Towards a New Approach in Complex Neutrosophic Soft Ring Theory

Abstract

This paper focuses on investigating the algebraic structures within the complex neutrosophic soft model. Two key concepts, the complex neutrosophic soft ring (CNSR) and the complex neutrosophic soft ideal (CNSI), are introduced. The former integrates the characteristics of complex neutrosophic soft sets with the foundational principles of ring theory, to effectively address the issue of uncertainty and indeterminacy in a ring environment using complex neutrosophic membership values. The latter, on the other hand, are specific subsets of CNSRs with unique properties that play important roles in the study of ring theory. We examine and validate the algebraic properties of both CNSRs and CNSIs, enhancing the understanding of their algebraic behaviors and interactions. Specifically, we explore the connection between CNSRs and soft rings, offering insights into how CNSRs align with the broader framework of soft rings while emphazising the distinctive features of complex neutrosophic soft structures in algebraic analyses. Furthermore, we determine the relationships between CNSRs and neutrosophic soft rings, as well as between CNSIs and neutrosophic soft ideals. This thorough analysis aims to advance the knowledge of CNSRs and CNSIs, contributing to algebraic analysis and its application in managing uncertainty and vagueness.

Keywords

complex neutrosophic soft ring complex neutrosophic soft ideal complex neutrosophic soft set neutrosophic logic

1. Introduction

Although significant progress has been made in managing uncertain data, uncertainty and vagueness remain persistent challenges for many researchers in their daily work. To address these issues and to manage the associated uncertainties, Smarandache (1998) introduced the concept of the neutrosophic set (NS) to overcome the inadequacy related to the available fuzzy set tools. Meanwhile, Molodtsov (1999) developed the soft set (SS) as an additional mathematical tool to provide a better precision of everyday data. Both NS and SS research have gained significant momentum, leading to the publication of numerous studies and applications. Tran et al. (2022) added a key contribution by establishing the connection between topologies on neutrosophic soft sets (NSSs) and fuzzy soft sets. Specifically, the topology on NSSs which can be used to parameterize topologies on fuzzy soft sets; however, the converse does not necessarily hold. Some models combining NS and SS have resulted in a more enhanced ability to address real-world problems.

The previously mentioned models that deal with uncertainties, have been applied in various areas, including soft computing (Jumaili et al., 2019), graph-related problems (Yow et al., 2021, 2023), algebraic structures (Abed et al., 2021), and medical diagnosis (Karataş & Ozturk, 2023) where a three-step neutrosophic algorithm was used. Al-Sharqi et al. (2022) presented their model with the ability to provide a comprehensive representation of two-dimensional interval neutrosophic information (amplitude terms and phase terms), as well as adequate parameterization and opinions of experts, all in the form of an interval. Ali and Smarandache (2017) extended the concept of NS into complex space, by defining the complex neutrosophic set (CNS) that is characterized by three membership functions, namely truth, indeterminacy and falsity, each of which is assigned a complex value that takes periodicity into account. Smarandache et al. (2017) further integrated the SS and CNS models to introduce the concept of complex neutrosophic soft sets (CNSSs). A complex neutrosophic soft decision-making algorithm was constructed and used in signal processing applications to improve the CNS theory (Khan et al., 2021).

Rahoumah et al. (2024) introduced the concept of complex neutrosophic soft groups (CNSGs) and conducted an extensive investigation, which included the verification of their properties and the exploration of their fundamental characteristics. The study of CNSGs allows the extension to the ring and field environments within the scope of complex neutrosophic soft settings. Alsarahead and Ahmad (2017) introduced complex fuzzy sub-ring and defined some new concepts such as $π$ -fuzzy sub-ring and $π$ -fuzzy ideal, in addition to the characteristics of complex fuzzy sub-ring and complex fuzzy ideal., but three years later they introduced the complex fuzzy soft ring (Alsarahead & Ahmad, 2020). The extension of the neutrosophic soft group (NSG) to the complex plane to produce CNSG has resulted in a comprehensive study of their algebraic structures within group theory (Smarandache et al., 2017). Neutrosophic soft ring (NSR) was introduced by Bera and Mahapatra (2017) where the properties of neutrosophic soft set (NSS) were examined in the ring environment. In this study, the aim is to extend the complex neutrosophic soft group to the ring environment and to introduce neutrosophic soft rings (NSRs) and neutrosophic soft ideals (NSIs) in the complex space. Special attention is given to some algebraic structures within the CNSS models by introducing the concepts of CNSR and CNSI. The verification of some specific algebraic properties of CNSRs and CNSIs will be carried on to gain a detailed insight into their algebraic behavior and to highlight the relationship between CNSRs and soft rings. The connections between CNSRs and NSRs, and between CNSIs and NSIs are also investigated.

2. Preliminaries

We first present definitions of different components related to neutrosophic theory and describe the gradual evolution of concepts related to soft set (SS), soft ring (SR), soft ideal (SI), fuzzy subring, fuzzy ideal (FI), neutrosophic subring, CNS, CFSR and complex fuzzy soft ideal (CFSI). We then include definitions related to the properties and operations within neutrosophic theory.

Definition 2.1
Molodtsov (1999) Let $U$ represent a universal set and $E$ be a collection of parameters. The power set of $U$ is denoted by $P (U)$ . For a subset $A$ of $E$ , the pair $(k, A)$ is defined as a soft set over $U$ , where $k$ is a mapping from $A$ to $P (U)$ .
Definition 2.2
Acar et al. (2010) Let $(k, A)$ be a soft set defined over the ring $R$ . For every $a \in A$ , if $K_{a}$ is a sub-ring of $R$ , then the pair $(K, A)$ is referred to as a soft ring.
Definition 2.3
Ghosh et al. (2011) Let $(k, A)$ be a soft set over the ring $R$ . For every $a \in A$ , if $k_{a}$ is both a left and right ideal of $R$ , then the pair $(k, A)$ is referred to as a soft ideal. 1.
$x, y \in k_{a} \Rightarrow x - y \in k_{a}$ .
2.
$x \in k_{a}, r \in R \Rightarrow x \cdot r \in k_{a}, r \cdot x \in k_{a}$ .

Definition 2.4
Liu (1982) Let $R$ be a ring and $k = {(x, μ_{k} (x) : x \in U}$ be a fuzzy set where, $μ_{k} (x)$ is the degree of membership and $μ_{K} \in [0, 1]$ . Then $k$ is called a fuzzy sub-ring if the following conditions are satisfied: 1.
$μ_{k} (x - y) \geq min {μ_{k} (x), μ_{k} (y)} for all x, y \in R$ .
2.
$μ_{k} (x \cdot y) \geq min {μ_{k} (x), μ_{k} (y)} for all x, y \in R$ .

Definition 2.5
Liu (1982) Let $R$ be a ring and $k = {(x, μ_{K} (x)) : x \in U}$ be a fuzzy set where, $μ_{K} (x)$ is the membership degree and $μ_{K} \in [0, 1]$ . The set $k$ is referred to as a fuzzy ideal if the following conditions are satisfied: 1.
$μ_{k} (x - y) \geq min {μ_{k} (x), μ_{k} (y)} for all x, y \in R$ .
2.
$μ_{k} (x \cdot y) \geq max {μ_{k} (x), μ_{k} (y)} for all x, y \in R$ .

Definition 2.6
Ali and Smarandache (2017) Let S be a CNS in the universe $U$ , represented by three membership functions $T_{S} (u), I_{S} (u) and F_{S} (u)$ . For every $u \in U$ , these three membership terms lie within the unit circle in the complex plane, as illustrated below: 1.
$T_{S} (u) = p_{S} (u) e^{j μ_{S} (u)}$ .
2.
$I_{S} (u) = q_{S} (u) e^{j ν_{S} (u)}$ .
3.
$F_{S} (u) = r_{S} (u) e^{j ω_{S} (u)}$ .
where $p_{S} (u), q_{S} (u), r_{S} (u)$ are amplitude terms and $μ_{S} (u), ν_{S} (u), ω_{S} (u)$ are phase terms of truth, indeterminacy and falsity membership functions respectively and $p_{S} (u), q_{S} (u), r_{S} (u) \in [0, 1]$ such that $0 \leq p_{S} (u) + q_{S} (u), r_{S} (u) \leq 3$ .
Definition 2.7
Alsarahead and Ahmad (2020) Let $R$ be a ring and $(k, A)$ be a homogeneous complex fuzzy soft set over $R$ . The pair $(k, A)$ is called a CFSR if and only if the following conditions are satisfied: 1.
$μ_{k_{a}} (x - y) \geq min {μ_{k_{a}} (x), μ_{k_{a}} (y)} for all a \in A and x, y \in R$ .
2.
$μ_{K_{a}} (x \cdot y) \geq min {μ_{K_{a}} (x), μ_{K_{a}} (y)} for all a \in A and x, y \in R$ .

Definition 2.8
Alsarahead and Ahmad (2020) Let $R$ be a ring and $(k, A)$ be a homogeneous complex fuzzy soft set over $R$ . The set $(k, A)$ is called a complex fuzzy soft ideal if and only if the following conditions are met: 1.
$μ_{k_{a}} (x - y) \geq min {μ_{k_{a}} (x), μ_{k_{a}} (y)} for all a \in A and x, y \in R$ .
2.
$μ_{k_{a}} (x \cdot y) \geq max {μ_{k_{a}} (x), μ_{k_{a}} (y)} for all a \in A and x, y \in R$ .

3. Developing Complex Neutrosophic Soft Ring

We now investigate the properties of CNSR by providing detailed definitions and presenting a series of theorems to illustrate their importance and potential applications within this framework.

Definition 3.1
If $(k, A)$ and $(g, B)$ are two complex neutrosophic soft sets over the universe $U$ , they are defined by complex-valued membership functions as follows:
$\begin{aligned} T_{k_{a}} (x) = p_{k_{a}} (x) e^{j μ_{k_{a}} (x)}, I_{k_{a}} (x) = q_{k_{a}} (x) e^{j ν_{k_{a}} (x)}, F_{k_{a}} (x) = r_{k_{a}} (x) e^{j ω_{k_{a}} (x)} . \end{aligned}$

$\begin{aligned} T_{g_{a}} (x) = p_{g_{a}} (x) e^{j μ_{g_{a}} (x)}, I_{g_{a}} (x) = q_{g_{a}} (x) e^{j ν_{g_{a}} (x)}, F_{g_{a}} (x) = r_{g_{a}} (x) e^{j ω_{g_{a}} (x)} . \end{aligned}$
Then, 1.
The set $(k, A)$ is referred to as a homogeneous-CNSS if, for all $a \in A$ and $x, y \in U$ , the following conditions hold:
$p_{k_{a}} (x) \leq p_{k_{a}} (y) \Leftrightarrow μ_{k_{a}} (x) \leq μ_{k_{a}} (y)$ .

$q_{k_{a}} (x) \leq q_{k_{a}} (y) \Leftrightarrow ν_{k_{a}} (x) \leq ν_{k_{a}} (y)$ .

$r_{k_{a}} (x) \leq r_{k_{a}} (y) \Leftrightarrow ω_{k_{a}} (x) \leq ω_{k_{a}} (y)$ .

2.
A complex neutrosophic soft set $(k, A)$ is considered a completely homogeneous complex neutrosophic soft set if it is homogeneous and satisfies the following conditions for all $x \in U$ and for all $a, b \in A$ :
$p_{k_{a}} (x) \leq p_{k_{b}} (x) \Leftrightarrow μ_{k_{a}} (x) \leq μ_{k_{b}} (x)$ .

$q_{k_{a}} (x) \leq q_{k_{b}} (x) \Leftrightarrow ν_{k_{a}} (x) \leq ν_{k_{b}} (x)$ .

$r_{k_{a}} (x) \leq r_{k_{b}} (x) \Leftrightarrow ω_{k_{a}} (x) \leq ω_{k_{b}} (x)$ .

3.
A complex neutrosophic soft set $(k, A)$ is said to be homogeneous with $(g, B)$ if and only if, for all $a \in A \cap B$ and for all $x \in U$ , the following conditions hold:
$p_{k_{a}} (x) \leq p_{g_{a}} (x) \Leftrightarrow μ_{k_{a}} (x) \leq μ_{g_{a}} (x)$ .

$q_{k_{a}} (x) \leq q_{g_{a}} (x) \Leftrightarrow ν_{k_{a}} (x) \leq ν_{g_{a}} (x)$ .

$r_{k_{a}} (x) \leq r_{g_{a}} (x) \Leftrightarrow ω_{k_{a}} (x) \leq ω_{g_{a}} (x)$ .

Example 3.1
Let $X = {x_{1}, x_{2}, x_{3}}$ be a set top oil producing nations in Africa and let $E$ be a set of parameters describing each country’s economic indicators, with $A, B \subseteq E$ , where $A = {v_{1}, v_{2}}$ and $B = {v_{1}, v_{3}, v_{4}, v_{5}}$ , where these sets are defined as follows:
$\begin{aligned} X & = {x_{1} = L i b y a, x_{2} = A l g e r i a, x_{3} = N i g e r i a} . \end{aligned}$

$\begin{aligned} E & = {\begin{aligned} v_{1} = G D P g r o w t h, v_{2} = e x p o r t v o l u m e, v_{3} = p o p u l a t i o n g r o w t h, \\ v_{4} = u n e m p l o y m e n t r a t e, v_{5} = i n f l a t i o n r a t e . \end{aligned}} \end{aligned}$
In decision-making under uncertainty, complex numbers can represent probabilities or likelihoods with a phase component. In quantum decision theory, decisions are modelled similarly to quantum mechanics – where complex probability amplitudes are used.

Assume that $(k, A)$ and $(g, B)$ are two CNSSs defined as follows:
$\begin{aligned} (k, A) = {\begin{aligned} (v_{1}, \frac{x_{1}}{0.4 e^{j π (0.4)}, 0.5 e^{j π (0.6)}, 0.3 e^{j π (0.6)}}, \frac{x_{2}}{0.2 e^{j π (0.3)}, 0.6 e^{j π (0.8)}, 0.5 e^{j π (0.9)}}, \\ \frac{x_{3}}{0.4 e^{j π (0.5)}, 0.5 e^{j π (0.7)}, 0.9 e^{j π (0.9)}}), (v_{2}, \frac{x_{1}}{0.9 e^{j π (0.6)}, 0.7 e^{j π (0.7)}, 0.8 e^{j π (0.8)}}, \\ \frac{x_{2}}{0.3 e^{j π (0.4)}, 0.5 e^{j π (0.3)}, 0.2 e^{j π (0.8)}}, \frac{x_{3}}{0.9 e^{j π (0.6)}, 0.6 e^{j π (0.7)}, 0.4 e^{j π (0.8)}}) \end{aligned}} . \end{aligned}$

$\begin{aligned} (g, B) = {\begin{aligned} (v_{1}, \frac{x_{1}}{0.5 e^{j π (0.6)}, 0.5 e^{j π (0.7)}, 0.2 e^{j π (0.6)}}, \frac{x_{2}}{0.4 e^{j π (0.8)}, 0.7 e^{j π (0.9)}, 0.6 e^{j π (0.9)}}, \\ \frac{x_{3}}{0.9 e^{j π (0.9)}, 0.6 e^{j π (0.8)}, 0.9 e^{j π (0.8)}}), (v_{3}, \frac{x_{1}}{0.8 e^{j π (0.4)}, 0.2 e^{j π (0.9)}, 0.4 e^{j π (0.6)}}, \\ \frac{x_{2}}{0.3 e^{j π (0.2)}, 0.5 e^{j π (0.3)}, 0.8 e^{j π (0.7)}}, \frac{x_{3}}{0.2 e^{j π (0.9)}, 0.6 e^{j π (0.8)}, 0.8 e^{j π (0.3)}}), \\ (v_{4}, \frac{x_{1}}{0.3 e^{j π (0.4)}, 0.5 e^{j π (0.6)}, 0.4 e^{j π (0.7)}}, \frac{x_{2}}{0.2 e^{j π (0.3)}, 0.6 e^{j π (0.8)}, 0.5 e^{j π (0.9)}}, \\ \frac{x_{3}}{0.4 e^{j π (0.5)}, 0.5 e^{j π (0.7)}, 0.9 e^{j π (0.9)}}), (v_{5}, \frac{x_{1}}{0.9 e^{j π (0.6)}, 0.7 e^{j π (0.7)}, 0.8 e^{j π (0.8)}}, \\ \frac{x_{2}}{0.3 e^{j π (0.4)}, 0.5 e^{j π (0.3)}, 0.2 e^{j π (0.8)}}, \frac{x_{3}}{{0.9}^{j π (0.6)}, 0.6 e^{j π (0.7)}, 0.4 e^{j π (0.8)}}) \end{aligned}} . \end{aligned}$
According to Definition 3.1, the given conditions demonstrate that $(k, A)$ is both homogeneous and completely homogeneous. In contrast, $(g, B)$ does not satisfy the criteria for being homogeneous, and therefore, it is also not completely homogeneous.
Definition 3.2
Let $(k, A)$ be a neutrosophic soft set on the ring $(R, +, \cdot)$ . Then, $(k, A)$ is defined as a neutrosophic soft ring if, for all $a \in A$ and for all $x, y \in R$ , the following conditions are satisfied: 1.
$p_{k_{a}} (x - y) \geq min {p_{k_{a}} (x), p_{k_{a}} (y)}$ .
2.
$q_{k_{a}} (x - y) \leq max {q_{k_{a}} (x), q_{k_{a}} (y)}$ .
3.
$r_{k_{a}} (x - y) \leq max {r_{k_{a}} (x), r_{k_{a}} (y)}$ .
4.
$p_{k_{a}} (x \cdot y) \geq min {p_{k_{a}} (x), p_{k_{a}} (y)}$ .
5.
$q_{k_{a}} (x \cdot y) \leq max {q_{k_{a}} (x), q_{k_{a}} (y)}$ .
6.
$r_{k_{a}} (x \cdot y) \leq max {r_{k_{a}} (x), r_{k_{a}} (y)}$ .

Definition 3.3
Let $(k, A)$ be a neutrosophic soft set on the ring $(R, +, \cdot)$ . Then, $(k, A)$ is defined as a neutrosophic soft ideal if for all $a \in A$ and for all $x, y \in R$ , the following conditions are satisfied: 1.
$p_{k_{a}} (x - y) \geq min {p_{k_{a}} (x), p_{k_{a}} (y)}$ .
2.
$q_{k_{a}} (x - y) \leq max {q_{k_{a}} (x), q_{k_{a}} (y)}$ .
3.
$r_{k_{a}} (x - y) \leq max {r_{k_{a}} (x), r_{k_{a}} (y)}$ .
4.
$p_{k_{a}} (x \cdot y) \geq max {p_{k_{a}} (x), p_{k_{a}} (y)}$ .
5.
$q_{k_{a}} (x \cdot y) \leq min {q_{k_{a}} (x), q_{k_{a}} (y)}$ .
6.
$r_{k_{a}} (x \cdot y) \leq min {r_{k_{a}} (x), r_{k_{a}} (y)}$ .

Definition 3.4
Let $(k, A)$ be a homogenous complex neutrosophic soft set on the ring $(R, +, \cdot)$ . Then $(k, A)$ defines a complex neutrosophic soft ring (CNSR) if and only if for every $a \in A$ and $x, y \in R$ the following conditions are satisfied: 1.
$T_{k_{a}} (- x) \geq T_{k_{a}} (x)$ .
2.
$I_{k_{a}} (- x) \leq I_{k_{a}} (x)$ .
3.
$F_{k_{a}} (- x) \leq F_{k_{a}} (x)$ .
4.
$T_{k_{a}} (x \cdot y) \geq min {T_{k_{a}} (x), T_{k_{a}} (y)}$ .
5.
$I_{k_{a}} (x \cdot y) \leq max {I_{k_{a}} (x), I_{k_{a}} (y)}$ .
6.
$F_{k_{a}} (x \cdot y) \leq max {F_{k_{a}} (x), F_{k_{a}} (y)}$ .
7.
$T_{k_{a}} (x + y) \geq min {T_{k_{a}} (x), T_{k_{a}} (y)}$ .
8.
$I_{k_{a}} (x + y) \leq max {I_{k_{a}} (x), I_{k_{a}} (y)}$ .
9.
$F_{k_{a}} (x + y) \leq max {F_{k_{a}} (x), F_{k_{a}} (y)}$ .

Example 3.2
Let $E$ be the set of parameters and $N$ be the set of all positive integers, where $E = N$ and $R = (Z_{12}, +, \cdot)$ be a ring. Define a mapping $k : N \to S^{Z_{12}}$ , where for any $m \in N$ ,
$\begin{aligned} T_{k_{m}} (x) & = {\begin{cases} \frac{1}{m} e^{j π} & if x \in {0, 6}, \\ 0 & if x \in Z_{12} - {0, 6} . \end{cases} \end{aligned}$

$\begin{aligned} I_{k_{m}} (x) & = {\begin{cases} 0 & if x \in {0, 6}, \\ \frac{1}{2 m} e^{j π} & if x \in Z_{12} - {0, 6} . \end{cases} \end{aligned}$

$\begin{aligned} F_{k_{m}} (x) & = {\begin{cases} 1 - \frac{1}{m} e^{j π} & if x \in {0, 6}, \\ 1 & if x \in Z_{12} - {0, 6} . \end{cases} \end{aligned}$

Then $(k, N)$ is a complex neutrosophic soft ring over $R = [(Z_{12}, +, \cdot), N]$ . By examining Definition 3.4, it can be verified that $(k, N)$ is a complex neutrosophic subring of $Z_{12}$ . Therefore, $(k, N)$ is a complex neutrosophic soft ring of $Z_{12}$ .
Definition 3.5
Suppose $(k, A)$ and $(g, B)$ are two CNSSs. Then, $(k, A)$ is a complex neutrosophic soft subring of $(g, B)$ if it satisfies the following: 1.
$(k, A) \subseteq (g, B)$ where $\subseteq$ exemplify a CNS-subset.
2.
$(k, A)$ and $(g, B)$ are both CNS-rings.

The following theorem sets the reduced CNSR conditions when compared to Definition 3.4, where the nine conditions can be reduced to only six conditions as shown in the following result.
Theorem 3.1
Consider $(k, A)$ to be a homogenous complex neutrosophic soft set on the ring $(R, +, \cdot)$ . Then, $(k, A)$ is defined as a CNSR if and only if for every $a \in A$ , and all $x, y \in R$ , the following conditions hold: 1.
$T_{k_{a}} (x - y) \geq min {T_{k_{a}} (x), T_{k_{a}} (y)}$ .
2.
$I_{k_{a}} (x - y) \leq max {I_{k_{a}} (x), I_{k_{a}} (y)}$ .
3.
$F_{k_{a}} (x - y) \leq max {F_{k_{a}} (x), F_{k_{a}} (y)}$ .
4.
$T_{k_{a}} (x \cdot y) \geq min {T_{k_{a}} (x), T_{k_{a}} (y)}$ .
5.
$I_{k_{a}} (x \cdot y) \leq max {I_{k_{a}} (x), I_{k_{a}} (y)}$ .
6.
$F_{k_{a}} (x \cdot y) \leq max {F_{k_{a}} (x), F_{k_{a}} (y)}$ .

Proof.
$(\Rightarrow)$ Let $a \in A$ and $(k, A)$ be a CNSR for all $x, y \in R$ . This implies that $(k, A)$ satisfies all nine conditions of Definition 3.4 .

To prove the first condition in this theorem, we have
$\begin{aligned} T_{k_{a}} (x - y) & = T_{k_{a}} (x + (- y)) \\ \geq min {T_{k_{a}} (x), T_{k_{a}} (- y)} \\ \geq min {T_{k_{a}} (x), T_{k_{a}} (y)} . \end{aligned}$

To prove the second condition in this theorem, we have
$\begin{aligned} I_{k_{a}} (x - y) & = I_{k_{a}} (x + (- y)) \\ \leq max {I_{k_{a}} (x), I_{k_{a}} (- y)} \\ \leq max {I_{k_{a}} (x), I_{k_{a}} (y)} . \end{aligned}$

For the third condition in this theorem, we have
$\begin{aligned} F_{k_{a}} (x - y) & = F_{k_{a}} (x + (- y)) \\ \leq max {F_{k_{a}} (x), F_{k_{a}} (- y)} \\ \leq max {F_{k_{a}} (x), F_{k_{a}} (y)} . \end{aligned}$

Since $(k, A)$ is a CNSR, and $(k, A)$ satisfies the nine conditions stated in Definition 3.4 including Conditions 4-6, therefore, $(k, A)$ satisfies the six conditions in this theorem.

$(\Leftarrow)$ Assuming that $(k, A)$ satisfies the six conditions stated in Theorem 3.1. To validate Conditions 1-3 in Definition 3.4, we use the following property.

Utilizing the additive identity $‘ ‘ 0 "$ in the set $(R, +, \cdot)$ , we have
$\begin{aligned} T_{k_{a}} (0) & = T_{k_{a}} (x - x) \\ \geq min {T_{k_{a}} (x), T_{k_{a}} (x)} \\ = T_{k_{a}} (x) . \end{aligned}$

$\begin{aligned} T_{k_{a}} (- x) & = T_{k_{a}} (0 - x) \\ \geq min {T_{k_{a}} (0), T_{k_{a}} (x)} \\ \geq min {T_{k_{a}} (x), T_{k_{a}} (x)} \\ = T_{k_{a}} (x) . \end{aligned}$

The indeterminacy membership can be proven in the similar manner as the truth membership function has been demonstrated.
$\begin{aligned} I_{k_{a}} (0) & = I_{k_{a}} (x - x) \\ \leq max {I_{k_{a}} (x), I_{k_{a}} (x)} \\ = I_{k_{a}} (x) . \end{aligned}$

$\begin{aligned} I_{k_{a}} (- x) & = I_{k_{a}} (0 - x) \\ \leq max {I_{k_{a}} (0), I_{k_{a}} (x)} \\ \leq max {I_{k_{a}} (x), I_{k_{a}} (x)} \\ = I_{k_{a}} (x) . \end{aligned}$

Similarly, we prove that $F_{k_{a}} (0) \leq F_{k_{a}} (x)$ as follows:
$\begin{aligned} F_{k_{a}} (0) & = F_{k_{a}} (x - x) \\ \leq max {F_{k_{a}} (x), F_{k_{a}} (x)} \\ = F_{k_{a}} (x) . \end{aligned}$

$\begin{aligned} F_{k_{a}} (- x) & = F_{k_{a}} (0 - x) \\ \leq max {F_{k_{a}} (0), F_{k_{a}} (x)} \\ \leq max {F_{k_{a}} (x), F_{k_{a}} (x)} \\ = F_{k_{a}} (x) . \end{aligned}$

Therefore, Conditions 1-3 in Definition 3.4 are verified.

Since $(k, A)$ satisfies the six conditions in this theorem, Conditions 4-6 in Definition 3.4 follow immediately.

To establish the bases for Conditions 7-9 in Definition 3.4, we consider the following:
$\begin{aligned} T_{k_{a}} (x + y) & = T_{k_{a}} (x - (- y)) \\ \geq min {T_{k_{a}} (x), T_{k_{a}} (- y)} \\ \geq min {T_{k_{a}} (x), T_{k_{a}} (y)} . \end{aligned}$

$\begin{aligned} I_{k_{a}} (x + y) & = I_{k_{a}} (x - (- y)) \\ \leq max {I_{k_{a}} (x), I_{k_{a}} (- y)} \\ \leq max {I_{k_{a}} (x), I_{k_{a}} (y)} . \end{aligned}$

$\begin{aligned} F_{k_{a}} (x + y) & = F_{k_{a}} (x - (- y)) \\ \leq max {F_{k_{a}} (x), F_{k_{a}} (- y)} \\ \leq max {F_{k_{a}} (x), F_{k_{a}} (y)} . \end{aligned}$

Since $(k, A)$ satisfies the six conditions in this theorem, therefore $(k, A)$ is a CNSR.

The next theorem highlights the relation between CNSRs and NSRs.
Theorem 3.2
Let $(k, A) = {⟨ a; T_{k_{a}} (x), I_{k_{a}} (x), F_{k_{a}} (x) ⟩ : a \in A, x \in R}$ , be a homogenous CNSS in the ring $(R, +, \cdot)$ . Assuming that $(k, A)$ generates the two NSSs
$\begin{aligned} (m, A) & = {⟨ a; p_{m_{a}} (x), q_{m_{a}} (x), r_{m_{a}} (x) ⟩ : a \in A, x \in R} . \\ (n, A) & = {⟨ a; μ_{n_{a}} (x), ν_{n_{a}} (x), ω_{n_{a}} (x) ⟩ : a \in A, x \in R} . \end{aligned}$

Then, $(k, A)$ is a CNSR of $R$ if and only if both $(m, A)$ and $(n, A)$ are NSRs.
Proof.
$(\Rightarrow)$ To prove this direction, it is important to show how the six conditions outlined in Definition 3.2 are satisfied. Consider that $(k, A)$ is a CNSR of $R$ , then for all $a \in A$ and $x, y \in R$ , we have
$\begin{aligned} p_{k_{a}} (x - y) e^{j μ_{k_{a}} (x - y)} & = T_{k_{a}} (x - y) \\ \geq min {T_{k_{a}} (x), T_{k_{a}} (y)} \\ = min {p_{k_{a}} (x) e^{j μ_{k_{a}} (x)}, p_{k_{a}} (y) e^{j μ_{k_{a}} (y)}} \\ = min {p_{k_{a}} (x), p_{k_{a}} (y)} e^{j min {μ_{k_{a}} (x), μ_{k_{a}} (y)}} . \end{aligned}$
Since $(k, A)$ is homogenous, then we obtain the following:
$\begin{aligned} p_{k_{a}} (x - y) \geq min {p_{k_{a}} (x), p_{k_{a}} (y)}, \\ μ_{k_{a}} (x - y) \geq min {μ_{k_{a}} (x), μ_{k_{a}} (y)} . \end{aligned}$

In the same manner, we have
$\begin{aligned} q_{k_{a}} (x - y) e^{j ν_{k_{a}} (x - y)} & = I_{k_{a}} (x - y) \\ \leq max {I_{k_{a}} (x), I_{k_{a}} (y)} \\ = max {q_{k_{a}} (x) e^{j ν_{k_{a}} (x)}, q_{k_{a}} (y) e^{j ν_{k_{a}} (y)}} \\ = max {q_{k_{a}} (x), q_{k_{a}} (y)} e^{j max {ν_{k_{a}} (x), ν_{k_{a}} (y)}} . \end{aligned}$
Since $(k, A)$ is homogenous, then we obtain the following inequalities:
$\begin{aligned} q_{k_{a}} (x - y) & \leq max {q_{k_{a}} (x), q_{k_{a}} (y)}, \\ ν_{k_{a}} (x - y) & \leq max {ν_{k_{a}} (x), ν_{k_{a}} (y)} . \end{aligned}$

Next,
$\begin{aligned} r_{k_{a}} (x - y) e^{j ω_{k_{a}} (x - y)} & = F_{k_{a}} (x - y) \\ \leq max {F_{k_{a}} (x), F_{k_{a}} (y)} \\ = max {r_{k_{a}} (x) e^{j ω_{k_{a}} (x)}, r_{k_{a}} (y) e^{j ω_{k_{a}} (y)}} \\ = max {r_{k_{a}} (x), r_{k_{a}} (y)} e^{j max {ω_{k_{a}} (x), ω_{k_{a}} (y)}} . \end{aligned}$
Since $(k, A)$ is homogenous, then we get the following:
$\begin{aligned} r_{k_{a}} (x - y) & \leq max {r_{k_{a}} (x), r_{k_{a}} (y)}, \\ ω_{k_{a}} (x - y) & \leq max {ω_{k_{a}} (x), ω_{k_{a}} (y)} . \end{aligned}$
Therefore, Conditions 1-3 are verified.

To prove Conditions 4-6, we need to show that:
$\begin{aligned} p_{k_{a}} (x \cdot y) e^{j μ_{k_{a}} (x \cdot y)} & = T_{k_{a}} (x \cdot y) \\ \geq min {T_{k_{a}} (x), T_{k_{a}} (y)} \\ = min {p_{k_{a}} (x) e^{j μ_{k_{a}} (x)}, p_{k_{a}} (y) e^{j μ_{k_{a}} (y)}} \\ = min {p_{k_{a}} (x), p_{k_{a}} (y)} e^{j min {μ_{k_{a}} (x), μ_{k_{a}} (y)}} . \end{aligned}$
Since $(k, A)$ is homogenous, then
$\begin{aligned} p_{k_{a}} (x \cdot y) & \geq min {p_{k_{a}} (x), p_{k_{a}} (y)}, \\ μ_{k_{a}} (x \cdot y) & \geq min {μ_{k_{a}} (x), μ_{k_{a}} (y)} . \end{aligned}$

In the same manner, we have
$\begin{aligned} q_{k_{a}} (x \cdot y) e^{j ν_{k_{a}} (x \cdot y)} & = I_{k_{a}} (x \cdot y) \\ \leq max {I_{k_{a}} (x), I_{k_{a}} (y)} \\ = max {q_{k_{a}} (x) e^{j ν_{k_{a}} (x)}, q_{k_{a}} (y) e^{j ν_{k_{a}} (y)} \\ = max {q_{k_{a}} (x), q_{k_{a}} (y)} e^{j max {ν_{k_{a}} (x), ν_{k_{a}} (y)}} . \end{aligned}$
Since $(k, A)$ is homogenous, then we have
$\begin{aligned} q_{k_{a}} (x \cdot y) & \leq max {q_{k_{a}} (x), q_{k_{a}} (y)}, \\ ν_{k_{a}} (x \cdot y) & \leq max {ν_{k_{a}} (x), ν_{k_{a}} (y)} . \end{aligned}$

Next,
$\begin{aligned} r_{k_{a}} (x \cdot y) e^{j ω_{k_{a}} (x \cdot y)} & = F_{k_{a}} (x \cdot y) \\ \leq max {F_{k_{a}} (x), F_{k_{a}} (y)} \\ = max {r_{k_{a}} (x) e^{j ω_{k_{a}} (x)}, r_{k_{a}} (y) e^{j ω_{k_{a}} (y)}} \\ = max {r_{k_{a}} (x), r_{k_{a}} (y)} e^{j max {ω_{k_{a}} (x), ω_{k_{a}} (y)}} . \end{aligned}$
Since $(k, A)$ is homogenous, then
$\begin{aligned} r_{k_{a}} (x \cdot y) \leq max {r_{k_{a}} (x), r_{k_{a}} (y)}, \\ ω_{k_{a}} (x \cdot y) \leq max {ω_{k_{a}} (x), ω_{k_{a}} (y)} . \end{aligned}$
So, Conditions 4-6 are verified.

Therefore, (m,A) and (n,A) are NSRs.

$(\Leftarrow)$ To prove the other direction of this theorem, it must satisfy the pre-stated six conditions associated with Theorem 3.1.

Let $(m, A)$ and $(n, A)$ be NSRs. To prove that $(k, A)$ is a CNSR, we have to show that
$\begin{aligned} T_{k_{a}} (x - y) & = p_{k_{a}} (x - y) e^{j μ_{k_{a}} (x - y)} \\ \geq min {p_{k_{a}} (x), p_{k_{a}} (y)} e^{j min {μ_{k_{a}} (x), μ_{k_{a}} (y)}} \\ = min {p_{k_{a}} (x) e^{j μ_{k_{a}} (x)}, p_{k_{a}} (y) e^{j μ_{k_{a}} (y)}} \\ = min {T_{k_{a}} (x), T_{k_{a}} (y)} . \end{aligned}$
Since $(k, A)$ is homogenous, then the truth term is written as:
$\begin{aligned} T_{k_{a}} (x - y) \geq min {T_{k_{a}} (x), T_{k_{a}} (y)} . \end{aligned}$
In the same manner, we have,
$\begin{aligned} I_{k_{a}} (x - y) & = q_{k_{a}} (x - y) e^{j ν_{k_{a}} (x - y)} \\ \leq max {q_{k_{a}} (x), q_{k_{a}} (y)} e^{j max {ν_{k_{a}} (x), ν_{k_{a}} (y)}} \\ = max {q_{k_{a}} (x) e^{j ν_{k_{a}} (x)}, q_{k_{a}} (y) e^{j ν_{k_{a}} (y)}} \\ = max {I_{k_{a}} (x), I_{k_{a}} (y)} . \end{aligned}$
Since $(k, A)$ is homogenous, then indeterminacy membership function is expressed as:
$\begin{aligned} I_{k_{a}} (x - y) \leq max {I_{k_{a}} (x), I_{k_{a}} (y)} . \end{aligned}$
Next,
$\begin{aligned} F_{k_{a}} (x - y) & = r_{k_{a}} (x - y) e^{j ω_{k_{a}} (x - y)} \\ \leq max {r_{k_{a}} (x), r_{k_{a}} (y)} e^{j max {ω_{k_{a}} (x), ω_{k_{a}} (y)}} \\ = max {r_{k_{a}} (x) e^{j ω_{k_{a}} (x)}, r_{k_{a}} (y) e^{j ω_{k_{a}} (y)}} \\ = max {F_{k_{a}} (x), F_{k_{a}} (y)} . \end{aligned}$
Since $(k, A)$ is homogenous, then the falsity term is shown in the following inequality:
$\begin{aligned} F_{k_{a}} (x - y) \leq max {F_{k_{a}} (x), F_{k_{a}} (y)} . \end{aligned}$
So, Conditions 1-3 are verified.

To prove Conditions 4-6, we need to show that:
$\begin{aligned} T_{k_{a}} (x \cdot y) & = p_{k_{a}} (x \cdot y) e^{j μ_{k_{a}} (x \cdot y)} \\ \geq min {p_{k_{a}} (x), p_{k_{a}} (y)} e^{j min {μ_{k_{a}} (x), μ_{k_{a}} (y)}} \\ = min {p_{k_{a}} (x) e^{j μ_{k_{a}} (x)}, p_{k_{a}} (y) e^{j μ_{k_{a}} (y)}} \\ = min {T_{k_{a}} (x), T_{k_{a}} (y)} . \end{aligned}$
Since $(k, A)$ is homogenous, then
$\begin{aligned} T_{k_{a}} (x \cdot y) \geq min {T_{k_{a}} (x), T_{k_{a}} (y)} . \end{aligned}$
In the same manner, we have,
$\begin{aligned} I_{k_{a}} (x \cdot y) & = q_{k_{a}} (x \cdot y) e^{j ν_{k_{a}} (x \cdot y)} \\ \leq max {q_{k_{a}} (x), q_{k_{a}} (y)} e^{j max {ν_{k_{a}} (x), ν_{k_{a}} (y)}} \\ = max {q_{k_{a}} (x) e^{j ν_{k_{a}} (x)}, q_{k_{a}} (y) e^{j ν_{k_{a}} (y)}} \\ = max {I_{k_{a}} (x), I_{k_{a}} (y)} . \end{aligned}$
Since $(k, A)$ is homogenous, then we get
$\begin{aligned} I_{k_{a}} (x \cdot y) \leq max {I_{k_{a}} (x), I_{k_{a}} (y)} . \end{aligned}$
Next,
$\begin{aligned} F_{k_{a}} (x \cdot y) & = r_{k_{a}} (x \cdot y) e^{j ω_{k_{a}} (x \cdot y)} \\ \leq max {r_{k_{a}} (x), r_{k_{a}} (y)} e^{j max {ω_{k_{a}} (x), ω_{k_{a}} (y)}} \\ = max {r_{k_{a}} (x) e^{j ω_{k_{a}} (x)}, r_{k_{a}} (y) e^{j ω_{k_{a}} (y)}} \\ = max {F_{k_{a}} (x), F_{k_{a}} (y)} . \end{aligned}$
Since $(k, A)$ is homogenous, then we have
$\begin{aligned} F_{k_{a}} (x \cdot y) \leq max {F_{k_{a}} (x), F_{k_{a}} (y)} . \end{aligned}$
Therefore, the pre-stated six conditions in Theorem 3.1 have been satisfied, which implies that $(k, A)$ is a CNSR.

The next theorem deals with the six conditions associated with CNSRs in Theorem 3.1 to highlight the intersection property.
Theorem 3.3
Let $(k, A)$ and $(g, B)$ be two homogenous-NSSs on $(R, +, \cdot)$ , where $(k, A)$ is homogenous with $(g, B)$ . If $(k, A)$ and $(g, B)$ are two complex neutrosophic soft rings on $R$ , then the intersection $(k, A) \cap (g, B)$ is a CNSR.
Proof.
Let $x, y \in R$ and $ϵ \in A \cap B$ . Then, by Theorem 3.1, it is sufficient to demonstrate that 1.
$T_{k \cap g_{ϵ}} (x - y) \geq min {T_{k_{ϵ}} (x), T_{g_{ϵ}} (y)}$ .
2.
$I_{k \cap g_{ϵ}} (x - y) \leq max {I_{k_{ϵ}} (x), I_{g_{ϵ}} (y)}$ .
3.
$F_{k \cap g_{ϵ}} (x - y) \leq max {F_{k_{ϵ}} (x), F_{g_{ϵ}} (y)}$ .
4.
$T_{k \cap g_{ϵ}} (x \cdot y) \geq min {T_{k_{ϵ}} (x), T_{g_{ϵ}} (y)}$ .
5.
$I_{k \cap g_{ϵ}} (x \cdot y) \leq max {I_{k_{ϵ}} (x), I_{g_{ϵ}} (y)}$ .
6.
$F_{k \cap g_{ϵ}} (x \cdot y) \leq max {F_{k_{ϵ}} (x), F_{g_{ϵ}} (y)}$ .

Looking into the first condition, we have:
$\begin{aligned} T_{k \cap g_{ϵ}} (x - y) & = p_{k \cap g_{ϵ}} (x - y) e^{j μ_{k \cap g_{ϵ}} (x - y)} \\ = min {p_{k_{ϵ}} (x - y), p_{g_{ϵ}} (x - y)} e^{j min {μ_{k_{ϵ}} (x - y), μ_{g_{ϵ}} (x - y)}} \\ = min {p_{k_{ϵ}} (x - y) e^{j μ_{k_{ϵ}} (x - y)}, p_{g_{ϵ}} (x - y) e^{j μ_{g_{ϵ}} (x - y)}} \\ \geq min {min {p_{k_{ϵ}} (x) e^{j μ_{k_{ϵ}} (x)}, p_{k_{ϵ}} (y) e^{j μ_{k_{ϵ}} (y)}}, min {p_{g_{ϵ}} (x) e^{j μ_{g_{ϵ}} (x)}, p_{g_{ϵ}} (y) e^{j μ_{g_{ϵ}} (y)}}} \\ = min {min {p_{k_{ϵ}} (x) e^{j μ_{_{ϵ}} (x)}, p_{g_{ϵ}} (x) e^{j μ_{g_{ϵ}} (x)}}, min {p_{k_{ϵ}} (y) e^{j μ_{k_{ϵ}} (y)}, p_{g_{ϵ}} (y) e^{j μ_{g_{ϵ}} (y)}}} \\ = min {min {p_{k_{ϵ}} (x), p_{g_{ϵ}} (x)} e^{j min {μ_{k_{ϵ}} (x), μ_{g_{ϵ}} (x)}}}, min {p_{k_{ϵ}} (y), p_{g_{ϵ}} (y)} e^{j min {μ_{k_{ϵ}} (y), μ_{g_{ϵ}} (y)}}} \\ = min {p_{k \cap g_{ϵ}} (x) e^{j μ_{k \cap g_{ϵ}} (x)}, p_{k \cap g_{ϵ}} (y) e^{j μ_{k \cap g_{ϵ}} (y)}} \\ = min {T_{k \cap g_{ϵ}} (x), T_{k \cap g_{ϵ}} (y)} . \end{aligned}$

Considering the second condition, we have
$\begin{aligned} I_{k \cap g_{ϵ}} (x - y) & = q_{k \cap g_{ϵ}} (x - y) e^{j ν_{k \cap g_{ϵ}} (x - y)} \\ = max {q_{k_{ϵ}} (x - y), q_{g_{ϵ}} (x - y)} e^{j max {ν_{k_{ϵ}} (x - y), ν_{g_{ϵ}} (x - y)}} \\ = max {q_{k_{ϵ}} (x - y) e^{j ν_{k_{ϵ}} (x - y)}, q_{g_{ϵ}} (x - y) e^{j ν_{g_{ϵ}} (x - y)}} \\ \leq max {max {q_{k_{ϵ}} (x) e^{j ν_{k_{ϵ}} (x)}, q_{k_{ϵ}} (y) e^{j ν_{k_{ϵ}} (y)}}, max {q_{g_{ϵ}} (x) e^{j ν_{g_{ϵ}} (x)}, q_{g_{ϵ}} (y) e^{j ν_{g_{ϵ}} (y)}}} \\ = max {max {q_{k_{ϵ}} (x) e^{j ν_{_{ϵ}} (x)}, q_{g_{ϵ}} (x) e^{j ν_{g_{ϵ}} (x)}}, max {q_{k_{ϵ}} (y) e^{j ν_{k_{ϵ}} (y)}, q_{g_{ϵ}} (y) e^{j ν_{g_{ϵ}} (y)}}} \\ = max {max {q_{k_{ϵ}} (x), q_{g_{ϵ}} (x)} e^{j max {ν_{k_{ϵ}} (x), ν_{g_{ϵ}} (x)}}}, max {q_{k_{ϵ}} (y), q_{g_{ϵ}} (y)} e^{j max {ν_{k_{ϵ}} (y), ν_{g_{ϵ}} (y)}}} \\ = max {q_{k \cap g_{ϵ}} (x) e^{j ν_{k \cap g_{ϵ}} (x)}, q_{k \cap g_{ϵ}} (y) e^{j ν_{k \cap g_{ϵ}} (y)}} \\ = max {I_{k \cap g_{ϵ}} (x), I_{k \cap g_{ϵ}} (y)} . \end{aligned}$

Examining the third condition,
$\begin{aligned} F_{k \cap g_{ϵ}} (x - y) & = r_{k \cap g_{ϵ}} (x - y) e^{j ω_{k \cap g_{ϵ}} (x - y)} \\ = max {r_{k_{ϵ}} (x - y), r_{g_{ϵ}} (x - y)} e^{j max {ω_{k_{ϵ}} (x - y), ω_{g_{ϵ}} (x - y)}} \\ = max {r_{k_{ϵ}} (x - y) e^{j ω_{k_{ϵ}} (x - y)}, r_{g_{ϵ}} (x - y) e^{j ω_{g_{ϵ}} (x - y)}} \\ \leq max {max {r_{k_{ϵ}} (x) e^{j ω_{k_{ϵ}} (x)}, r_{k_{ϵ}} (y) e^{j ω_{k_{ϵ}} (y)}}, max {r_{g_{ϵ}} (x) e^{j ω_{g_{ϵ}} (x)}, r_{g_{ϵ}} (y) e^{j ω_{g_{ϵ}} (y)}}} \\ = max {max {r_{k_{ϵ}} (x) e^{j ω_{_{ϵ}} (x)}, r_{g_{ϵ}} (x) e^{j ω_{g_{ϵ}} (x)}}, max {r_{k_{ϵ}} (y) e^{j ω_{k_{ϵ}} (y)}, r_{g_{ϵ}} (y) e^{j ω_{g_{ϵ}} (y)}} \\ = max {max {r_{k_{ϵ}} (x), r_{g_{ϵ}} (x)} e^{j max {ω_{k_{ϵ}} (x), ω_{g_{ϵ}} (x)}}}, max {r_{k_{ϵ}} (y), r_{g_{ϵ}} (y)} e^{j max {ω_{k_{ϵ}} (y), ω_{g_{ϵ}} (y)}}} \\ = max {r_{k \cap g_{ϵ}} (x) e^{j ω_{k \cap g_{ϵ}} (x)}, r_{k \cap g_{ϵ}} (y) e^{j ω_{k \cap g_{ϵ}} (y)}} \\ = max {F_{k \cap g_{ϵ}} (x), F_{k \cap g_{ϵ}} (y)} . \end{aligned}$

Next, for the fourth condition, we have
$\begin{aligned} T_{k \cap g_{ϵ}} (x \cdot y) & = p_{k \cap g_{ϵ}} (x \cdot y) e^{j μ_{k \cap g_{ϵ}} (x \cdot y)} \\ = min {p_{k_{ϵ}} (x \cdot y), p_{g_{ϵ}} (x \cdot y)} e^{j min {μ_{k_{ϵ}} (x \cdot y), μ_{g_{ϵ}} (x \cdot y)}} \\ = min {p_{k_{ϵ}} (x \cdot y) e^{j μ_{k_{ϵ}} (x \cdot y)}, p_{g_{ϵ}} (x \cdot y) e^{j μ_{g_{ϵ}} (x \cdot y)}} \\ \geq min {min {p_{k_{ϵ}} (x) e^{j μ_{k_{ϵ}} (x)}, p_{k_{ϵ}} (y) e^{j μ_{k_{ϵ}} (y)}}, min {p_{g_{ϵ}} (x) e^{j μ_{g_{ϵ}} (x)}, p_{g_{ϵ}} (y) e^{j μ_{g_{ϵ}} (y)}}} \\ = min {min {p_{k_{ϵ}} (x) e^{j μ_{_{ϵ}} (x)}, p_{g_{ϵ}} (x) e^{j μ_{g_{ϵ}} (x)}}, min {p_{k_{ϵ}} (y) e^{j μ_{k_{ϵ}} (y)}, p_{g_{ϵ}} (y) e^{j μ_{g_{ϵ}} (y)}}} \\ = min {min {p_{k_{ϵ}} (x), p_{g_{ϵ}} (x)} e^{j min {μ_{k_{ϵ}} (x), μ_{g_{ϵ}} (x)}}}, min {p_{k_{ϵ}} (y), p_{g_{ϵ}} (y)} e^{j min {μ_{k_{ϵ}} (y), μ_{g_{ϵ}} (y)}}} \\ = min {p_{k \cap g_{ϵ}} (x) e^{j μ_{k \cap g_{ϵ}} (x)}, p_{k \cap g_{ϵ}} (y) e^{j μ_{k \cap g_{ϵ}} (y)}} \\ = min {T_{k \cap g_{ϵ}} (x), T_{k \cap g_{ϵ}} (y)} . \end{aligned}$

Considering the fifth condition, we obtain
$\begin{aligned} I_{k \cap g_{ϵ}} (x \cdot y) & = q_{k \cap g_{ϵ}} (x \cdot y) e^{j ν_{k \cap g_{ϵ}} (x \cdot y)} \\ = max {q_{k_{ϵ}} (x \cdot y), q_{g_{ϵ}} (x \cdot y)} e^{j max {ν_{k_{ϵ}} (x \cdot y), ν_{g_{ϵ}} (x \cdot y)}} \\ = max {q_{k_{ϵ}} (x \cdot y) e^{j ν_{k_{ϵ}} (x \cdot y)}, q_{g_{ϵ}} (x \cdot y) e^{j ν_{g_{ϵ}} (x \cdot y)}} \\ \leq max {max q_{k_{ϵ}} (x) e^{j ν_{k_{ϵ}} (x)}, q_{k_{ϵ}} (y) e^{j ν_{k_{ϵ}} (y)}}, max {q_{g_{ϵ}} (x) e^{j ν_{g_{ϵ}} (x)}, q_{g_{ϵ}} (y) e^{j ν_{g_{ϵ}} (y)}}} \\ = max {max q_{k_{ϵ}} (x) e^{j ν_{_{ϵ}} (x)}, q_{g_{ϵ}} (x) e^{j ν_{g_{ϵ}} (x)}}, max {q_{k_{ϵ}} (y) e^{j ν_{k_{ϵ}} (y)}, q_{g_{ϵ}} (y) e^{j ν_{g_{ϵ}} (y)}}} \\ = max {max {q_{k_{ϵ}} (x), q_{g_{ϵ}} (x)} e^{j max {ν_{k_{ϵ}} (x), ν_{g_{ϵ}} (x)}}}, max {q_{k_{ϵ}} (y), q_{g_{ϵ}} (y)} e^{j max {ν_{k_{ϵ}} (y), ν_{g_{ϵ}} (y)}}} \\ = max {q_{k \cap g_{ϵ}} (x) e^{j ν_{k \cap g_{ϵ}} (x)}, q_{k \cap g_{ϵ}} (y) e^{j ν_{k \cap g_{ϵ}} (y)}} \\ = max {I_{k \cap g_{ϵ}} (x), I_{k \cap g_{ϵ}} (y)} . \end{aligned}$

Lastly, for the final condition,
$\begin{aligned} F_{k \cap g_{ϵ}} (x \cdot y) & = r_{k \cap g_{ϵ}} (x \cdot y) e^{j ω_{k \cap g_{ϵ}} (x \cdot y)} \\ = max {r_{k_{ϵ}} (x \cdot y), r_{g_{ϵ}} (x \cdot y)} e^{j max {ω_{k_{ϵ}} (x \cdot y), ω_{g_{ϵ}} (x \cdot y)}} \\ = max {r_{k_{ϵ}} (x \cdot y) e^{j ω_{k_{ϵ}} (x \cdot y)}, r_{g_{ϵ}} (x \cdot y) e^{j ω_{g_{ϵ}} (x \cdot y)}} \\ \leq max {max {r_{k_{ϵ}} (x) e^{j ω_{k_{ϵ}} (x)}, r_{k_{ϵ}} (y) e^{j ω_{k_{ϵ}} (y)}}, max {r_{g_{ϵ}} (x) e^{j ω_{g_{ϵ}} (x)}, r_{g_{ϵ}} (y) e^{j ω_{g_{ϵ}} (y)}}} \\ = max {max {r_{k_{ϵ}} (x) e^{j ω_{_{ϵ}} (x)}, r_{g_{ϵ}} (x) e^{j ω_{g_{ϵ}} (x)}}, max {r_{k_{ϵ}} (y) e^{j ω_{k_{ϵ}} (y)}, r_{g_{ϵ}} (y) e^{j ω_{g_{ϵ}} (y)}}} \\ = max {max {r_{k_{ϵ}} (x), r_{g_{ϵ}} (x)} e^{j max {ω_{k_{ϵ}} (x), ω_{g_{ϵ}} (x)}}}, max {r_{k_{ϵ}} (y), r_{g_{ϵ}} (y)} e^{j max {ω_{k_{ϵ}} (y), ω_{g_{ϵ}} (y)}}} \\ = max {r_{k \cap g_{ϵ}} (x) e^{j ω_{k \cap g_{ϵ}} (x)}, r_{k \cap g_{ϵ}} (y) e^{j ω_{k \cap g_{ϵ}} (y)}} \\ = max {F_{k \cap g_{ϵ}} (x), F_{k \cap g_{ϵ}} (y)} . \end{aligned}$

Therefore, $(k, A) \cap (g, B)$ is a CNSR.
4. Complex Neutrosophic Soft Ideal Approach

Definition 4.1
Let $(k, A)$ be a homogenous complex neutrosophic soft set on the ring $(R, +, \cdot)$ . The set $(k, A)$ is considered as a complex neutrosophic soft ideal (CNSI) if and only if, for all $a \in A$ and for all $x, y \in R$ the following holds: 1.
$T_{k_{a}} (x - y) \geq min {T_{k_{a}} (x), T_{k_{a}} (y)}$ .
2.
$I_{k_{a}} (x - y) \leq max {I_{k_{a}} (x), I_{k_{a}} (y)}$ .
3.
$F_{k_{a}} (x - y) \leq max {F_{k_{a}} (x), F_{k_{a}} (y)}$ .
4.
$T_{k_{a}} (x \cdot y) \geq max {T_{k_{a}} (x), T_{k_{a}} (y)}$ .
5.
$I_{k_{a}} (x \cdot y) \leq min {I_{k_{a}} (x), I_{k_{a}} (y)}$ .
6.
$F_{k_{a}} (x \cdot y) \leq min {F_{k_{a}} (x), F_{k_{a}} (y)}$ .

The relation between CNSIs and NSIs is shown in the following theorem.
Theorem 4.1
Let $(k, A) = {⟨ a; T_{k_{a}} (x), I_{k_{a}} (x), F_{k_{a}} (x) ⟩ : a \in A, x \in R},$ be a homogenous CNSS in the ring $(R, +, \cdot)$ . Assuming that $(k, A)$ generates the two NSSs

$\begin{aligned} (m, A) & = {⟨ a; p_{m_{a}} (x), q_{m_{a}} (x), r_{m_{a}} (x) ⟩ : a \in A, x \in R}, \\ (n, A) & = {⟨ a; μ_{n_{a}} (x), ν_{n_{a}} (x), ω_{n_{a}} (x) ⟩ : a \in A, x \in R} . \end{aligned}$

Then, $(k, A)$ is a CNSI of $R$ if and only if both $(m, A)$ and $(n, A)$ are NSIs.
Proof.
$(\Rightarrow)$ To prove this direction, we show that the six conditions outlined in Definition 3.3 are satisfied. Suppose $(k, A)$ is a CNSI of $R$ , then for all $a \in A$ and $x, y \in R$ , we have
$\begin{aligned} p_{k_{a}} (x - y) e^{j μ_{k_{a}} (x - y)} & = T_{k_{a}} (x - y) \\ \geq min {T_{k_{a}} (x), T_{k_{a}} (y)} \\ = min {p_{k_{a}} (x) e^{j μ_{k_{a}} (x)}, p_{k_{a}} (y) e^{j μ_{k_{a}} (y)}} \\ = min {p_{k_{a}} (x), p_{k_{a}} (y)} e^{j min {μ_{k_{a}} (x), μ_{k_{a}} (y)}} . \end{aligned}$
Since $(k, A)$ is homogenous, then we get the following:
$\begin{aligned} p_{k_{a}} (x - y) & \geq min {p_{k_{a}} (x), p_{k_{a}} (y)}, \\ μ_{k_{a}} (x - y) & \geq min {μ_{k_{a}} (x), μ_{k_{a}} (y)} . \end{aligned}$

In the same manner, we have
$\begin{aligned} q_{k_{a}} (x - y) e^{j ν_{k_{a}} (x - y)} & = I_{k_{a}} (x - y) \\ \leq max {I_{k_{a}} (x), I_{k_{a}} (y)} \\ = max {q_{k_{a}} (x) e^{j ν_{k_{a}} (x)}, q_{k_{a}} (y) e^{j ν_{k_{a}} (y)}} \\ = max {q_{k_{a}} (x), q_{k_{a}} (y)} e^{j max {ν_{k_{a}} (x), ν_{k_{a}} (y)}} . \end{aligned}$
Since $(k, A)$ is homogenous, then we obtain the following two inequalities: inequalities:
$\begin{aligned} q_{k_{a}} (x - y) & \leq max {q_{k_{a}} (x), q_{k_{a}} (y)}, \\ ν_{k_{a}} (x - y) & \leq max {ν_{k_{a}} (x), ν_{k_{a}} (y)} . \end{aligned}$

Next,
$\begin{aligned} r_{k_{a}} (x - y) e^{j ω_{k_{a}} (x - y)} & = F_{k_{a}} (x - y) \\ \leq max {F_{k_{a}} (x), F_{k_{a}} (y)} \\ = max {r_{k_{a}} (x) e^{j ω_{k_{a}} (x)}, r_{k_{a}} (y) e^{j ω_{k_{a}} (y)}} \\ = max {r_{k_{a}} (x), r_{k_{a}} (y)} e^{j max {ω_{k_{a}} (x), ω_{k_{a}} (y)}} . \end{aligned}$

Since $(k, A)$ is homogenous, then we have
$\begin{aligned} r_{k_{a}} (x - y) & \leq max {r_{k_{a}} (x), r_{k_{a}} (y)}, \\ ω_{k_{a}} (x - y) & \leq max {ω_{k_{a}} (x), ω_{k_{a}} (y)} . \end{aligned}$
Therefore, Conditions 1-3 are verified.

To prove Conditions 4-6, we need to show that:
$\begin{aligned} p_{k_{a}} (x \cdot y) e^{j μ_{k_{a}} (x \cdot y)} & = T_{k_{a}} (x \cdot y) \\ \geq max {T_{k_{a}} (x), T_{k_{a}} (y)} \\ = max {p_{k_{a}} (x) e^{j μ_{k_{a}} (x)}, p_{k_{a}} (y) e^{j μ_{k_{a}} (y)}} \\ = max {p_{k_{a}} (x), p_{k_{a}} (y)} e^{j max {μ_{k_{a}} (x), μ_{k_{a}} (y)}} . \end{aligned}$

Since $(k, A)$ is homogenous, then we obtain the following two inequalities:
$\begin{aligned} p_{k_{a}} (x \cdot y) & \geq max {p_{k_{a}} (x), p_{k_{a}} (y)}, \\ μ_{k_{a}} (x \cdot y) & \geq max {μ_{k_{a}} (x), μ_{k_{a}} (y)} . \end{aligned}$

In the same manner, we have
$\begin{aligned} q_{k_{a}} (x \cdot y) e^{j ν_{k_{a}} (x \cdot y)} & = I_{k_{a}} (x \cdot y) \\ \leq min {I_{k_{a}} (x), I_{k_{a}} (y)} \\ = min {q_{k_{a}} (x) e^{j ν_{k_{a}} (x)}, q_{k_{a}} (y) e^{j ν_{k_{a}} (y)}} \\ = min {q_{k_{a}} (x), q_{k_{a}} (y)} e^{j min {ν_{k_{a}} (x), ν_{k_{a}} (y)}} . \end{aligned}$

Since $(k, A)$ is homogenous, then
$\begin{aligned} q_{k_{a}} (x \cdot y) & \leq min {q_{k_{a}} (x), q_{k_{a}} (y)}, \\ ν_{k_{a}} (x \cdot y) & \leq min {ν_{k_{a}} (x), ν_{k_{a}} (y)} . \end{aligned}$

Next,
$\begin{aligned} r_{k_{a}} (x \cdot y) e^{j ω_{k_{a}} (x \cdot y)} & = F_{k_{a}} (x \cdot y) \\ \leq min {F_{k_{a}} (x), F_{k_{a}} (y)} \\ = min {r_{k_{a}} (x) e^{j ω_{k_{a}} (x)}, r_{k_{a}} (y) e^{j ω_{k_{a}} (y)}} \\ = min {r_{k_{a}} (x), r_{k_{a}} (y)} e^{j min {ω_{k_{a}} (x), ω_{k_{a}} (y)}} . \end{aligned}$

Since $(k, A)$ is homogenous, then
$\begin{aligned} r_{k_{a}} (x \cdot y) & \leq min {r_{k_{a}} (x), r_{k_{a}} (y)}, \\ ω_{k_{a}} (x \cdot y) & \leq min {ω_{k_{a}} (x), ω_{k_{a}} (y)} . \end{aligned}$
Therefore, (m,A) and (n,A) are NSIs.

$(\Leftarrow)$ To prove the other direction of this theorem, the six pre-stated conditions associated with Definition 4.1 must be satisfied.

Let $(m, A)$ and $(n, A)$ be NSIs. To prove that $(k, A)$ is a CNSI, we have to show that
$\begin{aligned} T_{k_{a}} (x - y) & = p_{k_{a}} (x - y) e^{j μ_{k_{a}} (x - y)} \\ \geq min {p_{k_{a}} (x), p_{k_{a}} (y)} e^{j min {μ_{k_{a}} (x), μ_{k_{a}} (y)}} \\ = min {p_{k_{a}} (x) e^{j μ_{k_{a}} (x)}, p_{k_{a}} (y) e^{j μ_{k_{a}} (y)}} \\ = min {T_{k_{a}} (x), T_{k_{a}} (y)} . \end{aligned}$
Since $(k, A)$ is homogenous, then the truth membership term is written as:
$\begin{aligned} T_{k_{a}} (x - y) \geq min {T_{k_{a}} (x), T_{k_{a}} (y)} . \end{aligned}$
In the same manner, we have,
$\begin{aligned} I_{k_{a}} (x - y) & = q_{k_{a}} (x - y) e^{j ν_{k_{a}} (x - y)} \\ \leq max {q_{k_{a}} (x), q_{k_{a}} (y)} e^{j max {ν_{k_{a}} (x), ν_{k_{a}} (y)}} \\ = max {q_{k_{a}} (x) e^{j ν_{k_{a}} (x)}, q_{k_{a}} (y) e^{j ν_{k_{a}} (y)}} \\ = max {I_{k_{a}} (x), I_{k_{a}} (y)} . \end{aligned}$
Since $(k, A)$ is homogenous, then the indeterminacy term is written as follows:
$\begin{aligned} I_{k_{a}} (x - y) \leq max {I_{k_{a}} (x), I_{k_{a}} (y)} . \end{aligned}$
Next,
$\begin{aligned} F_{k_{a}} (x - y) & = r_{k_{a}} (x - y) e^{j ω_{k_{a}} (x - y)} \\ \leq max {r_{k_{a}} (x), r_{k_{a}} (y)} e^{j max {ω_{k_{a}} (x), ω_{k_{a}} (y)}} \\ = max {r_{k_{a}} (x) e^{j ω_{k_{a}} (x)}, r_{k_{a}} (y) e^{j ω_{k_{a}} (y)}} \\ = max {F_{k_{a}} (x), F_{k_{a}} (y)} . \end{aligned}$
Since $(k, A)$ is homogenous, then the falsity term is expressed as:
$\begin{aligned} F_{k_{a}} (x - y) \leq max {F_{k_{a}} (x), F_{k_{a}} (y)} . \end{aligned}$
So, Conditions 1-3 are verified.

To prove Conditions 4-6, we need to show that:
$\begin{aligned} T_{k_{a}} (x \cdot y) & = p_{k_{a}} (x \cdot y) e^{j μ_{k_{a}} (x \cdot y)} \\ \geq max {p_{k_{a}} (x), p_{k_{a}} (y)} e^{j max {μ_{k_{a}} (x), μ_{k_{a}} (y)}} \\ = max {p_{k_{a}} (x) e^{j μ_{k_{a}} (x)}, p_{k_{a}} (y) e^{j μ_{k_{a}} (y)}} \\ = max {T_{k_{a}} (x), T_{k_{a}} (y)} . \end{aligned}$
Since $(k, A)$ is homogenous, then
$\begin{aligned} T_{k_{a}} (x \cdot y) \geq max {T_{k_{a}} (x), T_{k_{a}} (y)} . \end{aligned}$
In the same manner, we have,
$\begin{aligned} I_{k_{a}} (x \cdot y) & = q_{k_{a}} (x \cdot y) e^{j ν_{k_{a}} (x \cdot y)} \\ \leq min {q_{k_{a}} (x), q_{k_{a}} (y)} e^{j min {ν_{k_{a}} (x), ν_{k_{a}} (y)}} \\ = min {q_{k_{a}} (x) e^{j ν_{k_{a}} (x)}, q_{k_{a}} (y) e^{j ν_{k_{a}} (y)}} \\ = min {I_{k_{a}} (x), I_{k_{a}} (y)} . \end{aligned}$
Since $(k, A)$ is homogenous, then
$\begin{aligned} I_{k_{a}} (x \cdot y) \leq min {I_{k_{a}} (x), I_{k_{a}} (y)} . \end{aligned}$
Next,
$\begin{aligned} F_{k_{a}} (x \cdot y) & = r_{k_{a}} (x \cdot y) e^{j ω_{k_{a}} (x \cdot y)} \\ \leq min {r_{k_{a}} (x), r_{k_{a}} (y)} e^{j min {ω_{k_{a}} (x), ω_{k_{a}} (y)}} \\ = min {r_{k_{a}} (x) e^{j ω_{k_{a}} (x)}, r_{k_{a}} (y) e^{j ω_{k_{a}} (y)}} \\ = min {F_{k_{a}} (x), F_{k_{a}} (y)} . \end{aligned}$
Since $(k, A)$ is homogenous, then
$\begin{aligned} F_{k_{a}} (x \cdot y) \leq min {F_{k_{a}} (x), F_{k_{a}} (y)} . \end{aligned}$
Therefore, the pre-stated six conditions in Definition 4.1 have been satisfied, which implies that $(k, A)$ is a CNSI.

The following theorem explains the intersection of two ideals using Definition 4.1.
Theorem 4.2
Let $(k, A)$ and $(g, B)$ be two complex neutrosophic soft ideals on the ring $(R, +, \cdot)$ , where $(k, A)$ is homogenous with $(g, B)$ . Then $(k, A) \cap (g, B)$ is a CNSI on $R$ .
Proof.
Let $x, y \in R$ and $ϵ \in A \cap B$ . Then, by Definition 4.1, it is sufficient to demonstrate that 1.
$T_{k \cap g_{ϵ}} (x - y) \geq min {T_{k \cap g_{ϵ}} (x), T_{k \cap g_{ϵ}} (y)}$ .
2.
$I_{k \cap g_{ϵ}} (x - y) \leq max {I_{k \cap g_{ϵ}} (x), I_{k \cap g_{ϵ}} (y)}$ .
3.
$F_{k \cap g_{ϵ}} (x - y) \leq max {F_{k \cap g_{ϵ}} (x), F_{k \cap g_{ϵ}} (y)}$ .
4.
$T_{k \cap g_{ϵ}} (x \cdot y) \geq max {T_{k \cap g_{ϵ}} (x), T_{k \cap g_{ϵ}} (y)}$ .
5.
$I_{k \cap g_{ϵ}} (x \cdot y) \leq min {I_{k \cap g_{ϵ}} (x), I_{k \cap g_{ϵ}} (y)}$ .
6.
$F_{k \cap g_{ϵ}} (x \cdot y) \leq min {F_{k \cap g_{ϵ}} (x), F_{k \cap g_{ϵ}} (y)}$ .

For the first condition, we have
$\begin{aligned} T_{k \cap g_{ϵ}} (x - y) & = p_{k \cap g_{ϵ}} (x - y) e^{j μ_{k \cap g_{ϵ}} (x - y)} \\ = min {p_{k_{ϵ}} (x - y), p_{g_{ϵ}} (x - y)} e^{j min {μ_{k_{ϵ}} (x - y), μ_{g_{ϵ}} (x - y)}} \\ = min {p_{k_{ϵ}} (x - y) e^{j μ_{k_{ϵ}} (x - y)}, p_{g_{ϵ}} (x - y) e^{j μ_{g_{ϵ}} (x - y)}} \\ \geq min {min {p_{k_{ϵ}} (x) e^{j μ_{k_{ϵ}} (x)}, p_{k_{ϵ}} (y) e^{j μ_{k_{ϵ}} (y)}}, min {p_{g_{ϵ}} (x) e^{j μ_{g_{ϵ}} (x)}, p_{g_{ϵ}} (y) e^{j μ_{g_{ϵ}} (y)}}} \\ = min {min {p_{k_{ϵ}} (x) e^{j μ_{_{ϵ}} (x)}, p_{g_{ϵ}} (x) e^{j μ_{g_{ϵ}} (x)}}, min {p_{k_{ϵ}} (y) e^{j μ_{k_{ϵ}} (y)}, p_{g_{ϵ}} (y) e^{j μ_{g_{ϵ}} (y)}}} \\ = min {min {p_{k_{ϵ}} (x), p_{g_{ϵ}} (x)} e^{j min {μ_{k_{ϵ}} (x), μ_{g_{ϵ}} (x)}}, min {p_{k_{ϵ}} (y), p_{g_{ϵ}} (y)} e^{j min {μ_{k_{ϵ}} (y), μ_{g_{ϵ}} (y)}}}} \\ = min {p_{k \cap g_{ϵ}} (x) e^{j μ_{k \cap g_{ϵ}} (x)}, p_{k \cap g_{ϵ}} (y) e^{j μ_{k \cap g_{ϵ}} (y)}} \\ = min {T_{k \cap g_{ϵ}} (x), T_{k \cap g_{ϵ}} (y)} . \end{aligned}$

Then, considering the second condition, we obtain
$\begin{aligned} I_{k \cap g_{ϵ}} (x - y) & = q_{k \cap g_{ϵ}} (x - y) e^{j ν_{k \cap g_{ϵ}} (x - y)} \\ = max {q_{k_{ϵ}} (x - y), q_{g_{ϵ}} (x - y)} e^{j max {ν_{k_{ϵ}} (x - y), ν_{g_{ϵ}} (x - y)}} \\ = max {q_{k_{ϵ}} (x - y) e^{j ν_{k_{ϵ}} (x - y)}, q_{g_{ϵ}} (x - y) e^{j ν_{g_{ϵ}} (x - y)}} \\ \leq max {max {q_{k_{ϵ}} (x) e^{j ν_{k_{ϵ}} (x)}, q_{k_{ϵ}} (y) e^{j ν_{k_{ϵ}} (y)}}, max {q_{g_{ϵ}} (x) e^{j ν_{g_{ϵ}} (x)}, q_{g_{ϵ}} (y) e^{j ν_{g_{ϵ}} (y)}}} \\ = max {max {q_{k_{ϵ}} (x) e^{j ν_{_{ϵ}} (x)}, q_{g_{ϵ}} (x) e^{j ν_{g_{ϵ}} (x)}}, max {q_{k_{ϵ}} (y) e^{j ν_{k_{ϵ}} (y)}, q_{g_{ϵ}} (y) e^{j ν_{g_{ϵ}} (y)}}} \\ = max {max {q_{k_{ϵ}} (x), q_{g_{ϵ}} (x)} e^{j max {ν_{k_{ϵ}} (x), ν_{g_{ϵ}} (x)}}, max {q_{k_{ϵ}} (y), q_{g_{ϵ}} (y)} e^{j max {ν_{k_{ϵ}} (y), ν_{g_{ϵ}} (y)}}} \\ = max {q_{k \cap g_{ϵ}} (x) e^{j ν_{k \cap g_{ϵ}} (x)}, q_{k \cap g_{ϵ}} (y) e^{j ν_{k \cap g_{ϵ}} (y)}} \\ = max {I_{k \cap g_{ϵ}} (x), I_{k \cap g_{ϵ}} (y)} . \end{aligned}$

Next, for the third condition,
$\begin{aligned} F_{k \cap g_{ϵ}} (x - y) & = r_{k \cap g_{ϵ}} (x - y) e^{j ω_{k \cap g_{ϵ}} (x - y)} \\ = max {r_{k_{ϵ}} (x - y), r_{g_{ϵ}} (x - y)} e^{j max {ω_{k_{ϵ}} (x - y), ω_{g_{ϵ}} (x - y)}}} \\ = max {r_{k_{ϵ}} (x - y) e^{j ω_{k_{ϵ}} (x - y)}, r_{g_{ϵ}} (x - y) e^{j ω_{g_{ϵ}} (x - y)}} \\ \leq max {max {r_{k_{ϵ}} (x) e^{j ω_{k_{ϵ}} (x)}, r_{k_{ϵ}} (y) e^{j ω_{k_{ϵ}} (y)}}, max {r_{g_{ϵ}} (x) e^{j ω_{g_{ϵ}} (x)}, r_{g_{ϵ}} (y) e^{j ω_{g_{ϵ}} (y)}}} \\ = max {max {r_{k_{ϵ}} (x) e^{j ω_{_{ϵ}} (x)}, r_{g_{ϵ}} (x) e^{j ω_{g_{ϵ}} (x)}}, max {r_{k_{ϵ}} (y) e^{j ω_{k_{ϵ}} (y)}, r_{g_{ϵ}} (y) e^{j ω_{g_{ϵ}} (y)}}} \\ = max {max {r_{k_{ϵ}} (x), r_{g_{ϵ}} (x)} e^{j max {ω_{k_{ϵ}} (x), ω_{g_{ϵ}} (x)}}}, max {r_{k_{ϵ}} (y), r_{g_{ϵ}} (y)} e^{j max {ω_{k_{ϵ}} (y), ω_{g_{ϵ}} (y)}}} \\ = max {r_{k \cap g_{ϵ}} (x) e^{j ω_{k \cap g_{ϵ}} (x)}, r_{k \cap g_{ϵ}} (y) e^{j ω_{k \cap g_{ϵ}} (y)}} \\ = max {F_{k \cap g_{ϵ}} (x), F_{k \cap g_{ϵ}} (y)} . \end{aligned}$

Addressing the fourth condition, we have
$\begin{aligned} T_{k \cap g_{ϵ}} (x \cdot y) & = p_{k \cap g_{ϵ}} (x \cdot y) e^{j μ_{k \cap g_{ϵ}} (x \cdot y)} \\ = min {p_{k_{ϵ}} (x \cdot y), p_{g_{ϵ}} (x \cdot y)} e^{j min {μ_{k_{ϵ}} (x \cdot y), μ_{g_{ϵ}} (x \cdot y)}} \\ = min {p_{k_{ϵ}} (x \cdot y) e^{j μ_{k_{ϵ}} (x \cdot y)}, p_{g_{ϵ}} (x \cdot y) e^{j μ_{g_{ϵ}} (x \cdot y)}} \\ \geq min {max {p_{k_{ϵ}} (x) e^{j μ_{k_{ϵ}} (x)}, p_{k_{ϵ}} (y) e^{j μ_{k_{ϵ}} (y)}}, max {p_{g_{ϵ}} (x) e^{j μ_{g_{ϵ}} (x)}, p_{g_{ϵ}} (y) e^{j μ_{g_{ϵ}} (y)}}} \\ = max {min {p_{k_{ϵ}} (x) e^{j μ_{_{ϵ}} (x)}, p_{g_{ϵ}} (x) e^{j μ_{g_{ϵ}} (x)}}, min {p_{k_{ϵ}} (y) e^{j μ_{k_{ϵ}} (y)}, p_{g_{ϵ}} (y) e^{j μ_{g_{ϵ}} (y)}}} \\ = max {min {p_{k_{ϵ}} (x), p_{g_{ϵ}} (x)} e^{j min {μ_{k_{ϵ}} (x), μ_{g_{ϵ}} (x)}}, min {p_{k_{ϵ}} (y), p_{g_{ϵ}} (y)} e^{j min {μ_{k_{ϵ}} (y), μ_{g_{ϵ}} (y)}}} \\ = max {p_{k \cap g_{ϵ}} (x) e^{j μ_{k \cap g_{ϵ}} (x)}, p_{k \cap g_{ϵ}} (y) e^{j μ_{k \cap g_{ϵ}} (y)}} \\ = max {T_{k \cap g_{ϵ}} (x), T_{k \cap g_{ϵ}} (y)} . \end{aligned}$

Considering the fifth condition, we obtain
$\begin{aligned} I_{k \cap g_{ϵ}} (x \cdot y) & = q_{k \cap g_{ϵ}} (x \cdot y) e^{j ν_{k \cap g_{ϵ}} (x \cdot y)} \\ = max {q_{k_{ϵ}} (x \cdot y), q_{g_{ϵ}} (x \cdot y)} e^{j max {ν_{k_{ϵ}} (x \cdot y), ν_{g_{ϵ}} (x \cdot y)}} \\ = max {q_{k_{ϵ}} (x \cdot y) e^{j ν_{k_{ϵ}} (x \cdot y)}, q_{g_{ϵ}} (x \cdot y) e^{j ν_{g_{ϵ}} (x \cdot y)}} \\ \leq max {min {q_{k_{ϵ}} (x) e^{j ν_{k_{ϵ}} (x)}, q_{k_{ϵ}} (y) e^{j ν_{k_{ϵ}} (y)}}, min {q_{g_{ϵ}} (x) e^{j ν_{g_{ϵ}} (x)}, q_{g_{ϵ}} (y) e^{j ν_{g_{ϵ}} (y)}}} \\ = min {max {q_{k_{ϵ}} (x) e^{j ν_{_{ϵ}} (x)}, q_{g_{ϵ}} (x) e^{j ν_{g_{ϵ}} (x)}}, max {q_{k_{ϵ}} (y) e^{j ν_{k_{ϵ}} (y)}, q_{g_{ϵ}} (y) e^{j ν_{g_{ϵ}} (y)}}} \\ = min {max {q_{k_{ϵ}} (x), q_{g_{ϵ}} (x)} e^{j max {ν_{k_{ϵ}} (x), ν_{g_{ϵ}} (x)}}}, max {q_{k_{ϵ}} (y), q_{g_{ϵ}} (y)} e^{j max {ν_{k_{ϵ}} (y), ν_{g_{ϵ}} (y)}}} \\ = min {q_{k \cap g_{ϵ}} (x) e^{j ν_{k \cap g_{ϵ}} (x)}, q_{k \cap g_{ϵ}} (y) e^{j ν_{k \cap g_{ϵ}} (y)}} \\ = min {I_{k \cap g_{ϵ}} (x), I_{k \cap g_{ϵ}} (y)} . \end{aligned}$

Lastly,
$\begin{aligned} F_{k \cap g_{ϵ}} (x \cdot y) & = r_{k \cap g_{ϵ}} (x \cdot y) e^{j ω_{k \cap g_{ϵ}} (x \cdot y)} \\ = max {r_{k_{ϵ}} (x \cdot y), r_{g_{ϵ}} (x \cdot y)} e^{j max {ω_{k_{ϵ}} (x \cdot y), ω_{g_{ϵ}} (x \cdot y)}} \\ = max {r_{k_{ϵ}} (x \cdot y) e^{j ω_{k_{ϵ}} (x \cdot y)}, r_{g_{ϵ}} (x \cdot y) e^{j ω_{g_{ϵ}} (x \cdot y)}} \\ \leq max {min {r_{k_{ϵ}} (x) e^{j ω_{k_{ϵ}} (x)}, r_{k_{ϵ}} (y) e^{j ω_{k_{ϵ}} (y)}}, min {r_{g_{ϵ}} (x) e^{j ω_{g_{ϵ}} (x)}, r_{g_{ϵ}} (y) e^{j ω_{g_{ϵ}} (y)}}} \\ = min {max r_{k_{ϵ}} (x) e^{j ω_{_{ϵ}} (x)}, r_{g_{ϵ}} (x) e^{j ω_{g_{ϵ}} (x)}}, max {r_{k_{ϵ}} (y) e^{j ω_{k_{ϵ}} (y)}, r_{g_{ϵ}} (y) e^{j ω_{g_{ϵ}} (y)}}} \\ = min {max {r_{k_{ϵ}} (x), r_{g_{ϵ}} (x)} e^{j max {ω_{k_{ϵ}} (x), ω_{g_{ϵ}} (x)}}}, {max {r_{k_{ϵ}} (y), r_{g_{ϵ}} (y)} e^{j max {ω_{k_{ϵ}} (y), ω_{g_{ϵ}} (y)}}} \\ = min {r_{k \cap g_{ϵ}} (x) e^{j ω_{k \cap g_{ϵ}} (x)}, r_{k \cap g_{ϵ}} (y) e^{j ω_{k \cap g_{ϵ}} (y)}} \\ = min {F_{k \cap g_{ϵ}} (x), F_{k \cap g_{ϵ}} (y)} . \end{aligned}$

Therefore, $(k, A) \cap (g, B)$ is a CNSI.
5. Conclusions and Future Work

Various algebraic structures within the CNSS model have been explored through the introduction of CNSR and CNSI concepts. The study focuses on examining and verifying the distinct algebraic properties of CNSRs and CNSIs to gain a deeper insight of their algebraic characteristics. The investigation highlights the relationship between CNSRs and soft rings, and explains the connections between CNSRs and NSRs, as well as between CNSIs and NSIs. This thorough and detailed analysis aims to enhance the comprehension of CNSRs and CNSIs while contributing to the advancement of algebraic analysis.

Moving forward, our future research will delve into additional algebraic structures related to CNSSs, with a particular focus on complex neutrosophic soft fields (CNSFs). By extending our study to CNSFs, we aim to deepen the understanding of the algebraic behaviors and properties of CNSSs within this context. In addition, incorporating CNSRs and CNSIs into more complex problems, where precise decision-making is crucial, will be highly beneficial. Since many real-world problems can be modeled by graphs, investigating the relationships between CNSRs, CNSIs and is another promising direction, enabling solutions through both conventional and learning-based frameworks (Yow et al., 2022, 2023). Additionally, exploring optimization problems involving uncertainty and imprecision with CNSRs or CNSIs could lead to the development of efficient algorithms, particularly for large-scale or real-time applications (Yow & Li, 2024).

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This research was partially supported under Putra Grant GP-IPS/2024/9787900 by Universiti Putra Malaysia.

ORCID iDs

Fatimah Rahoumah

Kai Siong Yow

Menshawi Gasim

Hoa Pham

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