researchers have been drawn to the field of soft set theory and its hybrid models. These researchers have primarily focused on two types of evaluations within these models. The first type involves binary evaluations, also known as standard soft sets, where elements are either included or excluded from a set based on certain criteria. The second type involves evaluations using real numbers between 0 and 1, which are referred to as fuzzy soft sets. However, practical problems often involve non-binary and discrete data structures. Motivated by this, the notion of N-soft (N-S) sets was introduced by Fatimah et al. [25] as a broader concept than soft sets. N-S sets embrace the idea of parameterized description of objects within the universe, relying on a finite number of ordered grades. Various examples, presented by Fatimah et al. [25] and studied by Alcantud [26], demonstrate the practical applicability and the value of investigating this generalization. Researchers developed many innovative hybrid models like N-bipolar soft sets [27], hesitant N-S sets [28], picture fuzzy N-S sets [29], complex picture fuzzy N-S sets [30], and Pythagorean fuzzy N-S sets [31], providing evidence that N-S sets can effectively handle hybrid situations. Furthermore, algebraic structures [32] and topological structures [33] through N-S sets have been established.

In 2018, Smarandache [34] proposed the concept of HS sets as an enhancement to handle ambiguous and uncertain data in soft set-like models. HS sets utilize a multi-argument approximation mapping approach, making them more adaptable and trustworthy than traditional soft sets. The fundamentals, characteristics, and operations of HS sets have been explored by researchers such as Saeed et al. [35,36] and Abbas et al. [37]. Extensions of HS sets, including plithogenic HS sets [38], complex fuzzy HS sets [39], interval-valued intuitionistic fuzzy HS sets [40], and complex Q-rung orthopair fuzzy hypersoft sets [41] have been investigated, and practical applications have been discussed. Musa and Asaad developed the idea of bipolar HS sets, as documented in their work [42], and used their applications in decision-making [43] and topological concepts [44].

In brief, the motivation behind this article can be summarized as follows:

• The N-HS set, a novel area of research, aims to overcome the drawbacks of the N-S set in handling multiargument approximate functions. Such functions map subparametric tuples to the power set of a universe. The N-HS set focuses on partitioning parameters into distinct subparametric values through disjoint sets. These characteristics render it a fresh mathematical tool for effectively addressing uncertainty-related problems with an inherent sense of structural symmetry, ensuring a balanced representation of diverse subparametric values;
• The N-HS set introduces a parameterized representation of the universe, which differs from the binary nature of HS sets and the continuous nature of fuzzy HS sets. On the contrary, it relies on a finite level of granularity in perceiving parameters while maintaining a symmetrical approach to represent and categorize these parameters.

The sections in this document are organized in the following manner. Section 2 provides the essential conceptual framework concerning different categories of sets, including soft sets (Section 2.1), N-S sets (Section 2.2), and HS sets (Section 2.3), to familiarize the reader with the underlying principles. Moving on, in Section 3, our proposal of extended HS sets, namely, N-HS sets, is introduced, accompanied by an exploration of the associated definitions. Following that, Section 4 elucidates the aggregate operations applied to N-HS sets and delves into their respective properties. Section 5 outlines the decision-making procedures applicable to N-HS sets. In Section 6, the feasibility and adaptability of our algorithms are demonstrated by comparing their results using Examples 1, 6, and 7. Finally, Section 7 serves as the concluding section of our presentation.

2. Soft Sets, N-Soft Sets, and Hypersoft Sets

We begin our paper by exploring essential concepts associated with our research before delving into its discussion. Throughout this work, let $\Omega$ be a universal set of objects and $P(\Omega )$ be the power set of $\Omega$.

2.1. Soft Sets

Definition 1 

([5]). Let $\mathcal{E}$ be a set representing parameters and $\zeta \subseteq \mathcal{E}$. A soft set is characterized as a pair $(\nabla ,\zeta )$, where ∇ represents a mapping $\nabla :\zeta \to P(\Omega )$.

Definition 2 

([13]). A relative null soft set is described as $(\tilde{\Phi },\zeta )$, where ∀$ϵ\in \zeta$, $\tilde{\Phi }\left(ϵ\right)=\varnothing$.

Definition 3 

([13]). A relative whole soft set is described as $(\tilde{\Omega },\zeta )$, where ∀$ϵ\in \zeta$, $\tilde{\Omega }\left(ϵ\right)=\Omega$.

Definition 4 

([13]). The soft complement of $(\nabla ,\zeta )$ is denoted and defined as ${(\nabla ,\zeta )}^c$=$({\nabla }^c,\zeta )$, where ${\nabla }^c\left(ϵ\right)=\Omega \setminus \nabla \left(ϵ\right)$, ∀$ϵ\in \zeta$.

2.2. N-Soft Sets

Here, we examine crucial definitions pertaining to N-S sets, which have been sourced from reference [25].

Definition 5. 

Let Ω be a set representing a universe of objects, $\mathcal{E}$ be a set representing parameters, and $\zeta \subseteq \mathcal{E}$. Consider R as a set of ordered grades, specifically, $R=\{0,1,\cdots ,N-1\}$, where $N\in \{2,3,\cdots \}$. An N-S set can be described as a triple $(\nabla ,\zeta ,N)$, where $\nabla :\zeta \to P(\Omega \times R)$ and satisfies the following condition: for each $ϵ\in \zeta$, there is a unique $(\omega ,r_{ϵ})$∈$\Omega \times R$ such that $(\omega ,r_{ϵ})\in \nabla \left(ϵ\right)$ or, equivalently, $\nabla \left(ϵ\right)\left(\omega \right)=r_{ϵ}$.

Definition 6. 

Let Ω be a set representing a universe of objects, $\mathcal{E}$ be a set representing parameters, and $\zeta \subseteq \mathcal{E}$. Consider R as a set of ordered grades, specifically, $R=\{0,1,\cdots ,N-1\}$, where $N\in \{2,3,\cdots \}$. An incomplete N-S set is defined as a triple $(\nabla ,\zeta ,N)$, where $\nabla :\zeta \to P(\Omega \times R)$ and satisfies the following condition: for each $ϵ\in \zeta$, there is at most one $(\omega ,r_{ϵ})$∈$\Omega \times R$ such that $(\omega ,r_{ϵ})\in \nabla \left(ϵ\right)$ or, equivalently, $\nabla \left(ϵ\right)\left(\omega \right)=r_{ϵ}$.

Definition 7. 

An N-S set $(\nabla ,\zeta ,N)$ is considered efficient if ∃$ϵ\in \zeta$ and $\omega \in \Omega$ such that $\nabla \left(ϵ\right)\left(\omega \right)=N-1$.

Definition 8. 

The normalized N-S set $({\nabla }^o,P,N)$ of an N-S set $(\nabla ,\zeta ,N)$ is defined as follows: for every ${ϵ}_j\in \zeta$ and ${\omega }_i\in \Omega$, ${\nabla }^o\left({ϵ}_j\right)\left({\omega }_i\right)=\nabla \left({ϵ}_j\right)\left({\omega }_i\right)-m$, where $m=min\nabla \left({ϵ}_j\right)\left({\omega }_i\right)$ and $P=\{1,2,\cdots ,p\}$ denotes the set of indices for parameters.

Definition 9. 

Two N-S sets $(\nabla ,\zeta ,N_1)$ and $({\nabla }_1,{\zeta }_1,N_2)$ are considered N-S equal if $\nabla ={\nabla }_1$, $\zeta ={\zeta }_1$, and $N_1=N_2$.

Definition 10. 

Two N-S sets $(\nabla ,\zeta ,N)$ and $({\nabla }_1,{\zeta }_1,N)$ are equivalent if their normalized N-S sets are equal, that is, if $({\nabla }^o,P,N)$=$({\nabla }_1^o,P_1,N)$.

Definition 11. 

An N-S weak complement of $(\nabla ,\zeta ,N)$ is any N-S set ${(\nabla ,\zeta ,N)}^{\varpi }=({\nabla }^{\varpi },\zeta ,N)$ where ${\nabla }^{\varpi }\left(ϵ\right)\left(\omega \right)\cap \nabla \left(ϵ\right)\left(\omega \right)=\varnothing$ for each $ϵ\in \zeta$ and $\omega \in \Omega$.

Definition 12. 

An N-S top weak complement of $(\nabla ,\zeta ,N)$ is ${(\nabla ,\zeta ,N)}^t$=$({\nabla }^t,\zeta ,N)$, where

$${\nabla }^t\left(ϵ\right)\left(\omega \right)=\left\{\begin{array}{cc}N-1,\hfill & if\nabla \left(ϵ\right)\left(\omega \right)<N-1\hfill \\ 0,\hfill & if\nabla \left(ϵ\right)\left(\omega \right)=N-1\hfill \end{array}\right.$$

for each $ϵ\in \zeta$ and $\omega \in \Omega$.

Definition 13. 

An N-S bottom weak complement of $(\nabla ,\zeta ,N)$ is ${(\nabla ,\zeta ,N)}^b$=$({\nabla }^b,\zeta ,N)$, where

$${\nabla }^b\left(ϵ\right)\left(\omega \right)=\left\{\begin{array}{cc}0,\hfill & if\nabla \left(ϵ\right)\left(\omega \right)>0\hfill \\ N-1,\hfill & if\nabla \left(ϵ\right)\left(\omega \right)=0\hfill \end{array}\right.$$

for each $ϵ\in \zeta$ and $\omega \in \Omega$.

Definition 14. 

For a given threshold $0<T<N$ and an N-S set $(\nabla ,\zeta ,N)$, the associated soft set is denoted as $({\nabla }^T,\zeta )$ and is defined as follows:

$${\nabla }^T\left(ϵ\right)\left(\omega \right)=\left\{\begin{array}{cc}1,\hfill & if \nabla \left(ϵ\right)\left(\omega \right)\ge T\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right..$$

Definition 15. 

An N-S set $(\nabla ,\zeta ,N)$ is considered an N-S subset of $({\nabla }_1,{\zeta }_1,N)$, denoted as $(\nabla ,\zeta ,N)\widetilde{⊑}({\nabla }_1,{\zeta }_1,N)$, if

1. 

$\zeta \subseteq {\zeta }_1$;

2. 

$\nabla \left(ϵ\right)\left(\omega \right)\le {\nabla }_1\left(ϵ\right)\left(\omega \right)$, ∀$ϵ\in \zeta$ and $\omega \in \Omega$.

Definition 16. 

An N-S extended union of $(\nabla ,\zeta ,N_1)$ and $({\nabla }_1,{\zeta }_1,N_2)$ is denoted and defined as $(\nabla ,\zeta ,N_1)$${\bigsqcup }_{\epsilon }$$({\nabla }_1,{\zeta }_1,N_2)$=$({\nabla }_2,\zeta \cup {\zeta }_1,\max (N_1,N_2))$, where ∀$ϵ\in \zeta \cup {\zeta }_1$ and ω∈Ω:

$${\nabla }_2\left(ϵ\right)\left(\omega \right)=\left\{\begin{array}{cc}\nabla \left(ϵ\right)\left(\omega \right)\hfill & ifϵ\in \zeta \setminus {\zeta }_1\hfill \\ {\nabla }_1\left(ϵ\right)\left(\omega \right)\hfill & ifϵ\in {\zeta }_1\setminus \zeta \hfill \\ max\{\nabla \left(ϵ\right)\left(\omega \right),{\nabla }_1\left(ϵ\right)\left(\omega \right)\}\hfill & ifϵ\in \zeta \cap {\zeta }_1\hfill \end{array}\right..$$

Definition 17. 

An N-S extended intersection of $(\nabla ,\zeta ,N_1)$ and $({\nabla }_1,{\zeta }_1,N_2)$ is denoted and defined as $(\nabla ,\zeta ,N_1)$${\sqcap }_{\epsilon }$$({\nabla }_1,{\zeta }_1,N_2)$=$({\nabla }_2,\zeta \cup {\zeta }_1,\max (N_1,N_2))$, where ∀$ϵ\in \zeta \cup {\zeta }_1$ and ω∈Ω:

$${\nabla }_2\left(ϵ\right)\left(\omega \right)=\left\{\begin{array}{cc}\nabla \left(ϵ\right)\left(\omega \right)\hfill & ifϵ\in \zeta \setminus {\zeta }_1\hfill \\ {\nabla }_1\left(ϵ\right)\left(\omega \right)\hfill & ifϵ\in {\zeta }_1\setminus \zeta \hfill \\ min\{\nabla \left(ϵ\right)\left(\omega \right),{\nabla }_1\left(ϵ\right)\left(\omega \right)\}\hfill & ifϵ\in \zeta \cap {\zeta }_1\hfill \end{array}\right..$$

Definition 18. 

An N-S restricted union of $(\nabla ,\zeta ,N_1)$ and $({\nabla }_1,{\zeta }_1,N_2)$ is denoted and defined as $(\nabla ,\zeta ,N_1)$${\bigsqcup }_{\Re }$$({\nabla }_1,{\zeta }_1,N_2)$=$({\nabla }_2,\zeta \cap {\zeta }_1,\max (N_1,N_2))$, where ∀$ϵ\in \zeta \cap {\zeta }_1$ and ω∈Ω: ${\nabla }_2\left(ϵ\right)\left(\omega \right)=max\{\nabla \left(ϵ\right)\left(\omega \right),{\nabla }_1\left(ϵ\right)\left(\omega \right)\}$.

Definition 19. 

An N-S restricted intersection of $(\nabla ,\zeta ,N_1)$ and $({\nabla }_1,{\zeta }_1,N_2)$ is denoted and defined as $(\nabla ,\zeta ,N_1)$${\sqcap }_{\Re }$$({\nabla }_1,{\zeta }_1,N_2)$=$({\nabla }_2,\zeta \cap {\zeta }_1,\max (N_1,N_2))$, where ∀$ϵ\in \zeta \cap {\zeta }_1$ and ω∈Ω: ${\nabla }_2\left(ϵ\right)\left(\omega \right)=min\{\nabla \left(ϵ\right)\left(\omega \right),{\nabla }_1\left(ϵ\right)\left(\omega \right)\}$.

2.3. Hypersoft Sets

In this subsection, we present essential definitions of HS sets that are integral to our research. From now on, let $E_1,E_2,\cdots ,E_n$ be pairwise disjoint sets of parameters. Let ${\tilde{F}}_i,{\tilde{G}}_i\subseteq E_i$ for $i=1,2,\cdots ,n$.

Definition 20 

([34]). A pair $(\nabla ,{\tilde{F}}_1\times {\tilde{F}}_2\times \dots \times {\tilde{F}}_n)$ is referred to as an HS set, where $\nabla :{\tilde{F}}_1\times {\tilde{F}}_2\times \dots \times {\tilde{F}}_n\to P(\Omega )$.

For simplicity, we will use the symbol E to represent $E_1\times E_2\times \dots \times E_n$, ${\xi }_1$ to represent ${\tilde{F}}_1\times {\tilde{F}}_2\times \dots \times {\tilde{F}}_n$, and ${\xi }_2$ to represent ${\tilde{G}}_1\times {\tilde{G}}_2\times \dots \times {\tilde{G}}_n$, where ${\xi }_1,{\xi }_2\subseteq E$. It is important to note that each element in ${\xi }_1$, ${\xi }_2$, and E is an N-tuple element.

Definition 21 

([35,36]). Let $(\nabla ,{\xi }_1)$ and $({\nabla }_1,{\xi }_2)$ be two HS sets. Then,

1. 

$(\nabla ,{\xi }_1)$ is an HS subset of $({\nabla }_1,{\xi }_2)$, denoted by $(\nabla ,{\xi }_1)$${⊑}_s$$({\nabla }_1,{\xi }_2)$, if ${\xi }_1\subseteq {\xi }_2$ and $\nabla \left(q\right)\subseteq {\nabla }_1\left(q\right)$, ∀$q\in {\xi }_1$;

2. 

$(\nabla ,{\xi }_1)$ and $({\nabla }_1,{\xi }_2)$ are HS equal, if $(\nabla ,{\xi }_1)$${⊑}_s$$({\nabla }_1,{\xi }_2)$ and $({\nabla }_1,{\xi }_2)$${⊑}_s$$(\nabla ,{\xi }_1)$;

3. 

If $\nabla \left(q\right)=\varnothing$, ∀$q\in {\xi }_1$, then $(\nabla ,{\xi }_1)$ is called a relative null HS set and is denoted by $(\tilde{\Phi },{\xi }_1)$;

4. 

If $\nabla \left(q\right)=\Omega$, ∀$q\in {\xi }_1$, then $(\nabla ,{\xi }_1)$ is called a relative whole HS set and is denoted by $(\tilde{\Omega },{\xi }_1)$;

5. 

The HS complement of $(\nabla ,{\xi }_1)$ is an HS set $(\acute{\nabla },{\xi }_1)$, where $\acute{\nabla }\left(q\right)=\Omega \setminus \nabla \left(q\right)$, ∀$q\in {\xi }_1$;

6. 

The HS extended union of $(\nabla ,{\xi }_1)$ and $({\nabla }_1,{\xi }_2)$, denoted by $(\nabla ,{\xi }_1)$${\bigsqcup }_e$$({\nabla }_1,{\xi }_2)$, is an HS set $({\nabla }_2,{\xi }_3)$, where ${\xi }_3={\xi }_1\cup {\xi }_2$ and ∀$q\in {\xi }_3$:

$${\nabla }_2\left(q\right)=\left\{\begin{array}{cc}\nabla \left(q\right)\hfill & ifq\in {\xi }_1\setminus {\xi }_2\hfill \\ {\nabla }_1\left(q\right)\hfill & ifq\in {\xi }_2\setminus {\xi }_1\hfill \\ \nabla \left(q\right)\cup {\nabla }_1\left(q\right)\hfill & ifq\in {\xi }_1\cap {\xi }_2\hfill \end{array}\right.;$$

7. 

The HS extended intersection of $(\nabla ,{\xi }_1)$ and $({\nabla }_1,{\xi }_2)$, denoted by $(\nabla ,{\xi }_1)$${\sqcap }_e$$({\nabla }_1,{\xi }_2)$, is an HS set $({\nabla }_2,{\xi }_3)$, where ${\xi }_3={\xi }_1\cup {\xi }_2$ and ∀$q\in {\xi }_3$:

$${\nabla }_2\left(q\right)=\left\{\begin{array}{cc}\nabla \left(q\right)\hfill & ifq\in {\xi }_1\setminus {\xi }_2\hfill \\ {\nabla }_1\left(q\right)\hfill & ifq\in {\xi }_2\setminus {\xi }_1\hfill \\ \nabla \left(q\right)\cap {\nabla }_1\left(q\right)\hfill & ifq\in {\xi }_1\cap {\xi }_2\hfill \end{array}\right.;$$

8. 

The HS restricted union of $(\nabla ,{\xi }_1)$ and $({\nabla }_1,{\xi }_2)$, denoted by $(\nabla ,{\xi }_1)$${\bigsqcup }_r$$({\nabla }_1,{\xi }_2)$, is an HS set $({\nabla }_2,{\xi }_3)$, where ${\xi }_3={\xi }_1\cap {\xi }_2$ and ∀$q\in {\xi }_3$: ${\nabla }_2\left(q\right)=\nabla \left(q\right)\cup {\nabla }_1\left(q\right)$;

9. 

The HS restricted intersection of $(\nabla ,{\xi }_1)$ and $({\nabla }_1,{\xi }_2)$, denoted by $(\nabla ,{\xi }_1)$${\sqcap }_r$$({\nabla }_1,{\xi }_2)$, is an HS set $({\nabla }_2,{\xi }_3)$, where ${\xi }_3={\xi }_1\cap {\xi }_2$ and ∀$q\in {\xi }_3$: ${\nabla }_2\left(q\right)=\nabla \left(q\right)\cap {\nabla }_1\left(q\right)$.

3. N-Hypersoft Sets

In this section, we introduce the concept of N-HS sets and provide several associated definitions.

Definition 22. 

Let Ω be a set representing a universe of objects, E be a set representing parameters, and ${\xi }_1\subseteq E$. We define R as a set of ordered grades, specifically, $R=\{0,1,\cdots ,N-1\}$, where $N\in \{2,3,\cdots \}$. An N-HS set can be described as a triple $(\nabla ,{\xi }_1,N)$, where $\nabla :{\xi }_1\to P(\Omega \times R)$ and satisfies the following condition: for each $q\in {\xi }_1$, there is a unique $(\omega ,r_q)$∈$\Omega \times R$ such that $(\omega ,r_q)\in \nabla \left(q\right)$.

Given each attribute q, it is ensured that every object $\omega \in \Omega$ receives a unique evaluation from the assessment space R, specifically denoted by $r_q$, where $(\omega ,r_q)\in \nabla \left(q\right)$. To simplify our notation and align it with the HS set case, we can use the shorthand $\nabla \left(q\right)\left(\omega \right)=r_q$ to represent $(\omega ,r_q)\in \nabla \left(q\right)$. From now on, unless stated otherwise, we assume that both $\Omega =\{{\omega }_i,i=1,2,\cdots ,m\}$ and ${\xi }_1=\{q_j,j=1,2,\cdots ,n\}$ are finite. In this case, the N-HS set can also be presented in tabular form, where $r_{ij}$ represents $({\omega }_i,r_{ij})\in \nabla \left(q_j\right)$ or $\nabla \left(q_j\right)\left({\omega }_i\right)=r_{ij}$. This tabular representation is illustrated in

Table 1. Tabular representation of an N-HS set $(\nabla ,{\xi }_1,N)$.

$(\mathbf{\nabla },{\mathit{\xi }}_1,\mathit{N})$	${\mathit{q}}_1$	${\mathit{q}}_2$	...	${\mathit{q}}_{\mathit{n}}$
${\omega }_1$	$r_{11}$	$r_{12}$	...	$r_{1n}$
${\omega }_2$	$r_{21}$	$r_{22}$	...	$r_{2n}$
...	...	...	...	...
${\omega }_m$	$r_{m1}$	$r_{m2}$	...	$r_{mn}$

.

The N-HS set $(\nabla ,{\xi }_1,N)$ can be presented as follows:

$$(\nabla ,{\xi }_1,N)=\{(q,\{\omega ,\nabla \left(q\right)\left(\omega \right)\}):q\in {\xi }_1,\omega \in \Omega ,\mathrm{and}\nabla \left(q\right)\left(\omega \right)\in R\}.$$

Now, we provide an actual example that illustrates Definition 22 and extends beyond the conventional HS set model.

Example 1. 

Suppose a restaurant is looking to hire a chef, and there are five candidates who have applied for the position. Let the set of candidates be denoted as $\Omega =\{{\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4,{\omega }_5\}$. The required skill set for this role includes Cooking Skills $={\tilde{F}}_1= \{e_1=$ culinary expertise, e₂ = food presentation, e₃ = menu planning}, Management Abilities $={\tilde{F}}_2=${e₄ = kitchenmanagement, e₅ =time management}, and Creative Aptitude $={\tilde{F}}_3=\{e_6=$creativity}. Combining these skills, we have the set  ${\xi }_1={\tilde{F}}_1\times {\tilde{F}}_2\times {\tilde{F}}_3=\{q_1=(e_1,e_4,e_6),q_2=(e_1,e_5,e_6),\dots ,q_6=(e_3,e_5,e_6)\}$. Let R =$\{0,1,2,3,4,5\}$ be a set of ordered grades where $0=NoviceLevel$, $1=BeginnerLevel$, $2=IntermediateLevel$, $3=AdvancedLevel$, $4=ExpertLevel$, and $5=MasterLevel$. We can use a 6-HS set to represent the assessment values of each candidate for their respective skills:

$$\begin{array}{c}(\nabla ,{\xi }_1,6)=\{(q_1,\{{\omega }_1,4\},\{{\omega }_2,2\},\{{\omega }_3,0\},\{{\omega }_4,0\},\{{\omega }_5,4\}),(q_2,\{{\omega }_1,5\},\{{\omega }_2,4\},\{{\omega }_3,1\},\\ \{{\omega }_4,3\},\{{\omega }_5,5\}),(q_3,\{{\omega }_1,3\},\{{\omega }_2,1\},\{{\omega }_3,3\},\{{\omega }_4,2\},\{{\omega }_5,0\}),(q_4,\{{\omega }_1,5\},\{{\omega }_2,3\},\{{\omega }_3,0\},\\ \{{\omega }_4,5\},\{{\omega }_5,1\}),(q_5,\{{\omega }_1,4\},\{{\omega }_2,4\},\{{\omega }_3,4\},\{{\omega }_4,5\},\{{\omega }_5,4\}),(q_6,\{{\omega }_1,3\},\{{\omega }_2,5\},\{{\omega }_3,3\},\\ \{{\omega }_4,4\},\{{\omega }_5,4\}\left)\right\}\end{array}$$

We can present this information in the form of

Table 2. $(\nabla ,{\xi }_1,6)$: The Candidate Skills Assessment.

$(\mathbf{\nabla },{\mathit{\xi }}_1,6)$	${\mathit{q}}_1$	${\mathit{q}}_2$	${\mathit{q}}_3$	${\mathit{q}}_4$	${\mathit{q}}_5$	${\mathit{q}}_6$
${\omega }_1$	4	5	3	5	4	3
${\omega }_2$	2	4	1	3	4	5
${\omega }_3$	0	1	3	0	4	3
${\omega }_4$	0	3	2	5	5	4
${\omega }_5$	4	5	0	1	4	4

.

Remark 1. 

We can naturally associate a 2-HS set $(\nabla ,{\xi }_1,2)$ with an HS set $(\nabla ,{\xi }_1)$. Formally, we identify the 2-HS set $(\nabla ,{\xi }_1,2)$, where $\nabla :{\xi }_1\to P(\Omega \times \{0,1\})$, with the HS set $(\nabla ,{\xi }_1)$, where $\nabla :{\xi }_1\to P(\Omega )$, defined as $\nabla \left(q\right)$ = $\{\omega \in \Omega :\nabla (q\left)\right(\omega )=1\}$.

This concept can be exemplified by using the next example.

Example 2. 

Let $\Omega =\{{\omega }_1,{\omega }_2,{\omega }_3\}$. Let ${\tilde{F}}_1=\{e_1,e_2\}$, ${\tilde{F}}_2=\left\{e_3\right\}$, and ${\tilde{F}}_3=\left\{e_4\right\}$; then, ${\xi }_1={\tilde{F}}_1\times {\tilde{F}}_2\times {\tilde{F}}_3=\{q_1=(e_1,e_3,e_4),q_2=(e_2,e_3,e_4)\}$. Consider the 2-HS set defined as

$$(\nabla ,{\xi }_1,2)=\{(q_1,\{{\omega }_1,1\},\{{\omega }_2,0\},\{{\omega }_3,1\}),(q_2,\{{\omega }_1,0\},\{{\omega }_2,0\},\{{\omega }_3,1\})\}.$$

Now, we can identify this 2-HS set with the HS set $(\nabla ,{\xi }_1)$, which is defined as:

$$(\nabla ,{\xi }_1)=\{(q_1,\{{\omega }_1,{\omega }_3\}),(q_2,\left\{{\omega }_3\right\}\}.$$

Remark 2. 

In Definition 22, the presence of the grade 0 ∈R does not indicate incomplete or no information. Rather, it represents the lowest grade in the ordered set of grades.

Based on this motivation, we propose the following definition.

Definition 23. 

Let Ω be a set representing a universe of objects, E be a set representing parameters, and ${\xi }_1\subseteq E$. Consider R as a set of ordered grades, specifically, $R=\{0,1,\cdots ,N-1\}$, where $N\in \{2,3,\cdots \}$. An incomplete N-HS set is defined as a triple $(\nabla ,{\xi }_1,N)$, where $\nabla :{\xi }_1\to P(\Omega \times R)$ and satisfies the following condition: for each $q\in {\xi }_1$, ∃ at most one $(\omega ,r_q)$∈$\Omega \times R$ such that $(\omega ,r_q)\in \nabla \left(q\right)$.

Remark 3. 

It is worth noting that any N-HS set can be naturally regarded as an $(N+1)$-HS set, or, more generally, as an M-HS set, with $M>N$ chosen arbitrarily.

The motivation behind introducing the latter notion is to accommodate situations where the top grades are present but remain unused. Therefore, we define the following concept.

Definition 24. 

An N-HS set $(\nabla ,{\xi }_1,N)$ is considered efficient if ∃$q\in {\xi }_1$ and $\omega \in \Omega$ such that $\nabla \left(q\right)\left(\omega \right)=N-1$.

We now provide a formal definition for the concept of the bottom grade.

Definition 25. 

The normalized N-HS set $({\nabla }^{\tilde{o}},P,N)$ of an N-HS set $(\nabla ,{\xi }_1,N)$ is defined as follows: ∀$q_j$∈${\xi }_1$, ${\omega }_i$∈Ω, ${\nabla }^{\tilde{o}}\left(q_j\right)\left({\omega }_i\right)$ = $\nabla \left(q_j\right)\left({\omega }_i\right)-m$, where m = $\min \nabla \left(q_j\right)\left({\omega }_i\right)$ and P = $\{1,2,\cdots ,p\}$ denotes the set of indices for parameters.

Definition 26. 

Two N-HS sets $(\nabla ,{\xi }_1,N_1)$ and $({\nabla }_1,{\xi }_2,N_2)$ are called N-HS equal if $\nabla ={\nabla }_1$, ${\xi }_1={\xi }_2$, and $N_1=N_2$.

Definition 27. 

Two N-HS sets $(\nabla ,{\xi }_1,N)$ and $({\nabla }_1,{\xi }_2,N)$ are said to be equivalent if their normalized N-HS sets are equal, that is, if $({\nabla }^{\tilde{o}},P,N)$=$({\nabla }_1^{\tilde{o}},P_1,N)$.

Example 3. 

Let $\Omega =\{{\omega }_1,{\omega }_2,{\omega }_3\}$. Let $E_1=\{e_1,e_2\}$, $E_2=\{e_3,e_4\}$, and $E_3=\left\{e_5\right\}$; then, $E=E_1\times E_2\times E_3=\{q_1=(e_1,e_3,e_5),q_2=(e_1,e_4,e_5),q_3=(e_2,e_3,e_5),q_4=(e_2,e_4,e_5)\}$. Let ${\xi }_1=\{q_1,q_2,q_3\}$ and ${\xi }_2=\{q_1,q_2,q_4\}$. The N-HS sets $(\nabla ,{\xi }_1,6)$ and $({\nabla }_1,{\xi }_2,6)$ are provided in

Table 3. The 6-HS set $(\nabla ,{\xi }_1,6)$ in Example 3.

$(\mathbf{\nabla },{\mathit{\xi }}_1,6)$	${\mathit{q}}_1$	${\mathit{q}}_2$	${\mathit{q}}_3$
${\omega }_1$	1	2	3
${\omega }_2$	2	3	1
${\omega }_3$	3	1	3

and

Table 4. The 6-HS set $({\nabla }_1,{\xi }_2,6)$ in Example 3.

$({\mathbf{\nabla }}_1,{\mathit{\xi }}_2,6)$	${\mathit{q}}_1$	${\mathit{q}}_2$	${\mathit{q}}_4$
${\omega }_1$	3	4	5
${\omega }_2$	4	5	3
${\omega }_3$	5	3	5

, respectively. The normalized 6-HS sets derived from Table 3 and Table 4 are identical, as shown in

Table 5. The equivalence of $(\nabla ,{\xi }_1,6)$ and $({\nabla }_1,{\xi }_2,6)$ under normalization in Example 3.

$({\mathbf{\nabla }}^{\tilde{\mathit{o}}},\mathit{P},6)$=$({\mathbf{\nabla }}_1^{\tilde{\mathit{o}}},{\mathit{P}}_1,6)$	1	2	3
${\omega }_1$	0	1	2
${\omega }_2$	1	2	0
${\omega }_3$	2	0	2

. Thus, $(\nabla ,{\xi }_1,6)$ and $({\nabla }_1,{\xi }_2,6)$ are equivalent.

Definition 28. 

An N-HS complement of $(\nabla ,{\xi }_1,N)$ is ${(\nabla ,{\xi }_1,N)}^{\tilde{c}}$=$({\nabla }^{\tilde{c}},{\xi }_1,N)$, where ${\nabla }^{\tilde{c}}\left(q\right)\left(\omega \right)$=$(N-1)\setminus \nabla \left(q\right)\left(\omega \right)$.

Definition 29. 

An N-HS weak complement of $(\nabla ,{\xi }_1,N)$ is any N-HS set ${(\nabla ,{\xi }_1,N)}^{\tilde{\varpi }}$ = $({\nabla }^{\tilde{\varpi }},{\xi }_1,N)$, where ${\nabla }^{\tilde{\varpi }}\left(q\right)\left(\omega \right)$∩$\nabla \left(q\right)\left(\omega \right)$ = ∅ for each q∈${\xi }_1$ and ω∈Ω.

Definition 30. 

An N-HS top weak complement of $(\nabla ,{\xi }_1,N)$ is ${(\nabla ,{\xi }_1,N)}^{\tilde{t}}$=$({\nabla }^{\tilde{t}},{\xi }_1,N)$, where

$${\nabla }^{\tilde{t}}\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}N-1,\hfill & if \nabla \left(q\right)\left(\omega \right)<N-1\hfill \\ 0,\hfill & if \nabla \left(q\right)\left(\omega \right)=N-1\hfill \end{array}\right.$$

for each q∈${\xi }_1$ and ω∈Ω.

Definition 31. 

An N-HS bottom weak complement of $(\nabla ,{\xi }_1,N)$ is ${(\nabla ,{\xi }_1,N)}^{\tilde{b}}$=$({\nabla }^{\tilde{b}},{\xi }_1,N)$, where

$${\nabla }^{\tilde{b}}\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}0,\hfill & if \nabla \left(q\right)\left(\omega \right)>0\hfill \\ N-1,\hfill & if \nabla \left(q\right)\left(\omega \right)=0\hfill \end{array}\right.$$

for each q∈${\xi }_1$ and ω∈Ω.

Example 4. 

Consider $(\nabla ,{\xi }_1,6)$, as given in Table 2. Then, its N-HS complement (weak complement, top weak complement, bottom weak complement) is provided in tabular form by

Table 6. An N-HS complement of $(\nabla ,{\xi }_1,6)$ in Example 1.

${(\mathbf{\nabla },{\mathit{\xi }}_1,6)}^{\tilde{\mathit{c}}}$	${\mathit{q}}_1$	${\mathit{q}}_2$	${\mathit{q}}_3$	${\mathit{q}}_4$	${\mathit{q}}_5$	${\mathit{q}}_6$
${\omega }_1$	1	0	2	0	1	2
${\omega }_2$	3	1	4	2	1	0
${\omega }_3$	5	4	2	5	1	2
${\omega }_4$	5	2	3	0	0	1
${\omega }_5$	1	0	5	4	1	1

,

Table 7. An N-HS weak complement of $(\nabla ,{\xi }_1,6)$ in Example 1.

${(\mathbf{\nabla },{\mathit{\xi }}_1,6)}^{\tilde{\mathit{\varpi }}}$	${\mathit{q}}_1$	${\mathit{q}}_2$	${\mathit{q}}_3$	${\mathit{q}}_4$	${\mathit{q}}_5$	${\mathit{q}}_6$
${\omega }_1$	3	1	5	0	2	1
${\omega }_2$	1	2	3	4	5	4
${\omega }_3$	4	5	2	4	1	0
${\omega }_4$	3	4	4	0	0	2
${\omega }_5$	2	1	1	3	3	3

,

Table 8. An N-HS top weak complement of $(\nabla ,{\xi }_1,6)$ in Example 1.

${(\mathbf{\nabla },{\mathit{\xi }}_1,6)}^{\tilde{\mathit{t}}}$	${\mathit{q}}_1$	${\mathit{q}}_2$	${\mathit{q}}_3$	${\mathit{q}}_4$	${\mathit{q}}_5$	${\mathit{q}}_6$
${\omega }_1$	5	0	5	0	5	5
${\omega }_2$	5	5	5	5	5	0
${\omega }_3$	5	5	5	5	5	5
${\omega }_4$	5	5	5	0	0	5
${\omega }_5$	5	0	5	5	5	5

and

Table 9. An N-HS bottom weak complement of $(\nabla ,{\xi }_1,6)$ in Example 1.

${(\mathbf{\nabla },{\mathit{\xi }}_1,6)}^{\tilde{\mathit{b}}}$	${\mathit{q}}_1$	${\mathit{q}}_2$	${\mathit{q}}_3$	${\mathit{q}}_4$	${\mathit{q}}_5$	${\mathit{q}}_6$
${\omega }_1$	0	0	0	0	0	0
${\omega }_2$	0	0	0	0	0	0
${\omega }_3$	5	0	0	5	0	0
${\omega }_4$	5	0	0	0	0	0
${\omega }_5$	0	0	5	0	0	0

.

Remark 4. 

For every value of $0<T<N$, there is a corresponding HS set associated with each N-HS set.

Definition 32. 

For a given threshold $0<T<N$ and an N-HS set $(\nabla ,{\xi }_1,N)$, the associated HS set is denoted as $({\nabla }^{\tilde{T}},{\xi }_1)$ and is defined by the following expressions,∀q∈${\xi }_1$ and ω∈Ω:

$${\nabla }^{\tilde{T}}\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}1,\hfill & if \nabla \left(q\right)\left(\omega \right)\ge T\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right..$$

Specifically, the bottom HS set associated with $(\nabla ,{\xi }_1,N)$ is denoted as $({\nabla }^{\tilde{T}=1},{\xi }_1)$, and the top HS set associated with $(\nabla ,{\xi }_1,N)$ is designated as $({\nabla }^{\tilde{T}=N-1},{\xi }_1)$.

4. Set-Theoretic Operations on N-Hypersoft Sets and Their Properties

This section is dedicated to the examination of operations derived from set theory and their properties within the framework of N-HS sets. We commence by introducing these operations and subsequently delve into an analysis of their pertinent properties and implications.

Definition 33. 

An N-HS set $(\tilde{\Phi },{\xi }_1,N)$ is called a relative null N-HS set if, ∀$q\in {\xi }_1$ and $\omega \in \Omega$, $\tilde{\Phi }\left(q\right)\left(\omega \right)=0$.

Definition 34. 

An N-HS set $(\tilde{\Omega },{\xi }_1,N)$ is called a relative whole N-HS set if, ∀$q\in {\xi }_1$ and $\omega \in \Omega$, $\tilde{\Omega }\left(q\right)\left(\omega \right)=N-1$.

Definition 35. 

An N-HS set $(\nabla ,{\xi }_1,N)$ is the N-HS subset of $({\nabla }_1,{\xi }_2,N)$, denoted by $(\nabla ,{\xi }_1,N)$$\widetilde{⊑}$$({\nabla }_1,{\xi }_2,N)$, if

1. 

${\xi }_1\subseteq {\xi }_2$;

2. 

$\nabla \left(q\right)\left(\omega \right)\le {\nabla }_1\left(q\right)\left(\omega \right)$, ∀$q\in {\xi }_1$ and $\omega \in \Omega$.

Definition 36. 

An N-HS extended union of $(\nabla ,{\xi }_1,N_1)$ and $({\nabla }_1,{\xi }_2,N_2)$ is denoted and defined as $(\nabla ,{\xi }_1,N_1)$${\tilde{\bigsqcup }}_{\epsilon }$$({\nabla }_1,{\xi }_2,N_2)$=$({\nabla }_2,{\xi }_1\cup {\xi }_2,max(N_1,N_2))$, where ∀$q\in {\xi }_1\cup {\xi }_2$ and ω∈Ω:

$${\nabla }_2\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}\nabla \left(q\right)\left(\omega \right)\hfill & if q\in {\xi }_1\setminus {\xi }_2\hfill \\ {\nabla }_1\left(q\right)\left(\omega \right)\hfill & if q\in {\xi }_2\setminus {\xi }_1\hfill \\ max\{\nabla \left(q\right)\left(\omega \right),{\nabla }_1\left(q\right)\left(\omega \right)\}\hfill & if q\in {\xi }_1\cap {\xi }_2\hfill \end{array}\right..$$

Definition 37. 

An N-HS extended intersection of $(\nabla ,{\xi }_1,N_1)$ and $({\nabla }_1,{\xi }_2,N_2)$ is denoted and defined as $(\nabla ,{\xi }_1,N_1)$${\tilde{\sqcap }}_{\epsilon }$$({\nabla }_1,{\xi }_2,N_2)$=$({\nabla }_2,{\xi }_1\cup {\xi }_2,max(N_1,N_2))$, where ∀$q\in {\xi }_1\cup {\xi }_2$ and $\omega \in \Omega$:

$${\nabla }_2\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}\nabla \left(q\right)\left(\omega \right)\hfill & if q\in {\xi }_1\setminus {\xi }_2\hfill \\ {\nabla }_1\left(q\right)\left(\omega \right)\hfill & if q\in {\xi }_2\setminus {\xi }_1\hfill \\ min\{\nabla \left(q\right)\left(\omega \right),{\nabla }_1\left(q\right)\left(\omega \right)\}\hfill & if q\in {\xi }_1\cap {\xi }_2\hfill \end{array}\right..$$

Definition 38. 

An N-HS restricted union of $(\nabla ,{\xi }_1,N_1)$ and $({\nabla }_1,{\xi }_2,N_2)$ is denoted and defined as $(\nabla ,{\xi }_1,N_1)$${\tilde{\bigsqcup }}_{\Re }$$({\nabla }_1,{\xi }_2,N_2)$=$({\nabla }_2,{\xi }_1\cap {\xi }_2,\max (N_1,N_2))$, where ∀$q\in {\xi }_1\cap {\xi }_2$ and ω∈Ω: ${\nabla }_2\left(q\right)\left(\omega \right)=max\{\nabla \left(q\right)\left(\omega \right),{\nabla }_1\left(q\right)\left(\omega \right)\}$.

Definition 39. 

An N-HS restricted intersection of $(\nabla ,{\xi }_1,N_1)$ and $({\nabla }_1,{\xi }_2,N_2)$ is denoted and defined as $(\nabla ,{\xi }_1,N_1)$${\tilde{\sqcap }}_{\Re }$$({\nabla }_1,{\xi }_2,N_2)$=$({\nabla }_2,{\xi }_1\cap {\xi }_2,\max (N_1,N_2))$, where ∀$q\in {\xi }_1\cap {\xi }_2$ and ω∈Ω: ${\nabla }_2\left(q\right)\left(\omega \right)=min\{\nabla \left(q\right)\left(\omega \right),{\nabla }_1\left(q\right)\left(\omega \right)\}$.

Example 5. 

Consider Example 3 and the two N-HS sets $(\nabla ,{\xi }_1,6)$ and $({\nabla }_1,{\xi }_2,7)$ given in

Table 10. The 6-HS set $(\nabla ,{\xi }_1,6)$ in Example 5.

$(\mathbf{\nabla },{\mathit{\xi }}_1,6)$	${\mathit{q}}_1$	${\mathit{q}}_2$	${\mathit{q}}_3$
${\omega }_1$	5	3	2
${\omega }_2$	4	3	1
${\omega }_3$	4	4	0

and

Table 11. The 7-HS set $({\nabla }_1,{\xi }_2,7)$ in Example 5.

$({\mathbf{\nabla }}_1,{\mathit{\xi }}_2,7)$	${\mathit{q}}_1$	${\mathit{q}}_2$	${\mathit{q}}_4$
${\omega }_1$	6	1	3
${\omega }_2$	4	0	6
${\omega }_3$	0	5	2

. The N-HS extended union (intersection) and the N-HS restricted union (intersection) are provided in

Table 12. The N-HS extended union of $(\nabla ,{\xi }_1,6)$ and $({\nabla }_1,{\xi }_2,7)$ in Example 5.

$(\mathbf{\nabla },{\mathit{\xi }}_1,6)$${\tilde{\bigsqcup }}_{\mathit{\epsilon }}$$({\mathbf{\nabla }}_1,{\mathit{\xi }}_2,7)$	${\mathit{q}}_1$	${\mathit{q}}_2$	${\mathit{q}}_3$	${\mathit{q}}_4$
${\omega }_1$	6	3	2	3
${\omega }_2$	4	3	1	6
${\omega }_3$	4	5	0	2

,

Table 13. The N-HS extended intersection of $(\nabla ,{\xi }_1,6)$ and $({\nabla }_1,{\xi }_2,7)$ in Example 5.

$(\mathbf{\nabla },{\mathit{\xi }}_1,6)$${\tilde{\sqcap }}_{\mathit{\epsilon }}$$({\mathbf{\nabla }}_1,{\mathit{\xi }}_2,7)$	${\mathit{q}}_1$	${\mathit{q}}_2$	${\mathit{q}}_3$	${\mathit{q}}_4$
${\omega }_1$	5	1	2	3
${\omega }_2$	4	0	1	6
${\omega }_3$	0	4	0	2

,

Table 14. The N-HS restricted union of $(\nabla ,{\xi }_1,6)$ and $({\nabla }_1,{\xi }_2,7)$ in Example 5.

$(\mathbf{\nabla },{\mathit{\xi }}_1,6)$${\tilde{\bigsqcup }}_{\mathit{\Re }}$$({\mathbf{\nabla }}_1,{\mathit{\xi }}_2,7)$	${\mathit{q}}_1$	${\mathit{q}}_2$
${\omega }_1$	6	3
${\omega }_2$	4	3
${\omega }_3$	4	5

and

Table 15. The N-HS restricted intersection of $(\nabla ,{\xi }_1,6)$ and $({\nabla }_1,{\xi }_2,7)$ in Example 5.

$(\mathbf{\nabla },{\mathit{\xi }}_1,6)$${\tilde{\sqcap }}_{\mathit{\Re }}$$({\mathbf{\nabla }}_1,{\mathit{\xi }}_2,7)$	${\mathit{q}}_1$	${\mathit{q}}_2$
${\omega }_1$	5	1
${\omega }_2$	4	0
${\omega }_3$	0	4

.

Definition 40. 

The N-HS extended T-union of $(\nabla ,{\xi }_1,N_1)$ and $({\nabla }_1,{\xi }_2,N_2)$, where T< min$(N_1,N_2)$ is the HS extended union of two HS sets $({\nabla }^{\tilde{T}},{\xi }_1)$ and $({\nabla }_1^{\tilde{T}},{\xi }_2)$, is denoted by $(\nabla ,{\xi }_1,N_1)$${\tilde{\bigsqcup }}_{\epsilon }^{\tilde{T}}$$({\nabla }_1,{\xi }_2,N_2)$=$({\nabla }^{\tilde{T}},{\xi }_1)$${\bigsqcup }_e$$({\nabla }_1^{\tilde{T}},{\xi }_2)$.

Definition 41. 

The N-HS extended T-intersection of $(\nabla ,{\xi }_1,N_1)$ and $({\nabla }_1,{\xi }_2,N_2)$ where T< min$(N_1,N_2)$ is the HS extended intersection of two HS sets $({\nabla }^{\tilde{T}},{\xi }_1)$ and $({\nabla }_1^{\tilde{T}},{\xi }_2)$, is denoted by $(\nabla ,{\xi }_1,N_1)$${\tilde{\sqcap }}_{\epsilon }^{\tilde{T}}$$({\nabla }_1,{\xi }_2,N_2)$=$({\nabla }^{\tilde{T}},{\xi }_1)$${\sqcap }_e$$({\nabla }_1^{\tilde{T}},{\xi }_2)$.

Definition 42. 

The N-HS restricted T-union of $(\nabla ,{\xi }_1,N_1)$ and $({\nabla }_1,{\xi }_2,N_2)$ where T< min$(N_1,N_2)$ is the HS restricted union of two HS sets $({\nabla }^{\tilde{T}},{\xi }_1)$ and $({\nabla }_1^{\tilde{T}},{\xi }_2)$, is denoted by $(\nabla ,{\xi }_1,N_1)$${\tilde{\bigsqcup }}_{\Re }^{\tilde{T}}$$({\nabla }_1,{\xi }_2,N_2)$=$({\nabla }^{\tilde{T}},{\xi }_1)$${\bigsqcup }_r$$({\nabla }_1^{\tilde{T}},{\xi }_2)$.

Definition 43. 

The N-HS restricted T-intersection of $(\nabla ,{\xi }_1,N_1)$ and $({\nabla }_1,{\xi }_2,N_2)$ where T< min$(N_1,N_2)$ is the HS restricted intersection of two HS sets $({\nabla }^{\tilde{T}},{\xi }_1)$ and $({\nabla }_1^{\tilde{T}},{\xi }_2)$, is denoted by $(\nabla ,{\xi }_1,N_1)$${\tilde{\sqcap }}_{\Re }^{\tilde{T}}$$({\nabla }_1,{\xi }_2,N_2)$=$({\nabla }^{\tilde{T}},{\xi }_1)$${\sqcap }_r$$({\nabla }_1^{\tilde{T}},{\xi }_2)$.

Proposition 1. 

Let $(\nabla ,{\xi }_1,N)$, $({\nabla }_1,{\xi }_1,N)$, and $({\nabla }_2,{\xi }_1,N)$ be three N-HS sets. Then,

1. 

$(\nabla ,{\xi }_1,N)$$\widetilde{⊑}$$(\nabla ,{\xi }_1,N)$;

2. 

$(\nabla ,{\xi }_1,N)$$\widetilde{⊑}$$(\tilde{\Omega },{\xi }_1,N)$;

3. 

$(\tilde{\Phi },{\xi }_1,N)$$\widetilde{⊑}$$(\nabla ,{\xi }_1,N)$;

4. 

If $(\nabla ,{\xi }_1,N)$$\widetilde{⊑}$$({\nabla }_1,{\xi }_1,N)$ and $({\nabla }_1,{\xi }_1,N)$$\widetilde{⊑}$$({\nabla }_2,{\xi }_1,N)$, then $(\nabla ,{\xi }_1,N)$$\widetilde{⊑}$$({\nabla }_2,{\xi }_1,N)$.

Proof. 

Straightforward.    □

Proposition 2. 

Suppose $(\nabla ,{\xi }_1,N)$ and $({\nabla }_1,{\xi }_2,N)$ are two N-HS sets. Then,

1. 

$(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\epsilon }$$({\nabla }_1,{\xi }_2,N)$ is the smallest N-HS set which contains both $(\nabla ,{\xi }_1,N)$ and $({\nabla }_1,{\xi }_2,N)$;

2. 

$(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\Re }$$({\nabla }_1,{\xi }_2,N)$ is the largest N-HS set which is contained in both $(\nabla ,{\xi }_1,N)$ and $({\nabla }_1,{\xi }_2,N)$.

Proof. 

Straightforward.    □

Proposition 3. 

Suppose $(\nabla ,{\xi }_1,N)$ and $({\nabla }_1,{\xi }_1,N)$ are two N-HS sets. Then,

1. 

${(\tilde{\Phi },{\xi }_1,N)}^{\alpha }$=$(\tilde{\Omega },{\xi }_1,N)$, where $\alpha \in \{\tilde{c},\tilde{t},\tilde{b}\}$;

2. 

${(\tilde{\Omega },{\xi }_1,N)}^{\alpha }$=$(\tilde{\Phi },{\xi }_1,N)$, where $\alpha \in \{\tilde{c},\tilde{t},\tilde{b}\}$;

3. 

If $(\nabla ,{\xi }_1,N)$$\widetilde{⊑}$$({\nabla }_1,{\xi }_1,N)$, then ${({\nabla }_1,{\xi }_1,N)}^{\alpha }$$\widetilde{⊑}$${(\nabla ,{\xi }_1,N)}^{\alpha }$, where $\alpha \in \{\tilde{c},\tilde{t},\tilde{b}\}$;

4. 

${\left({(\nabla ,{\xi }_1,N)}^{\tilde{c}}\right)}^{\tilde{c}}$=$(\nabla ,{\xi }_1,N)$, ${\left({(\nabla ,{\xi }_1,N)}^{\tilde{t}}\right)}^{\tilde{t}}$$\widetilde{⊑}$$(\nabla ,{\xi }_1,N)$, and $(\nabla ,{\xi }_1,N)$$\widetilde{⊑}$${\left({(\nabla ,{\xi }_1,N)}^{\tilde{b}}\right)}^{\tilde{b}}$;

5. 

$(\tilde{\Phi },{\xi }_1,N)$$\widetilde{⊑}$$(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\Re }$${(\nabla ,{\xi }_1,N)}^{\tilde{c}}$$\widetilde{⊑}$$(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\Re }$${(\nabla ,{\xi }_1,N)}^{\tilde{c}}$$\widetilde{⊑}$$(\tilde{\Omega },{\xi }_1,N)$;

6. 

$(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\Re }$${(\nabla ,{\xi }_1,N)}^{\tilde{b}}$=$(\tilde{\Phi },{\xi }_1,N)$;

7. 

$(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\Re }$${(\nabla ,{\xi }_1,N)}^{\tilde{t}}$=$(\tilde{\Omega },{\xi }_1,N)$;

8. 

If $(\nabla ,{\xi }_1,N)$$\widetilde{⊑}$$({\nabla }_1,{\xi }_1,N)$, then $(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\Re }$$({\nabla }_1,{\xi }_1,N)$=$(\nabla ,{\xi }_1,N)$;

9. 

If $(\nabla ,{\xi }_1,N)$$\widetilde{⊑}$$({\nabla }_1,{\xi }_1,N)$, then $(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\Re }$$({\nabla }_1,{\xi }_1,N)$=$({\nabla }_1,{\xi }_1,N)$.

Proof. 

Straightforward.    □

Proposition 4. 

Suppose $(\nabla ,{\xi }_1,N)$ and $({\nabla }_1,{\xi }_2,N)$ are two N-HS sets and $\alpha \in \{\tilde{c},\tilde{t},\tilde{b}\}$. Then,

1. 

$\left(\right(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\epsilon }$$({\nabla }_1,{\xi }_2,N){)}^{\alpha }$=${(\nabla ,{\xi }_1,N)}^{\alpha }$${\tilde{\sqcap }}_{\epsilon }$${({\nabla }_1,{\xi }_2,N)}^{\alpha }$;

2. 

$\left(\right(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\epsilon }$$({\nabla }_1,{\xi }_2,N){)}^{\alpha }$=${(\nabla ,{\xi }_1,N)}^{\alpha }$${\tilde{\bigsqcup }}_{\epsilon }$${({\nabla }_1,{\xi }_2,N)}^{\alpha }$;

3. 

$\left(\right(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\Re }$$({\nabla }_1,{\xi }_2,N){)}^{\alpha }$=${(\nabla ,{\xi }_1,N)}^{\alpha }$${\tilde{\sqcap }}_{\Re }$${({\nabla }_1,{\xi }_2,N)}^{\alpha }$;

4. 

$\left(\right(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\Re }$$({\nabla }_1,{\xi }_2,N){)}^{\alpha }$=${(\nabla ,{\xi }_1,N)}^{\alpha }$${\tilde{\bigsqcup }}_{\Re }$${({\nabla }_1,{\xi }_2,N)}^{\alpha }$.

Proof. 

(1) Here, we prove the case where $\alpha =\tilde{t}$. Suppose that $(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\epsilon }$$({\nabla }_1,{\xi }_2,N)$ = $({\nabla }_2,{\xi }_1\cup {\xi }_2,N)$. Then, ∀$q\in {\xi }_1\cup {\xi }_2$ and ω∈Ω:

$${\nabla }_2\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}\nabla \left(q\right)\left(\omega \right)\hfill & \mathrm{if} q\in {\xi }_1\setminus {\xi }_2\hfill \\ {\nabla }_1\left(q\right)\left(\omega \right)\hfill & \mathrm{if} q\in {\xi }_2\setminus {\xi }_1\hfill \\ max\{\nabla \left(q\right)\left(\omega \right),{\nabla }_1\left(q\right)\left(\omega \right)\}\hfill & \mathrm{if} q\in {\xi }_1\cap {\xi }_2\hfill \end{array}\right..$$

Now, $\left(\right(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\epsilon }$$({\nabla }_1,{\xi }_2,N){)}^{\tilde{t}}$=${({\nabla }_2,{\xi }_1\cup {\xi }_2,N)}^{\tilde{t}}$. Then, ∀$q\in {\xi }_1\cup {\xi }_2$ and ω∈Ω:

$${\nabla }_2^{\tilde{t}}\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}N-1,\hfill & \mathrm{if} {\nabla }_2\left(q\right)\left(\omega \right)<N-1\hfill \\ 0,\hfill & \mathrm{if} {\nabla }_2\left(q\right)\left(\omega \right)=N-1\hfill \end{array}\right..$$

Hence, we have

$${\nabla }_2^{\tilde{t}}\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}N-1,\hfill & \mathrm{if} \nabla \left(q\right)\left(\omega \right)<N-1 \mathrm{and} q\in {\xi }_1\setminus {\xi }_2\hfill \\ 0,\hfill & \mathrm{if} \nabla \left(q\right)\left(\omega \right)=N-1 \mathrm{and}q\in {\xi }_1\setminus {\xi }_2\hfill \\ N-1,\hfill & \mathrm{if} {\nabla }_1\left(q\right)\left(\omega \right)<N-1 \mathrm{and} q\in {\xi }_2\setminus {\xi }_1\hfill \\ 0,\hfill & \mathrm{if} {\nabla }_1\left(q\right)\left(\omega \right)=N-1\mathrm{and} q\in {\xi }_2\setminus {\xi }_1\hfill \\ N-1,\hfill & \mathrm{if} max\{\nabla \left(q\right)\left(\omega \right),{\nabla }_1\left(q\right)\left(\omega \right)\}<N-1 \mathrm{and} q\in {\xi }_1\cap {\xi }_2\hfill \\ 0,\hfill & \mathrm{if} max\{\nabla \left(q\right)\left(\omega \right),{\nabla }_1\left(q\right)\left(\omega \right)\}=N-1 \mathrm{and} q\in {\xi }_1\cap {\xi }_2\hfill \end{array}\right.$$

$$=\left\{\begin{array}{cc}N-1,\hfill & \mathrm{if}\nabla \left(q\right)\left(\omega \right)<N-1\mathrm{and}q\in {\xi }_1\setminus {\xi }_2\hfill \\ 0,\hfill & \mathrm{if}\nabla \left(q\right)\left(\omega \right)=N-1\mathrm{and}q\in {\xi }_1\setminus {\xi }_2\hfill \\ N-1,\hfill & \mathrm{if}{\nabla }_1\left(q\right)\left(\omega \right)<N-1\mathrm{and}q\in {\xi }_2\setminus {\xi }_1\hfill \\ 0,\hfill & \mathrm{if}{\nabla }_1\left(q\right)\left(\omega \right)=N-1\mathrm{and}q\in {\xi }_2\setminus {\xi }_1\hfill \\ N-1,\hfill & \mathrm{if}\nabla \left(q\right)\left(\omega \right)<N-1\mathrm{and}{\nabla }_1\left(q\right)\left(\omega \right)<N-1\mathrm{and}q\in {\xi }_1\cap {\xi }_2\hfill \\ 0,\hfill & \mathrm{if}\nabla \left(q\right)\left(\omega \right)=N-1\mathrm{and}{\nabla }_1\left(q\right)\left(\omega \right)=N-1\hfill \\ & \mathrm{or}\nabla \left(q\right)\left(\omega \right)<N-1\mathrm{and}{\nabla }_1\left(q\right)\left(\omega \right)=N-1\hfill \\ & \mathrm{or}\nabla \left(q\right)\left(\omega \right)=N-1\mathrm{and}{\nabla }_1\left(q\right)\left(\omega \right)<N-1\mathrm{and}q\in {\xi }_1\cap {\xi }_2\hfill \end{array}\right..$$

On the other hand, let ${(\nabla ,{\xi }_1,N)}^{\tilde{t}}$${\tilde{\sqcap }}_{\epsilon }$${({\nabla }_1,{\xi }_2,N)}^{\tilde{t}}$=$({\nabla }_3,{\xi }_1\cup {\xi }_2,N)$, where

$${\nabla }^{\tilde{t}}\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}N-1,\hfill & \mathrm{if} \nabla \left(q\right)\left(\omega \right)<N-1\hfill \\ 0,\hfill & \mathrm{if} \nabla \left(q\right)\left(\omega \right)=N-1\hfill \end{array}\right.$$

and

$${\nabla }_1^{\tilde{t}}\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}N-1,\hfill & \mathrm{if} {\nabla }_1\left(q\right)\left(\omega \right)<N-1\hfill \\ 0,\hfill & \mathrm{if} {\nabla }_1\left(q\right)\left(\omega \right)=N-1\hfill \end{array}\right..$$

Then, ∀$q\in {\xi }_1\cup {\xi }_2$ and ω∈Ω:

$${\nabla }_3\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}{\nabla }^{\tilde{t}}\left(q\right)\left(\omega \right)\hfill & \mathrm{if} q\in {\xi }_1\setminus {\xi }_2\hfill \\ {\nabla }_1^{\tilde{t}}\left(q\right)\left(\omega \right)\hfill & \mathrm{if} q\in {\xi }_2\setminus {\xi }_1\hfill \\ min\{{\nabla }^{\tilde{t}}\left(q\right)\left(\omega \right),{\nabla }_1^{\tilde{t}}\left(q\right)\left(\omega \right)\}\hfill & \mathrm{if} q\in {\xi }_1\cap {\xi }_2\hfill \end{array}\right..$$

Hence, we have

$${\nabla }_3^{\tilde{t}}\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}N-1,\hfill & \mathrm{if} \nabla \left(q\right)\left(\omega \right)<N-1 \mathrm{and} q\in {\xi }_1\setminus {\xi }_2\hfill \\ 0,\hfill & \mathrm{if} \nabla \left(q\right)\left(\omega \right)=N-1 \mathrm{and} q\in {\xi }_1\setminus {\xi }_2\hfill \\ N-1,\hfill & \mathrm{if} {\nabla }_1\left(q\right)\left(\omega \right)<N-1 \mathrm{and} q\in {\xi }_2\setminus {\xi }_1\hfill \\ 0,\hfill & \mathrm{if} {\nabla }_1\left(q\right)\left(\omega \right)=N-1 \mathrm{and} q\in {\xi }_2\setminus {\xi }_1\hfill \\ N-1,\hfill & \mathrm{if} min\{{\nabla }^{\tilde{t}}\left(q\right)\left(\omega \right),{\nabla }_1^{\tilde{t}}\left(q\right)\left(\omega \right)\}<N-1 \mathrm{and} q\in {\xi }_1\cap {\xi }_2\hfill \\ 0,\hfill & \mathrm{if} min\{{\nabla }^{\tilde{t}}\left(q\right)\left(\omega \right),{\nabla }_1^{\tilde{t}}\left(q\right)\left(\omega \right)\}=N-1 \mathrm{and} q\in {\xi }_1\cap {\xi }_2\hfill \end{array}\right.$$

$$=\left\{\begin{array}{cc}N-1,\hfill & \mathrm{if}\nabla \left(q\right)\left(\omega \right)<N-1\mathrm{and}q\in {\xi }_1\setminus {\xi }_2\hfill \\ 0,\hfill & \mathrm{if}\nabla \left(q\right)\left(\omega \right)=N-1\mathrm{and}q\in {\xi }_1\setminus {\xi }_2\hfill \\ N-1,\hfill & \mathrm{if}{\nabla }_1\left(q\right)\left(\omega \right)<N-1\mathrm{and}q\in {\xi }_2\setminus {\xi }_1\hfill \\ 0,\hfill & \mathrm{if}{\nabla }_1\left(q\right)\left(\omega \right)=N-1\mathrm{and}q\in {\xi }_2\setminus {\xi }_1\hfill \\ N-1,\hfill & \mathrm{if}\nabla \left(q\right)\left(\omega \right)<N-1\mathrm{and}{\nabla }_1\left(q\right)\left(\omega \right)<N-1\mathrm{and}q\in {\xi }_1\cap {\xi }_2\hfill \\ 0,\hfill & \mathrm{if}\nabla \left(q\right)\left(\omega \right)=N-1\mathrm{and}{\nabla }_1\left(q\right)\left(\omega \right)=N-1\hfill \\ & \mathrm{or}\nabla \left(q\right)\left(\omega \right)<N-1\mathrm{and}{\nabla }_1\left(q\right)\left(\omega \right)=N-1\hfill \\ & \mathrm{or}\nabla \left(q\right)\left(\omega \right)=N-1\mathrm{and}{\nabla }_1\left(q\right)\left(\omega \right)<N-1\mathrm{and}q\in {\xi }_1\cap {\xi }_2\hfill \end{array}\right..$$

Since $({\nabla }_2,{\xi }_1\cup {\xi }_2,N)$ and $({\nabla }_3,{\xi }_1\cup {\xi }_2,N)$ are equivalent, ∀$q\in {\xi }_1\cup {\xi }_2$ and ω∈Ω, the proof is concluded.

The other cases and parts can be proven in a similar manner.    □

Proposition 5. 

Suppose $(\nabla ,{\xi }_1,N)$ and $({\nabla }_1,{\xi }_1,N)$ are two N-HS sets. Then,

1. 

$(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\epsilon }$$({\nabla }_1,{\xi }_1,N)$=$(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\Re }$$({\nabla }_1,{\xi }_1,N)$;

2. 

$(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\epsilon }$$({\nabla }_1,{\xi }_1,N)$=$(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\Re }$$({\nabla }_1,{\xi }_1,N)$;

3. 

$(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\Re }$$(\nabla ,{\xi }_1,N)$=$(\nabla ,{\xi }_1,N)$ and $(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\Re }$$(\nabla ,{\xi }_1,N)$=$(\nabla ,{\xi }_1,N)$;

4. 

$(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\Re }$$(\tilde{\Phi },{\xi }_1,N)$=$(\nabla ,{\xi }_1,N)$ and $(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\Re }$$(\tilde{\Phi },{\xi }_1,N)$=$(\tilde{\Phi },{\xi }_1,N)$;

5. 

$(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\Re }$$(\tilde{\Omega },{\xi }_1,N)$=$(\tilde{\Omega },{\xi }_1,N)$ and $(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\Re }$$(\tilde{\Omega },{\xi }_1,N)$=$(\nabla ,{\xi }_1,N)$.

Proof. 

Straightforward.    □

Proposition 6. 

Let $(\nabla ,{\xi }_1,N_1)$, $({\nabla }_1,{\xi }_2,N_2)$, and $({\nabla }_2,{\xi }_3,N_3)$ be three N-HS sets and ⊕∈ $\{{\tilde{\bigsqcup }}_{\Re },{\tilde{\sqcap }}_{\Re },{\tilde{\bigsqcup }}_{\epsilon },{\tilde{\sqcap }}_{\epsilon }\}$. Then,

1. 

$(\nabla ,{\xi }_1,N_1)$⊕$({\nabla }_1,{\xi }_2,N_2)$=$({\nabla }_1,{\xi }_2,N_2)$⊕$(\nabla ,{\xi }_1,N_1)$;

2. 

$(\nabla ,{\xi }_1,N_1)$⊕$\left(\right({\nabla }_1,{\xi }_2,N_2)$⊕$({\nabla }_2,{\xi }_3,N_3))$=$\left(\right(\nabla ,{\xi }_1,N_1)$⊕$({\nabla }_1,{\xi }_2,N_2))$⊕

$({\nabla }_2,{\xi }_3,N_3)$.

Proof. 

Straightforward.    □

Proposition 7. 

Suppose $(\nabla ,{\xi }_1,N)$ and $({\nabla }_1,{\xi }_2,N)$ are two N-HS sets. Then,

1. 

$(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\epsilon }$$\left(\right(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\Re }$$({\nabla }_1,{\xi }_2,N))$=$(\nabla ,{\xi }_1,N)$;

2. 

$(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\epsilon }$$\left(\right(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\Re }$$({\nabla }_1,{\xi }_2,N))$=$(\nabla ,{\xi }_1,N)$;

3. 

$(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\Re }$$\left(\right(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\epsilon }$$({\nabla }_1,{\xi }_2,N))$=$(\nabla ,{\xi }_1,N)$;

4. 

$(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\Re }$$\left(\right(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\epsilon }$$({\nabla }_1,{\xi }_2,N))$=$(\nabla ,{\xi }_1,N)$.

Proof. 

(1) Suppose that $(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\Re }$$({\nabla }_1,{\xi }_2,N)$ = $({\nabla }_2,{\xi }_1\cap {\xi }_2,N)$. Then, ∀$q\in {\xi }_1\cap {\xi }_2$ and ω∈Ω:

$${\nabla }_2\left(q\right)\left(\omega \right)=min\{\nabla \left(q\right)\left(\omega \right),{\nabla }_1\left(q\right)\left(\omega \right)\}.$$

Now, let $(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\epsilon }$$({\nabla }_2,{\xi }_1\cap {\xi }_2,N)$=$({\nabla }_3,{\xi }_1\cup ({\xi }_1\cap {\xi }_2),N)$. Then, ∀$q\in {\xi }_1\cup ({\xi }_1\cap {\xi }_2)$ and ω∈Ω:

$${\nabla }_3\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}\nabla \left(q\right)\left(\omega \right)\hfill & \mathrm{if} q\in {\xi }_1\setminus ({\xi }_1\cap {\xi }_2)\hfill \\ {\nabla }_2\left(q\right)\left(\omega \right)\hfill & \mathrm{if} q\in ({\xi }_1\cap {\xi }_2)\setminus {\xi }_1=\varnothing \hfill \\ max\{\nabla \left(q\right)\left(\omega \right),{\nabla }_2\left(q\right)\left(\omega \right)\}\hfill & \mathrm{if} q\in {\xi }_1\cap ({\xi }_1\cap {\xi }_2)\hfill \end{array}\right..$$

Hence, we obtain

$${\nabla }_3\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}\nabla \left(q\right)\left(\omega \right)\hfill & \mathrm{if} q\in {\xi }_1\setminus ({\xi }_1\cap {\xi }_2)\hfill \\ \nabla \left(q\right)\left(\omega \right)\hfill & \mathrm{if} q\in {\xi }_1\cap ({\xi }_1\cap {\xi }_2)\hfill \end{array}\right..$$

Therefore, $(\nabla ,{\xi }_1,N)$${\tilde{\bigsqcup }}_{\epsilon }$$\left(\right(\nabla ,{\xi }_1,N)$${\tilde{\sqcap }}_{\Re }$$({\nabla }_1,{\xi }_2,N))$=$(\nabla ,{\xi }_1,N)$.

The other parts can be proven in a similar manner.    □

Proposition 8. 

Let $(\nabla ,{\xi }_1,N_1)$, $({\nabla }_1,{\xi }_2,N_2)$, and $({\nabla }_2,{\xi }_3,N_3)$ be three N-HS sets. Then,

1. 

$(\nabla ,{\xi }_1,N_1)$${\tilde{\bigsqcup }}_{\epsilon }$$\left(\right({\nabla }_1,{\xi }_2,N_2)$${\tilde{\sqcap }}_{\Re }$$({\nabla }_2,{\xi }_3,N_3))$=$\left(\right(\nabla ,{\xi }_1,N_1)$${\tilde{\bigsqcup }}_{\epsilon }$$({\nabla }_1,{\xi }_2,N_2))$${\tilde{\sqcap }}_{\Re }$$\left(\right(\nabla ,{\xi }_1,N_1)$${\tilde{\bigsqcup }}_{\epsilon }$$({\nabla }_2,{\xi }_3,N_3))$;

2. 

$(\nabla ,{\xi }_1,N_1)$${\tilde{\sqcap }}_{\epsilon }$$\left(\right({\nabla }_1,{\xi }_2,N_2)$${\tilde{\bigsqcup }}_{\Re }$$({\nabla }_2,{\xi }_3,N_3))$=$\left(\right(\nabla ,{\xi }_1,N_1)$${\tilde{\sqcap }}_{\epsilon }$$({\nabla }_1,{\xi }_2,N_2))$${\tilde{\bigsqcup }}_{\Re }$

$\left(\right(\nabla ,{\xi }_1,N_1)$${\tilde{\sqcap }}_{\epsilon }$$({\nabla }_2,{\xi }_3,N_3))$;

3. 

$(\nabla ,{\xi }_1,N_1)$⊕$\left(\right({\nabla }_1,{\xi }_2,N_2)$⊖$({\nabla }_2,{\xi }_3,N_3))$=$\left(\right(\nabla ,{\xi }_1,N_1)$⊕$({\nabla }_1,{\xi }_2,N_2))$⊖

$\left(\right(\nabla ,{\xi }_1,N_1)$⊕$({\nabla }_2,{\xi }_3,N_3))$, where $\oplus \ne \ominus$, $\oplus \in \{{\tilde{\bigsqcup }}_{\Re },{\tilde{\sqcap }}_{\Re }\}$, and $\ominus \in \{{\tilde{\bigsqcup }}_{\Re },{\tilde{\sqcap }}_{\Re },{\tilde{\bigsqcup }}_{\epsilon },{\tilde{\sqcap }}_{\epsilon }\}$.

Proof. 

(2) Suppose that $\left(\right({\nabla }_1,{\xi }_2,N_2)$${\tilde{\sqcap }}_{\epsilon }$$({\nabla }_2,{\xi }_3,N_3))$ = $({\nabla }_3,{\xi }_2\cup {\xi }_3,max(N_2,N_3))$; then, ∀$q\in {\xi }_2\cup {\xi }_3$ and ω∈Ω:

$${\nabla }_3\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}{\nabla }_1\left(q\right)\left(\omega \right)\hfill & \mathrm{if} q\in {\xi }_2\setminus {\xi }_3\hfill \\ {\nabla }_2\left(q\right)\left(\omega \right)\hfill & \mathrm{if} q\in {\xi }_3\setminus {\xi }_2\hfill \\ min\{{\nabla }_1\left(q\right)\left(\omega \right),{\nabla }_2\left(q\right)\left(\omega \right)\}\hfill & \mathrm{if} q\in {\xi }_2\cap {\xi }_3\hfill \end{array}\right..$$

Let $(\nabla ,{\xi }_1,N_1)$${\tilde{\bigsqcup }}_{\Re }$$({\nabla }_3,{\xi }_2\cup {\xi }_3,max(N_2,N_3))$=$({\nabla }_4,{\xi }_1\cap ({\xi }_2\cup {\xi }_3)$, $max(N_1,max(N_2,N_3)))$=$({\nabla }_4,L\cup K,max(N_1,N_2,N_3))$, where $L={\xi }_1\cap {\xi }_2$ and $K={\xi }_1\cap {\xi }_3$; then, ∀$q\in L\cup K$ and ω∈Ω:

$${\nabla }_4\left(q\right)\left(\omega \right)=max\{\nabla \left(q\right)\left(\omega \right),{\nabla }_3\left(q\right)\left(\omega \right)\}.$$

Hence, we obtain

$${\nabla }_4\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}max\{\nabla \left(q\right)\left(\omega \right),{\nabla }_1\left(q\right)\left(\omega \right)\}\hfill & \mathrm{if} q\in L\setminus K\hfill \\ max\{\nabla \left(q\right)\left(\omega \right),{\nabla }_2\left(q\right)\left(\omega \right)\}\hfill & \mathrm{if} q\in K\setminus L\hfill \\ max\{\nabla \left(q\right)\left(\omega \right),min\{{\nabla }_1\left(q\right)\left(\omega \right),{\nabla }_2\left(q\right)\left(\omega \right)\}\}\hfill & \mathrm{if} q\in L\cap K\hfill \end{array}\right..$$

On the other hand, let $(\nabla ,{\xi }_1,N_1)$${\tilde{\bigsqcup }}_{\Re }$$({\nabla }_1,{\xi }_2,N_2)$=$({\nabla }_5,{\xi }_1\cap {\xi }_2,max(N_1,N_2))$; then, ∀$q\in {\xi }_1\cap {\xi }_2$ and ω∈Ω:

$${\nabla }_5\left(q\right)\left(\omega \right)=max\{\nabla \left(q\right)\left(\omega \right),{\nabla }_1\left(q\right)\left(\omega \right)\}.$$

Let $(\nabla ,{\xi }_1,N_1)$${\tilde{\bigsqcup }}_{\Re }$$({\nabla }_2,{\xi }_3,N_3)$ = $({\nabla }_6,{\xi }_1\cap {\xi }_3,max(N_1,N_3))$; then, ∀$q\in {\xi }_1\cap {\xi }_3$ and ω∈Ω:

$${\nabla }_6\left(q\right)\left(\omega \right)=max\{\nabla \left(q\right)\left(\omega \right),{\nabla }_2\left(q\right)\left(\omega \right)\}.$$

Now, suppose that $({\nabla }_5,{\xi }_1\cap {\xi }_2,max(N_1,N_2))$${\tilde{\sqcap }}_{\epsilon }$$({\nabla }_6,{\xi }_1\cap {\xi }_3,max(N_1,N_3))$ = $({\nabla }_7$, $L\cup K),max(N_1,N_2,N_3))$, where $L={\xi }_1\cap {\xi }_2$ and $K={\xi }_1\cap {\xi }_3$; then, ∀$q\in L\cup K$ and ω∈Ω:

$${\nabla }_7\left(q\right)\left(\omega \right)=\left\{\begin{array}{cc}{\nabla }_5\left(q\right)\left(\omega \right)\hfill & \mathrm{if} q\in L\setminus K\hfill \\ {\nabla }_6\left(q\right)\left(\omega \right)\hfill & \mathrm{if} q\in K\setminus L\hfill \\ min\{{\nabla }_5\left(q\right)\left(\omega \right),{\nabla }_6\left(q\right)\left(\omega \right)\}\hfill & \mathrm{if} q\in L\cap K\hfill \end{array}\right.$$

$$=\left\{\begin{array}{cc}max\{\nabla \left(q\right)\left(\omega \right),{\nabla }_1\left(q\right)\left(\omega \right)\}\hfill & \mathrm{if} q\in L\setminus K\hfill \\ max\{\nabla \left(q\right)\left(\omega \right),{\nabla }_2\left(q\right)\left(\omega \right)\}\hfill & \mathrm{if} q\in L\setminus K\hfill \\ min\{max\{\nabla \left(q\right)\left(\omega \right),{\nabla }_1\left(q\right)\left(\omega \right)\},max\{\nabla \left(q\right)\left(\omega \right),{\nabla }_2\left(q\right)\left(\omega \right)\}\}\hfill & \mathrm{if} q\in L\cap K\hfill \end{array}\right..$$

Since $({\nabla }_4,L\cup K,max(N_1,N_2,N_3))$ and $({\nabla }_7,L\cup K,max(N_1,N_2,N_3))$ are equivalent, ∀$q\in L\cup K$ and ω∈Ω; the proof is concluded.

The other parts can be proven in a similar manner.   □

5. Enhancing Decision-Making with N-Hypersoft Sets

The decision-making approach introduced in this context extends the decision-making approach for soft sets outlined in [6], while maintaining all the parameters. Algorithms 1 and 2 rank the alternatives based on their extended choice values and extended weight choice values, respectively. Moreover, Algorithm 3 showcases how the alternatives in Ω can be ranked using the information from $(\nabla ,{\xi }_1,N)$ and a threshold value T. Here, we present the detailed steps involved in these algorithms.

Applying Algorithm 1 to Example 1 in

Table 16. The extended choice values in Example 1.

$(\mathbf{\nabla },{\mathit{\xi }}_1,6)$	${\mathit{q}}_1$	${\mathit{q}}_2$	${\mathit{q}}_3$	${\mathit{q}}_4$	${\mathit{q}}_5$	${\mathit{q}}_6$	${\mathit{\sigma }}_{\mathit{i}}$
${\omega }_1$	4	5	3	5	4	3	24
${\omega }_2$	2	4	1	3	4	5	19
${\omega }_3$	0	1	3	0	4	3	11
${\omega }_4$	0	3	2	5	5	4	19
${\omega }_5$	4	5	0	1	4	4	18

, we observe that the candidate ${\omega }_1$ is chosen, and the ranking decision is determined as ${\omega }_1>{\omega }_2={\omega }_4>{\omega }_5>{\omega }_3$.

Algorithm 1: Extended Choice Values.

Step 1  Input Ω=$\{{\omega }_i,i=1,2,\cdots ,m\}$ and ${\xi }_1$=$\{q_j,j=1,2,\cdots ,n\}$. Step 2  Input N-HS set $(\nabla ,{\xi }_1,N)$. Step 3  Evaluate ${\sigma }_i$ = ${\sum }_{j=1}^n$$r_{ij}$ for each ${\omega }_i$. Step 4  Find k such that ${\sigma }_k$ = $max$${\sigma }_i$. Step 5  The output is any ${\omega }_k$ which satisfies Step 4.

Algorithm 2: Extended Weight Choice Values.

Step 1  Input Ω= $\{{\omega }_i,i=1,2,\cdots ,m\}$, ${\xi }_1$ = $\{q_j,j=1,2,\cdots ,n\}$, and a weight $W_j$ for each parameter. Step 2  Input N-HS set $(\nabla ,{\xi }_1,N)$. Step 3  Evaluate ${\sigma }_i^W$ = ${\sum }_{j=1}^n$$r_{ij}$$W_j$ for each ${\omega }_i$. Step 4  Find k such that ${\sigma }_k^W$ = $max$${\sigma }_i^W$. Step 5  The output is any ${\omega }_k$ which satisfies Step 4.

Example 6. 

Given the weights $W_1=0.8$, $W_2=0.6$, $W_3=0.6$, $W_4=0.5$, $W_5=0.7$, and $W_6=0.6$ assigned to the skills $q_j$ in Example 1, the analysis from

Table 17. The extended weight choice values in Example 6.

$(\mathbf{\nabla },{\mathit{\xi }}_1,6)$	${\mathit{W}}_1=0.8$	${\mathit{W}}_2=0.6$	${\mathit{W}}_3=0.6$	${\mathit{W}}_4=0.5$	${\mathit{W}}_5=0.7$	${\mathit{W}}_6=0.6$	${\mathit{\sigma }}_{\mathit{i}}^{\mathit{W}}$
${\omega }_1$	3.2	3	1.8	2.5	2.8	1.8	15.1
${\omega }_2$	1.6	2.4	0.6	1.5	2.8	3	11.9
${\omega }_3$	0	0.6	1.8	0	2.8	1.8	7
${\omega }_4$	0	1.8	1.2	2.5	3.5	2.4	11.4
${\omega }_5$	3.2	3	0	0.5	2.8	2.4	11.9

indicates that the candidate ${\omega }_1$ is chosen, and the ranking decision is ${\omega }_1>{\omega }_2={\omega }_5>{\omega }_4>{\omega }_3$.

Remark 5. 

The preservation of the ranking of extended choice values for Ω is evident when two N-HS sets, say $(\nabla ,{\xi }_1,N)$ and $({\nabla }_1,{\xi }_2,N)$, are equivalent through normalization.

Example 7. 

By referring to

Table 18. The 4-choice values in Example 1.

$({\mathbf{\nabla }}^4,{\mathit{\xi }}_1)$	${\mathit{q}}_1$	${\mathit{q}}_2$	${\mathit{q}}_3$	${\mathit{q}}_4$	${\mathit{q}}_5$	${\mathit{q}}_6$	${\mathit{\sigma }}_{\mathit{i}}^4$
${\omega }_1$	1	1	0	1	1	0	4
${\omega }_2$	0	1	0	0	1	1	3
${\omega }_3$	0	0	0	0	1	0	1
${\omega }_4$	0	0	0	1	1	1	3
${\omega }_5$	1	1	0	0	1	1	4

, which presents $({\nabla }^4,{\xi }_1)$ of $(\nabla ,{\xi }_1,6)$ in Example 1, it can be inferred that the candidates ${\omega }_1$ and ${\omega }_5$ are chosen and the ranking decision is ${\omega }_1={\omega }_5>{\omega }_2={\omega }_4>{\omega }_3$.

Algorithm 3: T-Choice Values.

Step 1  Input Ω = $\{{\omega }_i,i=1,2,\cdots ,m\}$ and ${\xi }_1$ = $\{q_j,j=1,2,\cdots ,n\}$. Step 2  Input N-HS set $(\nabla ,{\xi }_1,N)$. Step 3  Evaluate $r_{ij}^{\tilde{T}}$ where $r_{ij}^{\tilde{T}}=\left\{\begin{array}{cc}1,\hfill & if r_{ij}\ge T\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right.$ Step 4  Evaluate ${\sigma }_i^{\tilde{T}}$ = ${\sum }_{j=1}^n$$r_{ij}^{\tilde{T}}$ for each ${\omega }_i$. Step 5  Find k such that ${\sigma }_k^{\tilde{T}}$ = $max$ ${\sigma }_i^{\tilde{T}}$. Step 6  The output is any ${\omega }_k$ which satisfies Step 5.

6. Comparisons

In this section, we utilize Examples 1, 6, and 7 as a means to evaluate and contrast the outcomes yielded by our algorithms. Upon comparing the three algorithms (

Table 19. Comparison of Algorithms 1–3 using Examples 1, 6, and 7.

Algorithm No.	Algorithm Name	Decision Ranking
Algorithm 1	Extended Choice Values	${\omega }_1>{\omega }_2={\omega }_4>{\omega }_5>{\omega }_3$
Algorithm 2	Extended Weight Choice Values	${\omega }_1>{\omega }_2={\omega }_5>{\omega }_4>{\omega }_3$
Algorithm 3	T-Choice Values	${\omega }_1={\omega }_5>{\omega }_2={\omega }_4>{\omega }_3$

), notable dissimilarities in the results become apparent. This serves as evidence showcasing the versatility and adjustability inherent in our decision-making algorithms.

7. Concluding Remarks

This paper introduced N-HS sets, an enhanced version of HS sets that adeptly handle both binary and non-binary evaluations while inherently embodying a structural symmetry. It presented algebraic definitions such as 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, accompanied by illustrative examples. The paper also delved into set-theoretic operations, encompassing relative null/whole N-HS sets, N-HS subsets, and N-HS extended/restricted union and intersection, all intricately tied to the symmetrical balance inherent in the N-HS framework. Finally, decision-making procedures for N-HS sets were proposed and compared.

In addition to the substantial contributions detailed in this paper, numerous promising avenues for future exploration exist to expand upon the proposed model and amplify its capabilities. An intriguing path forward involves the integration of bipolarity, as elucidated in the work by Dubois and Prade [45]. Bipolarity, centering on the dual consideration of parameters and their opposites, promises to enrich information and preference representation. By weaving bipolarity into the tapestry of the N-HS sets framework, a more nuanced and comprehensive assessment of evaluations can be achieved, further harmonizing with the symmetrical ideals underlying our model. It is with great anticipation that we envisage revisiting and expanding upon this avenue in our future research endeavors.