A Study on Rough Ideal Statistical Convergence in Neutrosophic Normed Spaces

Abstract

In this paper, we introduce and study the concept of rough I–$\alpha \beta$–statistical convergence of order $\gamma$ in neutrosophic normed spaces. This new mode of convergence combines the principles of rough convergence, statistical convergence with respect to an ideal, and the flexible structure of neutrosophic norms to handle indeterminacy and vagueness in sequence behavior. We establish fundamental properties of this convergence type and investigate the structure of its limit set. Specifically, we prove that the set of rough I–$\alpha \beta$–statistical limit points of order $\gamma$ is convex and closed under certain conditions. We further analyze the relationship between cluster points and rough statistical limits in this context. The theoretical results are supported by illustrative examples to demonstrate the validity and applicability of the proposed notions. Our findings generalize several existing convergence concepts and contribute to the growing body of research in neutrosophic functional analysis.

1. Introduction

The concept of convergence plays a fundamental role in mathematical analysis. Classical notions such as pointwise, uniform, and norm convergence have been extended to address situations involving uncertainty or imprecision in data. One such extension is statistical convergence, first introduced by Fast [1] and further developed by Steinhaus [2], which relies on the natural density of the set of indices rather than the conventional limit. Subsequently, Šalát [3] investigated statistically convergent sequences of real numbers, while Maddox [4,5] extended the framework to locally convex spaces and established Tauberian theorems for statistical convergence.

Over time, the idea of statistical convergence was broadened through various summability techniques, including A–statistical convergence [6], ideal convergence [7], lacunary statistical convergence [8], $\lambda$–statistical convergence [9], and deferred statistical convergence [10]. Following the developments of statistical and ideal convergence, these methods were applied in the setting of intuitionistic fuzzy normed spaces [11,12], which later led to the introduction of I–statistical convergence in the same framework [13,14]. A more recent advancement is the concept of $\alpha \beta$–statistical convergence of order $\gamma$ [15], which has been employed in proving Korovkin-type approximation theorems and, more recently, in the study of modified discrete operator approximations [16]. Further details and applications of $\alpha \beta$–statistical convergence of order $\gamma$ can be found in [17,18,19].

In certain cases, the exact evaluation of terms in a convergent sequence, such as $u_k={\left(1+{\displaystyle \frac{2}{k}}\right)}^{k+{(-1)}^k}$, becomes impractical for large k, since computational procedures rely on rounded values. To address such issues, Phu [20] introduced the notion of rough convergence in finite-dimensional normed linear spaces, later extending it to infinite-dimensional spaces [21]. This idea was subsequently generalized to rough statistical convergence [22] and rough I–convergence [23], and studied within the contexts of metric spaces [24] and intuitionistic fuzzy normed spaces [25].

The foundation for this line of research can be traced back to Zadeh’s seminal work on fuzzy set theory [26], later generalized by Atanassov [27] through the notion of intuitionistic fuzzy sets, which provided tools for handling uncertainty and incomplete information. Building on these frameworks, Kramosil and Michalek [28] introduced fuzzy metric spaces, which were refined by George and Veeramani [29] through the development of a Hausdorff topology. Saadati and Vaezpour [30] then introduced fuzzy normed spaces, which were further extended by Saadati and Park [31,32] to intuitionistic fuzzy normed (metric) spaces. Smarandache [33] advanced these concepts into the neutrosophic domain, where truth, indeterminacy, and falsity are treated independently.

These generalizations paved the way for extensive studies of convergence in intuitionistic fuzzy and neutrosophic normed spaces. Contributions by Mursaleen et al. [34] on double sequence convergence and recent investigations by Jeyaraman and collaborators [35,36,37,38] and Vakeel et al. [39] further enriched the field within neutrosophic settings.

Motivated by these advancements, the present article introduces and explores the concept of rough I–$\alpha \beta$–statistical convergence of order $\gamma$ in neutrosophic normed spaces. This new framework integrates rough convergence, ideal statistical convergence, and neutrosophic logic, thereby generalizing several earlier results. We establish fundamental properties of this convergence, including the structure of limit sets, convexity, and cluster point behavior, thus extending the works of Antal et al. [40] and others.

2. Preliminaries

Some of the fundamental definitions and notations that are needed for the following section are provided in this section.

Definition 1 

([1,2]). Let $\mathcal{A}\subseteq \mathbb{N}$. The asymptotic density of the set $\mathcal{A}$ is denoted by $\delta \left(\mathcal{A}\right)$ and defined as

$$\delta \left(\mathcal{A}\right)=\underset{k\to \infty }{lim}{\displaystyle \frac{1}{k}}\left|\{1\le n\le k:n\in \mathcal{A}\}\right|,$$

A sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ is said to statistically converge (denoted by $\mathcal{STC}$) to a number l if, for every $ϵ>0$, the set of indices $\mathfrak{j}\in \mathbb{N}$ for which $|{\mathfrak{x}}_{\mathfrak{j}}-l|>ϵ$ has asymptotic density zero. This convergence is denoted by ${\mathfrak{x}}_{\mathfrak{j}}\stackrel{st}{\to }l.$

Definition 2 

([7]). Let Γ be a nonempty set and let $I\subseteq 2^{\Gamma }$. Then, I is said to be an ideal on Γ if it satisfies the following properties: (a) $\varphi \in I$; (b) If $\mathcal{A},\mathcal{B}\in I$, then $\mathcal{A}\cup \mathcal{B}\in I$; (c) If $\mathcal{A}\in I$ and $\mathcal{B}\subseteq \mathcal{A}$, then $\mathcal{B}\in I$.

An ideal I is termed nontrivial if $I\ne 2^{\Gamma }$. Furthermore, a nontrivial ideal is called admissible if it contains all singleton subsets of Γ

A subset $\mathfrak{F}\subseteq 2^{\Gamma }$ is called a filter on Γ if the following conditions are met: (i) $\varphi \notin \mathfrak{F}$; (ii) For all $\mathcal{A},\mathcal{B}\in \mathfrak{F}$, we have $\mathcal{A}\cap \mathcal{B}\in \mathfrak{F}$; (iii) If $\mathcal{A}\in \mathfrak{F}$ and $\mathcal{A}\subset \mathcal{B}$, then $\mathcal{B}\in \mathfrak{F}$. Given an ideal I on Γ, the filter associated with I, denoted by $\mathfrak{F}\left(I\right)$, is defined by $\mathfrak{F}\left(I\right)=\{\mathcal{A}\subseteq \Gamma :{\mathcal{A}}^c\in I\},$ where ${\mathcal{A}}^c$ is the complement of $\mathcal{A}$ in Γ.

Definition 3 

([7]). Let $I\subset 2^{\mathbb{N}}$ be a nontrivial admissible ideal (denoted by $\mathcal{AI}$). A sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ is said to be I–convergent to a real number l, if for every $ϵ>0$, the set of indices $\{\mathfrak{j}\in \mathbb{N}:|{\mathfrak{x}}_{\mathfrak{j}}-l|\ge ϵ\}$ belongs to the ideal I.

Definition 4 

([14]). Let $I\subset 2^{\mathbb{N}}$ be an $\mathcal{AI}$. A sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ is said to be I–statistically convergent (denoted by $\mathcal{I}-\mathcal{STC}$) to a number l if, for every pair of positive numbers $\eta >0$ and $ϵ>0$, the set of natural numbers $\mathfrak{n}$ such that ${\displaystyle \frac{1}{\mathfrak{n}}}\left|\left\{\mathfrak{j}\le \mathfrak{n}:|{\mathfrak{x}}_{\mathfrak{j}}-l|\ge \eta \right\}\right|\ge ϵ$ belongs to the ideal I.

Moreover, when the ideal I is chosen as $I_f$, the set of all finite subsets of $\mathbb{N}$, this definition reduces to the statistical convergence.

Definition 5 

([36]). Assume that $\mathfrak{X}$ is a real vector space, that $\mu ,\nu$, and ω are fuzzy subsets of $\mathfrak{X}\times (0,\infty )$, and that *, ∘, and ⊛ are continuous t-norm and continuous t-conorm, respectively. A neutrosophic normed space, or $\mathfrak{NNS}$, is the seven-tuple $(\mathfrak{X},\mu ,\nu ,\omega ,\ast ,\circ ,⊛)$ if the following criteria are satisfied for all $\mathfrak{x},\mathfrak{y}\in \mathfrak{X}$ and $\mathfrak{s},\varsigma >0$:

(1) $\mu (\mathfrak{x},\varsigma )+\nu (\mathfrak{x},\varsigma )+\omega (\mathfrak{x},\varsigma )\le 3$,

(2) $\mu (\mathfrak{x},\varsigma )>0$, $\nu (\mathfrak{x},\varsigma )<1$ and $\omega (\mathfrak{x},\varsigma )<1$,

(3) $\mu (\mathfrak{x},\varsigma )=1$, $\nu (\mathfrak{x},\varsigma )=0$ and $\omega (\mathfrak{x},\varsigma )=0\iff \mathfrak{x}=0$,

(4) $\mu (a\mathfrak{x},s)=\mu \left(\mathfrak{x},{\displaystyle \frac{\varsigma }{\left|a\right|}}\right)$, $\nu (a\mathfrak{x},\varsigma )=\nu \left(\mathfrak{x},{\displaystyle \frac{\varsigma }{\left|a\right|}}\right)$ and $\omega (a\mathfrak{x},\varsigma )=\omega \left(\mathfrak{x},{\displaystyle \frac{\varsigma }{\left|a\right|}}\right)$ for any $0\ne a\in \mathbb{R}$,

(5) $\mu (\mathfrak{x},\varsigma )\ast \mu (\mathfrak{y},\mathfrak{s})\le \mu (\mathfrak{x}+\mathfrak{y},\varsigma +\mathfrak{s})$, $\nu (\mathfrak{x},\varsigma )\circ \nu (\mathfrak{y},\mathfrak{s})\ge \nu (\mathfrak{x}+\mathfrak{y},\varsigma +\mathfrak{s})$ and $\omega (\mathfrak{x},\varsigma )⊛\omega (\mathfrak{y},\mathfrak{s})\ge \omega (\mathfrak{x}+\mathfrak{y},\varsigma +\mathfrak{s})$,

(6) $\mu (\mathfrak{x},.):(0,\infty )\to (0,1]$, $\nu (\mathfrak{x},.):(0,\infty )\to (0,1]$ and $\omega (\mathfrak{x},.):(0,\infty )\to (0,1]$ are continuous,

(7) $\mu (\mathfrak{x},\varsigma )\to 1$ as $\varsigma \to \infty$ and $\mu (\mathfrak{x},\varsigma )\to 0$ as $\varsigma \to 0$,

(8) $\nu (\mathfrak{x},\varsigma )\to 0$ as $\varsigma \to \infty$ and $\nu (\mathfrak{x},\varsigma )\to 1$ as $\varsigma \to 0$,

(9) $\omega (\mathfrak{x},\varsigma )\to 0$ as $\varsigma \to \infty$ and $\omega (\mathfrak{x},\varsigma )\to 1$ as $\varsigma \to 0$.

In this instance, $(\mu ,\nu ,\omega )$ is referred to as the neutrosophic norm (or $\mathfrak{NN}$) on $\mathfrak{X}$.

Definition 6 

([36]). Let $(\mathfrak{X},\mu ,\nu ,\omega ,\ast ,\circ ,⊛)$ be an $\mathfrak{NNS}$. For any point $\mathfrak{x}\in \mathfrak{X}$, a radius $\mathfrak{r}>0$, and a threshold $\rho \in (0,1)$, the open ball centered at $\mathfrak{x}$ with radius $\mathfrak{r}$, defined and denoted by

$${\mathcal{B}}_{\mathfrak{x}}^{(\mu ,\nu ,\omega )}(\mathfrak{r},\rho )=\left\{\mathfrak{y}\in \mathfrak{X}:\mu (\mathfrak{x}-\mathfrak{y},\mathfrak{r})>1-\rho ,\nu (\mathfrak{x}-\mathfrak{y},\mathfrak{r})<\rho and\omega (\mathfrak{x}-\mathfrak{y},\mathfrak{r})<\rho \right\}.$$

Definition 7 

([35]). Let $(\mathfrak{X},\mu ,\nu ,\omega ,\ast ,\circ ,⊛)$ be an $\mathfrak{NNS}$. A sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ in $\mathfrak{X}$ is said to converge to a point $\mathfrak{x}\in \mathfrak{X}$ with respect to the neutrosophic functions $(\mu ,\nu ,\omega )$ if, for every $\varsigma >0$, the following conditions are satisfied ${\displaystyle \underset{\mathfrak{j}\to \infty }{lim}\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )=1,}$ ${\displaystyle \underset{\mathfrak{j}\to \infty }{lim}\nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )=0\mathrm{and}}$ ${\displaystyle \underset{\mathfrak{j}\to \infty }{lim}\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )=0.}$ We denote this convergence by ${\mathfrak{x}}_{\mathfrak{j}}\stackrel{(\mu ,\nu ,\omega )}{\to }\mathfrak{x}.$

The concept of convergence has been generalized in the $\mathfrak{NNS}$ in a variety of ways, including the following:

Definition 8 

([37]). Let $(\mathfrak{X},\mu ,\nu ,\omega ,\ast ,\circ ,⊛)$ be an $\mathfrak{NNS}$. A sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ in $\mathfrak{X}$ is said to be $\mathcal{STC}$ to a point $\mathfrak{x}\in \mathfrak{X}$ with respect to the neutrosophic components $(\mu ,\nu ,\omega )$ if, for every $\rho \in (0,1)$ and every $\varsigma >0$, $\delta \left(\left\{\mathfrak{j}\in \mathbb{N}:\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\le 1-\rho ,or\nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \rho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \rho \right\}\right)=0.$

Definition 9 

([35]). Let I be a $\mathcal{AI}$ on $\mathbb{N}$. A sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ in an $\mathfrak{NNS}$$(\mathfrak{X},\mu ,\nu ,\omega ,\ast ,\circ ,⊛)$ is said to be I–statistically convergent (denoted by $\mathcal{I}-\mathcal{STC}$) to a point $\mathfrak{x}\in \mathfrak{X}$ with respect to $(\mu ,\nu ,\omega )$, if for every $ϵ>0$, $\varsigma >0$, and $\rho \in (0,1)$, the set of indices

$$\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{\mathfrak{n}}}\left|\left\{\mathfrak{j}\le \mathfrak{n}:\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\le 1-\rho ,or\nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \rho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \rho \right\}\right|\ge ϵ\right\}$$

belongs to the ideal I.

Definition 10 

([15]). Let $\left({\alpha }_{\mathfrak{n}}\right)$ and $\left({\beta }_{\mathfrak{n}}\right)$ be two sequences of positive numbers that satisfy the following conditions

(a₁)

Both sequences are nondecreasing;

(a₂)

For each $\mathfrak{n}$, we have ${\alpha }_{\mathfrak{n}}\le {\beta }_{\mathfrak{n}}$;

(a₃)

$\underset{\mathfrak{n}\to \infty }{lim}({\beta }_{\mathfrak{n}}-{\alpha }_{\mathfrak{n}})=\infty$.

Let Λ denote the collection of all such pairs $(\alpha ,\beta )$ that satisfy conditions (a₁), (a₂), and (a₃).

Let $\mathfrak{E}\subset \mathbb{N}$ be a subset of the natural numbers. For any $(\alpha ,\beta )\in \Lambda$ and any real number $\gamma \in (0,1]$, the $\alpha \beta$–density of order γ of the set $\mathfrak{E}$ is defined by

$${\delta }^{\alpha ,\beta }(\mathfrak{E},\gamma )=\underset{n\to \infty }{lim}{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}]:\mathfrak{j}\in \mathfrak{E}\right\}\right|.$$

Definition 11 

([15]). A sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ is said to be $\alpha \beta$–statistically convergent (denoted by $\alpha \beta$–$\mathcal{STC}$) of order γ to a real number l, if for every $ϵ>0$,

$${\delta }^{\alpha ,\beta }\left(\right\{\mathfrak{j}:|{\mathfrak{x}}_{\mathfrak{j}}-l|\ge ϵ\},\gamma )=\underset{\mathfrak{n}\to \infty }{lim}{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}]:|{\mathfrak{x}}_{\mathfrak{j}}-l|\ge ϵ\right\}\right|=0.$$

We denote this type of convergence by $s{t}_{\alpha \beta }^{\gamma }\underset{\mathfrak{j}\to \infty }{lim}{\mathfrak{x}}_{\mathfrak{j}}=l.$

In the special case when $\gamma =1$, the sequence is simply called $\alpha \beta$–$\mathcal{STC}$ to l, and we write $s{t}_{\alpha \beta }\underset{\mathfrak{j}\to \infty }{lim}{\mathfrak{x}}_{\mathfrak{j}}=l.$

To extend the classical concept of convergence, Phu [20] and Ayter [22] independently introduced the notions of rough convergence and rough statistical convergence, respectively, which are defined as follows.

Definition 12. 

Let $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ be a sequence in a normed space $(\mathfrak{X},\parallel ·\parallel )$.

The sequence is said to be rough convergent (denoted by $\mathcal{RC}$) to a point $\mathfrak{x}\in \mathfrak{X}$ with roughness degree $\mathfrak{t}\ge 0$, if for every $\rho >0$, there exists a natural number ${\mathfrak{n}}_0$ such that

$$\parallel {\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x}\parallel <\mathfrak{t}+\rho forall\mathfrak{j}\ge {\mathfrak{n}}_0.$$

The sequence is said to be rough statistically convergent (denoted by $\mathcal{RSTC}$) to $\mathfrak{x}\in \mathfrak{X}$ with roughness degree $\mathfrak{t}\ge 0$, if for every $\rho >0$, the set of indices $\mathfrak{j}\in \mathbb{N}$ for which $\parallel {\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x}\parallel \ge \mathfrak{t}+\rho$ has asymptotic density zero.

3. Rough $\mathbf{I}$–$\mathbf{\alpha }\mathbf{\beta }$–Statistical Convergence of Order $\mathbf{\gamma }$ in $\mathfrak{NNS}$

The notion of rough I–$\alpha \beta$–statistical convergence of order $\gamma$ in $\mathfrak{NNS}$ is formally introduced in this section. In order to better represent uncertainty and variability in data sequences, especially in environments driven by indeterminacy, this idea generalizes current convergence conceptions. After defining the novel convergence and demonstrating its applicability, we use theorems to investigate its fundamental characteristics. Illustrative examples are provided for each finding to help explain its importance and possible uses.

Definition 13. 

Let $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$ be an $\mathfrak{NNS}$. A sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ in $\mathfrak{X}$ is said to be $\alpha \beta$–$\mathcal{STC}$ of order γ to a point $\mathfrak{x}\in \mathfrak{X}$ with respect to the neutrosophic components $(\mu ,\nu ,\omega )$, if for every $\varsigma >0$ and every $\rho \in (0,1)$, the following condition is satisfied:

$$\underset{\mathfrak{n}\to \infty }{lim}{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}]:\begin{array}{c}\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\le 1-\rho ,or\hfill \\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \rho ,\hfill \\ \omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \rho \hfill \end{array}\right\}\right|=0.$$

(1)

In this case, we denote the limit as ${(\mu ,\nu ,\omega )}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x}.$

From Definition 13, it is evident that every sequence that converges with respect to $(\mu ,\nu ,\omega )$ is also $\alpha \beta$–$\mathcal{STC}$ of order $\gamma$ with respect to the same neutrosophic components.

Remark 1. 

For $\gamma =1$ in (1), the sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ is referred to as $\alpha \beta$–$\mathcal{STC}$ to u relative to the neutrosophic components $(\mu ,\nu ,\omega )$. Furthermore, by specifying ${\alpha }_{\mathfrak{n}}=1$ and ${\beta }_{\mathfrak{n}}=\mathfrak{n}$, Definition 13 reduces to the classical notion of $\mathcal{STC}$ in the neutrosophic normed structure $(\mu ,\nu ,\omega )$.

Definition 14. 

Let $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$ be an $\mathfrak{NNS}$ and $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ be a sequence in $\mathfrak{X}$. For a fixed nonnegative real number $\mathfrak{t}\ge 0$, the sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ is said to be rough I–$\alpha \beta$–statistically convergent (denoted by $\mathcal{RI}$–$\alpha \beta$–$\mathcal{STC}$) of order γ to $\mathfrak{x}$ with respect to the neutrosophic components $(\mu ,\nu ,\omega )$, if for every $ϵ>0$, $\varsigma >0$, and $\rho \in (0,1)$,

$$\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}]:\begin{array}{c}\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\le 1-\rho ,or\hfill \\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho ,\hfill \\ \omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho \hfill \end{array}\right\}\right|\ge ϵ\right\}\in I.$$

(2)

This convergence is denoted by ${(\mu ,\nu ,\omega )}_{I_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x}.$

It follows directly from Definition 14 that any sequence which is $\mathcal{I}-\mathcal{STC}$ with respect to $(\mu ,\nu ,\omega )$ is necessarily rough $\mathcal{RI}$–$\alpha \beta$–$\mathcal{STC}$ of order $\gamma$ with respect to the same neutrosophic structure.

Remark 2. 

Let $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$ be an $\mathfrak{NNS}$, and let $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ be a sequence in $\mathfrak{X}$. Then, the following special cases of $\mathcal{RI}$–$\alpha \beta$–$\mathcal{STC}$ of order γ with respect to $(\mu ,\nu ,\omega )$ are identified

(1) 

If the ideal I is taken as $I_f$ in (2), then the sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ is said to be $\mathcal{RI}$–$\alpha \beta$–$\mathcal{STC}$ of order γ to $\mathfrak{x}$ with respect to $(\mu ,\nu ,\omega )$.

(2) 

If $\mathfrak{t}=0$ in (2), then the sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ is referred to as $\mathcal{I}-\mathcal{STC}$ of order γ to $\mathfrak{x}$ with respect to $(\mu ,\nu ,\omega )$, and this is denoted by ${(\mu ,\nu ,\omega )}_I^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x}.$

Let $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$ be an $\mathfrak{NNS}$. Suppose $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ a sequence in $\mathfrak{X}$ and $\mathfrak{t}\ge 0$. Then,

(a)

The limit ${(\mu ,\nu ,\omega )}_{I_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}$ may not be unique, provided it exists. We write

$${(\mu ,\nu ,\omega )}_{I_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)=\left\{\mathfrak{x}\in \mathfrak{X}:{(\mu ,\nu ,\omega )}_{I_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x}\right\}$$

to denote the set of all limits of $\mathcal{RI}$–$\alpha \beta$–$\mathcal{STC}$ of order $\gamma$ of the sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$. We say that $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ is $\mathcal{RI}$–$\alpha \beta$–$\mathcal{STC}$ of order $\gamma$ if ${(\mu ,\nu ,\omega )}_{I_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)\ne \varphi$ for some $\mathfrak{t}\ge 0$.

(b)

From Definition 14, it is clear that

(1)

If $0\le {\mathfrak{t}}_1\le {\mathfrak{t}}_2$ for a fixed $\gamma \in (0,1]$, then

$${(\mu ,\nu ,\omega )}_{I_{{\mathfrak{t}}_1}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)\subseteq {(\mu ,\nu ,\omega )}_{I_{{\mathfrak{t}}_2}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right).$$

(2)

If $0<\gamma \le \delta \le 1$ for a fixed $\mathfrak{t}\ge 0$, then ${(\mu ,\nu ,\omega )}_{I_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x}$ implies ${(\mu ,\nu ,\omega )}_{I_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x}$.

Example 1. 

Consider the $\mathfrak{NNS}$$(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$, where $(\mathfrak{X},∥.∥)$ is the usual normed space, ${\rho }_1★{\rho }_2={\rho }_1{\rho }_2,{\rho }_1\circ {\rho }_2=min\left\{{\rho }_1+{\rho }_2,1\right\},{\rho }_1⊛{\rho }_2=min\left\{{\rho }_1+{\rho }_2,1\right\}$ for all ${\rho }_1,{\rho }_2\in [0,1]$ and $\mu ,\nu ,\omega$ are defined by $\mu (\mathfrak{x},\varsigma )={\displaystyle \frac{\varsigma }{\varsigma +∥\mathfrak{x}∥}}$, $\nu (\mathfrak{x},\varsigma )={\displaystyle \frac{∥\mathfrak{x}∥}{\varsigma +∥\mathfrak{x}∥}}$ and $\omega (\mathfrak{x},\varsigma )={\displaystyle \frac{∥\mathfrak{x}∥}{\varsigma }},\forall \mathfrak{x}\in \mathfrak{X}$ and $\varsigma >0$. For $i=1,2,3,\dots$, define ${\mathfrak{x}}_{\mathfrak{j}}=\left\{\begin{array}{cc}\mathfrak{j}\hfill & \mathit{if}\mathfrak{j}=i^2\hfill \\ 0\hfill & \mathit{otherwise}\hfill \end{array}\right.$ and ${\mathfrak{y}}_{\mathfrak{j}}=\left\{\begin{array}{cc}0\hfill & \mathit{if}\mathfrak{j}=i^2\hfill \\ -1\hfill & \mathit{if}{2}^{2i}+1\le \mathfrak{j}\le 2^{2i}+2^i\hfill \\ 1\hfill & \mathit{otherwise}\hfill \end{array}\right.$

Assign ${\beta }_{\mathfrak{n}}={\mathfrak{n}}^2+\mathfrak{n}$ and ${\alpha }_{\mathfrak{n}}={\mathfrak{n}}^2+1$. Then, for any $\mathfrak{t}>0$, $\gamma \in (0,1]$, and $\rho \in (0,1),$

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mu ({\mathfrak{x}}_{\mathfrak{j}},\varsigma )\le 1-\rho or\nu ({\mathfrak{x}}_{\mathfrak{j}},\varsigma )\ge \rho ,\omega ({\mathfrak{x}}_{\mathfrak{j}},\varsigma )\ge \rho \right\}\right|\hfill \\ \hfill & ={\displaystyle \frac{1}{{\mathfrak{n}}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\mathfrak{n}}^2+1,{\mathfrak{n}}^2+\mathfrak{n}];\mu ({\mathfrak{x}}_{\mathfrak{j}},\varsigma )\le 1-\rho or\nu ({\mathfrak{x}}_{\mathfrak{j}},\varsigma )\ge \rho ,\omega ({\mathfrak{x}}_{\mathfrak{j}},\varsigma )\ge \rho \right\}\right|\hfill \\ \hfill & ={\displaystyle \frac{1}{{\mathfrak{n}}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\mathfrak{n}}^2+1,{\mathfrak{n}}^2+\mathfrak{n}];∥{\mathfrak{x}}_x∥\ge {\displaystyle \frac{\varsigma \rho }{1-\rho }}>0\right\}\right|={\displaystyle \frac{0}{{\mathfrak{n}}^{\gamma }}}=0.\hfill \end{array}$$

Hence ${\displaystyle {(\mu ,\nu ,\omega )}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=0}$. Now, given $\mathfrak{t}\ge 0$ and $l\in \mathfrak{X}$, consider

$${\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\\ \mu ({\mathfrak{y}}_{\mathfrak{j}}-l,\varsigma +\mathfrak{t})\le 1-\rho or\\ \nu ({\mathfrak{y}}_{\mathfrak{j}}-l,\varsigma +\mathfrak{t})\ge \rho ,\\ \omega ({\mathfrak{y}}_{\mathfrak{j}}-l,\varsigma +\mathfrak{t})\ge \rho \end{array}\right\}\right|={\displaystyle \frac{1}{{\mathfrak{n}}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\mathfrak{n}}^2+1,{\mathfrak{n}}^2+\mathfrak{n}];\\ ∥{\mathfrak{y}}_{\mathfrak{j}}-l∥\ge {\displaystyle \frac{\rho }{1-\rho }}(\varsigma +\mathfrak{t})\end{array}\right\}\right|.$$

(3)

We obtain ${\mathfrak{t}}_1={\displaystyle \frac{\rho \mathfrak{t}}{1-\rho }}\ge 0$ since ${\displaystyle \frac{\rho }{1-\rho }}>0$. Assign an infinitely small value to $0<\zeta ={\displaystyle \frac{\varsigma \rho }{1-\rho }}$. The R.H.S. of (3) then reduces to

$${\displaystyle \frac{1}{{\mathfrak{n}}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\mathfrak{n}}^2+1,{\mathfrak{n}}^2+\mathfrak{n}];∥{\mathfrak{y}}_{\mathfrak{j}}-1∥\ge {\mathfrak{t}}_1+\zeta \right\}\right|={\mathfrak{z}}_{\mathfrak{n}}\left(assume\right).$$

This means that

$$\begin{array}{cc}\hfill {\mathfrak{z}}_{\mathfrak{n}}& ={\displaystyle \frac{1}{{\mathfrak{n}}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\mathfrak{n}}^2+1,{\mathfrak{n}}^2+\mathfrak{n}];{\mathfrak{y}}_{\mathfrak{j}}\ge 1+{\mathfrak{t}}_1+\zeta \mathrm{or}{\mathfrak{y}}_{\mathfrak{j}}\le 1-{\mathfrak{t}}_1-\zeta \right\}\right|\hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle \frac{\mathfrak{n}}{{\mathfrak{n}}^{\gamma }}}\hfill & \mathit{if}\mathfrak{n}=2^i\hfill \\ 0\hfill & \mathit{otherwise},\hfill \end{array}\right.i=1,2,3,\dots .\hfill \end{array}$$

Consider the following: $\mathrm{I}={\mathrm{I}}_d=\left\{\mathfrak{E}\subset \mathbb{N}:\delta \left(\mathfrak{E}\right)=0\right\}$. We have got

$$\left\{\mathfrak{n}\in \mathbb{N}:{\mathfrak{z}}_{\mathfrak{n}}>ϵ\right\}=\left\{\mathfrak{n}\in \mathbb{N}:\mathfrak{n}=2^i,i=1,2,3,\dots \right\}\in {\mathrm{I}}_d$$

for any given arbitrary small $ϵ>0$. Using $l=-1$ again from (3), we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{{\mathfrak{n}}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\mathfrak{n}}^2+1,{\mathfrak{n}}^2+\mathfrak{n}];{\mathfrak{y}}_{\mathfrak{j}}\ge -1+{\mathfrak{t}}_1+\zeta \mathrm{or}{\mathfrak{y}}_{\mathfrak{j}}\le -1-{\mathfrak{t}}_1-\zeta \right\}\right|\hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle \frac{\mathfrak{n}}{{\mathfrak{n}}^{\gamma }}}\hfill & \mathit{if}\mathfrak{n}\ne 2^i\hfill \\ 0\hfill & \mathit{otherwise},\hfill \end{array}\right.i=1,2,3,\dots ={\mathfrak{m}}_{\mathfrak{n}}\left(Assume\right),\hfill \end{array}$$

but $\left\{\mathfrak{n}\in \mathbb{N}:{\mathfrak{m}}_{\mathfrak{n}}>ϵ\right\}=\left\{\mathfrak{n}\in \mathbb{N}:\mathfrak{n}\ne 2^i,i=1,2,3,\dots \right\}$ does not belong to ${\mathrm{I}}_d$. In the same way, if $l=0$,

$$\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{\mathfrak{n}}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\mathfrak{n}}^2+1,{\mathfrak{n}}^2+\mathfrak{n}];{\mathfrak{y}}_{\mathfrak{j}}>{\mathfrak{t}}_1+\zeta \mathrm{or}{\mathfrak{y}}_{\mathfrak{j}}<-{\mathfrak{t}}_1-\zeta \right\}\right|>ϵ\right\}=\mathbb{N}\notin {\mathrm{I}}_d.$$

Therefore, ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-LIM\left({\mathfrak{y}}_{\mathfrak{j}}\right)=\left\{\begin{array}{cc}[1-\mathfrak{t},1+\mathfrak{t}]& \mathit{if}\mathfrak{t}\ge 0\hfill \\ \varphi & \mathit{otherwise}.\hfill \end{array}\right.$

It is evident that neither of the sequences $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ nor $\left({\mathfrak{y}}_{\mathfrak{j}}\right)$ exhibits convergence with respect to the neutrosophic structure $(\mu ,\nu ,\omega )$.

In classical convergence within a neutrosophic normed structure $\left(\mathfrak{NNS}\right)$, it is a well-established fact that every subsequence $\left({\mathfrak{x}}_{{\mathfrak{j}}_i}\right)$ of a convergent sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ remains convergent with respect to the neutrosophic triplet $(\mu ,\nu ,\omega )$. However, this property does not extend to the framework of $\mathcal{RI}$–$\alpha \beta$–$\mathcal{STC}$ of order $\gamma$. Specifically, the condition ${(\mu ,\nu ,\omega )}_{I_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-LIM\left({\mathfrak{x}}_{\mathfrak{j}}\right)\ne \varphi$ does not necessarily imply that ${(\mu ,\nu ,\omega )}_{I_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-LIM\left({\mathfrak{x}}_{{\mathfrak{j}}_i}\right)\ne \varphi$.

To illustrate this, consider the sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$, the index sequences $\left({\alpha }_{\mathfrak{n}}\right)$, $\left({\beta }_{\mathfrak{n}}\right)$, and a fixed $\gamma \in (0,1]$ as specified in Example 1. Then, for any nontrivial admissible ideal I and $\mathfrak{t}\ge 0$, the rough I–$\alpha \beta$–statistical limit set of order $\gamma$ satisfies ${(\mu ,\nu ,\omega )}_{I_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-LIM\left({\mathfrak{x}}_{\mathfrak{j}}\right)=[-\mathfrak{t},\mathfrak{t}]$.

Nonetheless, for the subsequence $\left({\mathfrak{m}}_{\mathfrak{j}}\right)=\left(\mathfrak{j}\right)$, which is trivially a subsequence of $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$, the corresponding rough I–$\alpha \beta$–statistical limit set is empty, i.e., ${(\mu ,\nu ,\omega )}_{I_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-LIM\left({\mathfrak{m}}_{\mathfrak{j}}\right)=\varphi$ for all $\mathfrak{t}\ge 0$.

Lemma 1. 

Let $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ be a sequence in a neutrosophic normed space $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$. If the sequence is $\alpha \beta$–$\mathcal{STC}$ of order γ to $\mathfrak{x}$, then the sequence is also $\mathcal{RI}$–$\alpha \beta$–$\mathcal{STC}$ of order γ to $\mathfrak{x}$ holds for every roughness parameter $\mathfrak{t}\ge 0$.

Proof. 

Assigning $\mathfrak{t}=0$ and taking the ideal I as the finite ideal $I_f$, Equation (2) becomes identical to Equation (1). Consequently, the result is established. □

Lemma 1 does not admit a converse in general. This is demonstrated through the subsequent example.

Example 2. 

Consider $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$ as defined in Example 1. Define ${\mathfrak{x}}_{\mathfrak{j}}=\left\{\begin{array}{cc}1\hfill & \mathit{if}\mathfrak{j}\mathit{is}\mathit{even}\hfill \\ -1\hfill & \mathit{if}\mathfrak{j}\mathit{is}\mathit{odd}.\hfill \end{array}\right.$ Take ${\alpha }_n=1$ and ${\beta }_n=n^{{\displaystyle \frac{1}{\gamma }}}$, where $\gamma ={\displaystyle \frac{1}{2}}$. Then ${(\mu ,\nu ,\omega )}^{s{t}_{\alpha ,\beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}$ does not exist, whereas

$${(\mu ,\nu ,\omega )}_{I_{\mathfrak{t}}}^{s{t}_{\alpha ,\beta }^{\gamma }}-LIM\left({\mathfrak{x}}_{\mathfrak{j}}\right)=\left\{\begin{array}{cc}[1-\mathfrak{t},-1+\mathfrak{t}]\hfill & \mathit{if}\mathfrak{t}\ge 1\hfill \\ \varphi \hfill & \mathit{otherwise}\hfill \end{array}\right.$$

for every I.

Lemma 2. 

Let $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$ be an $\mathfrak{NNS}$, and let $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ be a sequence in $\mathfrak{X}$. Suppose a roughness parameter $\mathfrak{t}\ge 0$ is fixed. Then, for any given positive real numbers ϵ and ς, and for any $\rho \in (0,1)$, the following conditions are equivalent:

(a) 

${(\mu ,\nu ,\omega )}_{I_{\mathfrak{t}}}^{s{t}_{\alpha ,\beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x}$.

(b) 

$\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}]:\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\le 1-\rho \right\}\right|\ge ϵ\right\}\in I$,

$\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}]:\nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho \right\}\right|\ge ϵ\right\}\in I$ and

$\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}]:\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho \right\}\right|\ge ϵ\right\}\in I$.

(c) 

$\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}]:\begin{array}{c}\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\le 1-\rho ,or\hfill \\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho ,\hfill \\ \omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho \hfill \end{array}\right\}\right|\ge ϵ\right\}\in \mathfrak{F}\left(\mathrm{I}\right).$

(d) 

$\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}]:\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\le 1-\rho \right\}\right|<ϵ\right\}\in \mathfrak{F}\left(\mathrm{I}\right)$,

$\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}]:\nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho \right\}\right|<ϵ\right\}\in \mathfrak{F}\left(\mathrm{I}\right)$ and

$\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}]:\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho \right\}\right|<ϵ\right\}\in \mathfrak{F}\left(\mathrm{I}\right)$.

(e) 

$I-\underset{\mathfrak{n}\to \infty }{lim}{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}]:\begin{array}{c}\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\le 1-\rho ,or\hfill \\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho ,\hfill \\ \omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho \hfill \end{array}\right\}\right|=0.$

In [40], Theorem 2.9 states that in an $\mathfrak{NNS}$, the set of rough statistically convergent sequences (with fixed roughness degree $\mathfrak{t}\ge 0$) is closed under addition and scalar multiplication of sequences. However, when considering rough $\mathcal{RI}$–$\alpha \beta$–$\mathcal{STC}$ of order $\gamma$ in an $\mathfrak{NNS}$, this analogous closure property does not hold in general. To illustrate this limitation, we present the following proposition along with a corresponding example.

Proposition 1. 

Let $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$ be an $\mathfrak{NNS}$. Consider two sequences $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ and $\left({\mathfrak{y}}_{\mathfrak{j}}\right)$ in $\mathfrak{X}$. Then, for certain nonnegative parameters ${\mathfrak{t}}_1$ and ${\mathfrak{t}}_2$, the statements below are satisfied.

(1) 

If ${\displaystyle {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{{\mathfrak{t}}_1}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x}}$ and ${\displaystyle {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{{\mathfrak{t}}_2}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{y}}_{\mathfrak{j}}=\mathfrak{y}}$, then ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{({\mathfrak{t}}_1+{\mathfrak{t}}_2)}}^{s{t}_{\alpha \beta }^{\gamma }}-lim[{\mathfrak{x}}_{\mathfrak{j}}+{\mathfrak{y}}_{\mathfrak{j}}]=\mathfrak{x}+\mathfrak{y}$.

(2) 

If ${\displaystyle {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{{\mathfrak{t}}_1}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x}}$, then ${\displaystyle {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\left|\mathfrak{u}\right|{\mathfrak{t}}_1}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{ux}}$ for any $\mathfrak{u}\in \mathbb{R}$.

Proof. 

The proof of part (1) is trivial. We only prove part (2). For $\mathfrak{u}=0$, there is nothing to prove. Suppose $\mathfrak{u}\ne 0$. For given $\rho \in (0,1)$, $\exists \varrho \in (0,1)$ such that $1-\varrho \ge 1-\rho$. For given $\mathfrak{t}>0$, consider

$$\mathfrak{G}=\left\{\mathfrak{j}\in \mathbb{N}:\begin{array}{c}\mu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},{\displaystyle \frac{\varsigma }{2\left|\mathfrak{u}\right|}}+{\mathfrak{t}}_1\right)\le 1-\varrho or\hfill \\ \nu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},{\displaystyle \frac{\varsigma }{2\left|\mathfrak{u}\right|}}+{\mathfrak{t}}_1\right)\ge \varrho ,\hfill \\ \omega \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},{\displaystyle \frac{\varsigma }{2\left|\mathfrak{u}\right|}}+{\mathfrak{t}}_1\right)\ge \varrho \hfill \end{array}\right\}.$$

Since ${\displaystyle {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{{\mathfrak{t}}_1}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x}}$, the set

$$\mathfrak{Q}=\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mathfrak{j}\in \mathfrak{G}\right\}\right|<ϵ\right\}\in \mathfrak{F}\left(\mathrm{I}\right)$$

(4)

for each $ϵ>0$. Take $m\in \mathfrak{Q}$. Then, ${\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mathfrak{j}\in \mathfrak{G}\right\}\right|<ϵ$

$\Rightarrow {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mathfrak{j}\in {\mathfrak{G}}^c\right\}\right|\ge 1-ϵ.$

Now, for $\mathfrak{j}\in {\mathfrak{G}}^c$,

$\mu \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)=\mu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},{\mathfrak{t}}_1+{\displaystyle \frac{\varsigma }{\left|\mathfrak{u}\right|}}\right)\ge \mu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},{\mathfrak{t}}_1+{\displaystyle \frac{\varsigma }{2\left|\mathfrak{u}\right|}}\right)>1-\varrho \ge 1-\rho ,$

$\nu \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)=\nu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},{\mathfrak{t}}_1+{\displaystyle \frac{\varsigma }{\left|\mathfrak{u}\right|}}\right)\le \nu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},{\mathfrak{t}}_1+{\displaystyle \frac{\varsigma }{2\left|\mathfrak{u}\right|}}\right)<\varrho \le \rho$

and $\omega \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)=\omega \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},{\mathfrak{t}}_1+{\displaystyle \frac{\varsigma }{\left|\mathfrak{u}\right|}}\right)\le \omega \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},{\mathfrak{t}}_1+{\displaystyle \frac{\varsigma }{2\left|\mathfrak{u}\right|}}\right)<\varrho \le \rho$

Hence, ${\mathfrak{G}}^c\subseteq \left\{\begin{array}{c}\mathfrak{j}\in \mathbb{N}:\mu \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)>1-\rho ,\\ \nu \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)<\rho \mathrm{and}\omega \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)<\rho \end{array}\right\}$.

Consequently, for $m\in \mathfrak{Q}$, this means that

$1-ϵ\le {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mathfrak{j}\in {\mathfrak{G}}^c\right\}\right|$

$\le {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mu \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)>1-\rho ,\\ \nu \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)<\rho \mathrm{and}\omega \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)<\rho \end{array}\right\}\right|$.

Hence, ${\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mu \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)\le 1-\rho ,\\ \nu \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)\ge \rho \mathrm{and}\omega \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)\ge \rho \end{array}\right\}\right|<ϵ$.

Therefore,

$\mathfrak{Q}\subseteq \left\{\begin{array}{c}\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mu \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)\le 1-\rho \mathrm{or}\right.\right.\\ \left.\left.\nu \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)\ge \rho ,\omega \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)\ge \rho \right\}\right|<ϵ\end{array}\right\}$.

From (4), it follows that

$\left\{\begin{array}{c}\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mu \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)\le 1-\rho \mathrm{or}\right.\right.\\ \left.\left.\nu \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)\ge \rho ,\omega \left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{ux},\left|\mathfrak{u}\right|{\mathfrak{t}}_1+\varsigma \right)\ge \rho \right\}\right|<ϵ\end{array}\right\}\in \mathfrak{F}\left(\mathrm{I}\right)$.

Hence, by Lemma 2, ${\displaystyle {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\left|\mathfrak{u}\right|{\mathfrak{t}}_1}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{ux}}$. □

Remark 3. 

Let $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$ be an $\mathfrak{NNS}$, and let $\mathfrak{u}\in \mathbb{R}$ with $\mathfrak{u}\ne 0$. Then, the following assertions hold:

(a) 

If ${\displaystyle {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{{\mathfrak{t}}_1}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x}}$ and ${\displaystyle {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{{\mathfrak{t}}_2}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{y}}_{\mathfrak{j}}=\mathfrak{y}}$, where one of ${\mathfrak{t}}_1$ and ${\mathfrak{t}}_2$ is positive, then there exists $0<\mathfrak{t}<{\mathfrak{t}}_1+{\mathfrak{t}}_2$ such that ${\displaystyle {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}[{\mathfrak{x}}_{\mathfrak{j}}+{\mathfrak{y}}_{\mathfrak{j}}]\ne \mathfrak{x}+\mathfrak{y}}$.

(b) 

If ${\displaystyle {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x},}$ for $\mathfrak{t}>0,$ then there exists $0<l<\left|\mathfrak{u}\right|\mathfrak{t}$ such that ${\displaystyle {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_l}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}\ne \mathfrak{ux}.}$

Example 3. 

Consider $(\mathbb{R},\mu ,\nu ,\omega ,★,\circ ,⊛)$ as defined in Example 1. Define

${\mathfrak{x}}_{\mathfrak{j}}=\left\{\begin{array}{ccc}\mathfrak{j}\hfill & \mathit{if}\hfill & \mathfrak{j}=7^{\mathfrak{n}}\hfill \\ 0\hfill & \mathit{if}\hfill & \mathfrak{j}=4\mathfrak{n}\hfill \\ 1\hfill & \mathit{otherwise}\hfill \end{array}\right.$ and ${\mathfrak{y}}_{\mathfrak{j}}=\left\{\begin{array}{ccc}0\hfill & \mathit{if}\hfill & \mathfrak{j}=7^{\mathfrak{n}}\hfill \\ -1\hfill & \mathit{if}\hfill & \mathfrak{j}=4\mathfrak{n}\hfill \\ 1\hfill & \mathit{otherwise}\hfill \end{array}\right.$

Consider $\gamma =0.2$, ${\beta }_{\mathfrak{n}}=\mathfrak{n}$, and ${\alpha }_{\mathfrak{n}}=1$, then ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{{\mathfrak{t}}_1}}^{s{t}_{\alpha \beta }^{\gamma }}-LIM\left({\mathfrak{x}}_{\mathfrak{j}}\right)=\left\{\begin{array}{cc}[2-{\mathfrak{t}}_1,{\mathfrak{t}}_1]\hfill & \mathit{if}{\mathfrak{t}}_1\ge 1\hfill \\ \phi \hfill & \mathit{otherwise}\hfill \end{array}\right.$ and ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{{\mathfrak{t}}_2}}^{s{t}_{\alpha \beta }^{\gamma }}-LIM\left({\mathfrak{y}}_{\mathfrak{j}}\right)=\left\{\begin{array}{cc}[3-{\mathfrak{t}}_2,{\mathfrak{t}}_2-3]\hfill & \mathit{if}{\mathfrak{t}}_2\ge 2\hfill \\ \phi \hfill & \mathit{otherwise}\hfill \end{array}\right.$ for any I.

Now, ${\mathfrak{x}}_{\mathfrak{j}}+{\mathfrak{y}}_{\mathfrak{j}}=\left\{\begin{array}{ccc}\mathfrak{j}\hfill & \mathit{if}\hfill & \mathfrak{j}=7^{\mathfrak{n}}\hfill \\ -1\hfill & \mathit{if}\hfill & \mathfrak{j}=4\mathfrak{n}\hfill \\ 2\hfill & \mathit{otherwise}\hfill \end{array}\right.$.

Then ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-LIM({\mathfrak{x}}_{\mathfrak{j}}+{\mathfrak{y}}_{\mathfrak{j}})=\left\{\begin{array}{cc}[5-\mathfrak{t},\mathfrak{t}-3]\hfill & \mathit{if}\mathfrak{t}\ge 4\hfill \\ \phi \hfill & \mathit{otherwise}\hfill \end{array}\right.$.

Put ${\mathfrak{t}}_1=1$ and ${\mathfrak{t}}_2=3$. Hence ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{{\mathfrak{t}}_1}}^{s{t}_{\alpha \beta }^{\gamma }}-lim{\mathfrak{x}}_{\mathfrak{j}}=1$ and ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{{\mathfrak{t}}_2}}^{s{t}_{\alpha \beta }^{\gamma }}-lim{\mathfrak{y}}_{\mathfrak{j}}=0$.

However, for any $0<\mathfrak{t}<{\mathfrak{t}}_1+{\mathfrak{t}}_2=4$, we obtain ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-LIM[{\mathfrak{x}}_{\mathfrak{j}}+{\mathfrak{y}}_{\mathfrak{j}}]=\phi$.

Now, take $\mathfrak{u}=4$. Clearly $\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}=\left\{\begin{array}{ccc}4\mathfrak{j}\hfill & \mathit{if}\hfill & \mathfrak{j}=7^{\mathfrak{n}}\hfill \\ 0\hfill & \mathit{if}\hfill & \mathfrak{j}=4\mathfrak{n}\hfill \\ 4\hfill & \mathit{otherwise}\hfill \end{array}\right.$ and

${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_l}^{s{t}_{\alpha \beta }^{\gamma }}-LIM\left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}\right)=\left\{\begin{array}{cc}[4-a,a]\hfill & \mathit{if}a\ge 2\hfill \\ \phi \hfill & \mathit{otherwise}.\hfill \end{array}\right.$

Now, for ${\mathfrak{t}}_1=4$,

${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{{\mathfrak{t}}_1}}^{s{t}_{\alpha \beta }^{\gamma }}-LIM\left({\mathfrak{x}}_{\mathfrak{j}}\right)=[0,4]$ and ${(\mu ,\nu ,\omega )}_{I_{\left|u\right|{\mathfrak{t}}_1}}^{s{t}_{\alpha \beta }^{\gamma }}-LIM\left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}\right)=[0,4]=\mathfrak{u}[0,1]$.

Now, take $2.5=a<\left|\mathfrak{u}\right|{\mathfrak{t}}_1=4$. Then ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_l}^{s{t}_{\alpha \beta }^{\gamma }}-LIM\left(\mathfrak{u}{\mathfrak{x}}_{\mathfrak{j}}\right)=[1.5,2.5]\ne u[0,1]$.

As established in Remark 3, it can be readily inferred that, unlike the space of classically convergent sequences, the set of sequences exhibiting $\mathcal{RI}$–$\alpha \beta$–$\mathcal{STC}$ of order $\gamma$ fails to satisfy the conditions of a linear space for any fixed $\mathfrak{t}>0$.

Theorem 1. 

Let $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$ be an $\mathfrak{NNS}$. Consider $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ in $\mathfrak{X}$. Then, ${\displaystyle {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x}}$ for some $\mathfrak{t}>0$, if ∃ a sequence $\left({\mathfrak{y}}_{\mathfrak{j}}\right)$ in $\mathfrak{X}$ with ${\displaystyle {(\mu ,\nu ,\omega )}_{\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{y}}_{\mathfrak{j}}=\mathfrak{x}}$ such that

$$\mu ({\mathfrak{x}}_{\mathfrak{j}}-{\mathfrak{y}}_{\mathfrak{j}},\mathfrak{t})>1-\rho ,\nu ({\mathfrak{x}}_{\mathfrak{j}}-{\mathfrak{y}}_{\mathfrak{j}},\mathfrak{t})<\rho and\omega ({\mathfrak{x}}_{\mathfrak{j}}-{\mathfrak{y}}_{\mathfrak{j}},\mathfrak{t})<\rho$$

(5)

for every $\rho \in (0,1)$ and ∀$\mathfrak{j}\in \mathbb{N}$.

Proof. 

For given $\rho \in (0,1),\exists \varrho \in (0,1)$ such that $(1-\varrho )★(1-\varrho )>1-\rho$ and $\varrho \circ \varrho <\rho ,$$\varrho ⊛\varrho <\rho$. Suppose ${\displaystyle {(\mu ,\nu ,\omega )}_{\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{y}}_{\mathfrak{j}}=\mathfrak{x}}$ and (5) holds. Then, ∀$ϵ,\varsigma >0$,

$\mathfrak{P}=\left\{\begin{array}{c}\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mu ({\mathfrak{y}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\le 1-\varrho \mathrm{or}\right.\right.\\ \left.\left.\nu ({\mathfrak{y}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \varrho ,\omega ({\mathfrak{y}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \varrho \right\}\right|\ge ϵ\end{array}\right\}\in \mathfrak{I}$.

Let $m\in {\mathfrak{P}}^c$. Then

${\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mu ({\mathfrak{y}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\le 1-\varrho \mathrm{or}\\ \nu ({\mathfrak{y}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \varrho ,\omega ({\mathfrak{y}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \varrho \end{array}\right\}\right|<ϵ$

$\Rightarrow {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mu ({\mathfrak{y}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )>1-\varrho ,\\ \nu ({\mathfrak{y}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )<\varrho \mathrm{and}\omega ({\mathfrak{y}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )<\varrho \end{array}\right\}\right|\ge 1-ϵ$.

Now, define $\mathfrak{J}=\left\{\mathfrak{j}\in \mathbb{N}:\mu ({\mathfrak{y}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )>1-\varrho ,\nu ({\mathfrak{y}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )<\varrho \mathrm{and}\omega ({\mathfrak{y}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )<\varrho \right\}$.

Then, for $\mathfrak{j}\in \mathfrak{J}$, we obtain

$\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \mu ({\mathfrak{x}}_{\mathfrak{j}}-{\mathfrak{y}}_{\mathfrak{j}},\mathfrak{t})★\mu ({\mathfrak{y}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )>(1-\varrho )★(1-\varrho )>1-\rho ,$

$\nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\le \nu ({\mathfrak{x}}_{\mathfrak{j}}-{\mathfrak{y}}_{\mathfrak{j}},\mathfrak{t})\circ \nu ({\mathfrak{y}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )<\varrho \circ \varrho <\rho$ and

$\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\le \omega ({\mathfrak{x}}_{\mathfrak{j}}-{\mathfrak{y}}_{\mathfrak{j}},\mathfrak{t})⊛\omega ({\mathfrak{y}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )<\varrho ⊛\varrho <\rho$.

Hence, $\mathfrak{J}\subseteq \left\{\begin{array}{c}\mathfrak{j}\in \mathbb{N}:\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})>1-\rho ,\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})<\rho \mathrm{and}\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})<\rho \end{array}\right\}$.

⇒$1-ϵ\le {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mathfrak{j}\in \mathfrak{J}\right\}\right|$

$\le {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})>1-\rho ,\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})<\rho \mathrm{and}\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})<\rho \end{array}\right\}\right|$

$\Rightarrow {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m],\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\le 1-\rho \mathrm{or}\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho \end{array}\right\}\right|<ϵ$.

Since $m\in {\mathfrak{P}}^c$ and ${\mathfrak{P}}^c\in \mathfrak{F}\left(\mathrm{I}\right)$, we get

${\mathfrak{P}}^c\subseteq \left\{\begin{array}{c}\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\le 1-\rho \right.\right.\mathrm{or}\\ \left.\left.\nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho \right\}\right|<ϵ\end{array}\right\}\in \mathfrak{F}\left(\mathrm{I}\right)$.

As a result, ${\displaystyle {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x}}$. □

In view of Theorem 1, for any fixed roughness threshold $\mathfrak{t}$, it follows that every sequence $\left(\mathfrak{x}\mathfrak{j}\right)$ exhibiting $\mathcal{RI}$–$\alpha \beta$–$\mathcal{STC}$ of order $\gamma$ in an $\mathfrak{NNS}$ can be approximated by a sequence $\left(\mathfrak{y}\mathfrak{j}\right)$ which is $\mathcal{I}$–$\alpha \beta$–$\mathcal{STC}$ of the same order $\gamma$, such that the deviation between corresponding terms does not exceed $\mathfrak{t}$. Moreover, the limit of the rough convergent sequence satisfies ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-{lim}_{\mathfrak{j}}{\mathfrak{x}}_{\mathfrak{j}}={(\mu ,\nu ,\omega )}_{\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}-{lim}_{\mathfrak{j}}{\mathfrak{y}}_{\mathfrak{j}}.$

It is important to note that, unlike standard convergence where the limit is unique, the limit of a $\mathcal{RI}$–$\alpha \beta$–$\mathcal{STC}$ sequence of order $\gamma$ generally forms a set. Therefore, an in-depth analysis of both the topological and geometric characteristics of the limit set ${(\mu ,\nu ,\omega )}^{s{t}^{\gamma }\alpha \beta }\mathrm{I}\mathfrak{t}-\mathrm{LIM}\left(\mathfrak{x}\mathfrak{j}\right)$ is of interest. The upcoming theorems examine its convexity and closedness properties.

Theorem 2. 

Let $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$ be an $\mathfrak{NNS}$. The set ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-LIM\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ is closed for every $\mathfrak{t}\ge 0$ for a sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ in $\mathfrak{X}$.

Proof. 

Given that $\rho \in (0,1),\exists \varrho \in (0,1)$ such that $(1-\varrho )★(1-\varrho )>(1-\rho )$,$\varrho \circ \varrho <\rho$ and $\varrho ⊛\varrho <\rho$.

Let $\mathfrak{x}\in cl\left({(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)\right)$, the closure of ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$.

Then, there exists a sequence $\left({\mathfrak{v}}_{\mathfrak{j}}\right)$ in ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ such that ${\mathfrak{v}}_{\mathfrak{j}}\stackrel{(\mu ,\nu ,\omega )}{⟶}\mathfrak{x}$,

, i.e., assuming $\varsigma >0,\exists {\mathfrak{n}}_0\in \mathbb{N}$:

$\mu \left({\mathfrak{v}}_{\mathfrak{j}}-\mathfrak{x},{\displaystyle \frac{\varsigma }{2}}\right)>1-\varrho ,$$\nu \left({\mathfrak{v}}_{\mathfrak{j}}-\mathfrak{x},{\displaystyle \frac{\varsigma }{2}}\right)<\varrho and\omega \left({\mathfrak{v}}_{\mathfrak{j}}-\mathfrak{x},{\displaystyle \frac{\varsigma }{2}}\right)<\varrho \forall \mathfrak{j}\ge n_0$.

Hence, for any $m_0>{\mathfrak{n}}_0$ and the set

$\mathfrak{Y}=\left\{\mathfrak{j}\in \mathbb{N}:\mu \left({\mathfrak{x}}_{\mathfrak{j}}-{\mathfrak{v}}_{m_0},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2}}\right)\le 1-\varrho \mathrm{or}\nu \left({\mathfrak{x}}_{\mathfrak{j}}-{\mathfrak{v}}_{m_0},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2}}\right)\ge \varrho ,\omega \left({\mathfrak{x}}_{\mathfrak{j}}-{\mathfrak{v}}_{m_0},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2}}\right)\ge \varrho \right\}$ we have

$\mathfrak{Z}=\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mathfrak{j}\in \mathfrak{Y}\right\}\right|<ϵ\right\}\in \mathfrak{F}\left(\mathrm{I}\right)$ for every $ϵ>0$. Let $m\in \mathfrak{Z}$. Then,

${\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mathfrak{j}\in \mathfrak{Y}\right\}\right|<ϵ$

$\Rightarrow {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mathfrak{j}\in {\mathfrak{Y}}^c\right\}\right|\ge 1-ϵ$.

Thus, for $\mathfrak{j}\in {\mathfrak{Y}}^c$,

$$\begin{array}{cc}\hfill \mu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t}\right)& \ge \mu \left({\mathfrak{x}}_{\mathfrak{j}}-{\mathfrak{v}}_{m_0},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2}}\right)★\mu \left({\mathfrak{v}}_{m_0}-\mathfrak{x},{\displaystyle \frac{\varsigma }{2}}\right)>(1-\varrho )★(1-\varrho )>1-\rho \hfill \\ \hfill \nu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t}\right)& \le \nu \left({\mathfrak{x}}_{\mathfrak{j}}-{\mathfrak{v}}_{m_0},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2}}\right)\circ \nu \left({\mathfrak{v}}_{m_0}-\mathfrak{x},{\displaystyle \frac{\varsigma }{2}}\right)<\varrho \circ \varrho <\rho \mathrm{and}\hfill \\ \hfill \omega \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t}\right)& \le \omega \left({\mathfrak{x}}_{\mathfrak{j}}-{\mathfrak{v}}_{m_0},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2}}\right)⊛\omega \left({\mathfrak{v}}_{m_0}-\mathfrak{x},{\displaystyle \frac{\varsigma }{2}}\right)<\varrho ⊛\varrho <\rho .\hfill \end{array}$$

Hence, ${\mathfrak{Y}}^c\subseteq \left\{\mathfrak{j}\in \mathbb{N}:\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})>1-\rho ,\nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})<\rho \mathrm{and}\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})<\rho \right\}$.

That being said, for $m\in \mathfrak{Z}$, we get

$1-ϵ\le {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mathfrak{j}\in {\mathfrak{Y}}^c\right\}\right|$

$\le {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})>1-\rho ,\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})<\rho \mathrm{and}\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})<\rho \end{array}\right\}\right|$

$\Rightarrow {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\le 1-\rho \mathrm{or}\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho \end{array}\right\}\right|<ϵ$.

Thus, $\mathfrak{Z}\subseteq \left\{\begin{array}{c}\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\\ \left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\le 1-\rho \mathrm{or}\right.\right.\\ \left.\left.\nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma +\mathfrak{t})\ge \rho \right\}\right|<ϵ\end{array}\right\}\in \mathfrak{F}\left(\mathrm{I}\right)$.

Hence, $\mathfrak{x}\in {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ and thus ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ is closed. □

When the roughness parameter $\mathfrak{t}$ is set to zero, the notion of $\mathcal{RI}$–$\alpha \beta$–$\mathcal{STC}$ of order $\gamma$ coincides with the standard $\mathcal{I}$–$\alpha \beta$–$\mathcal{STC}$ of order $\gamma$ for the sequence. Under this condition, the corresponding limit set ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ degenerates to a singleton, which is trivially a closed set.

Theorem 3. 

Let $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$ be an $\mathfrak{NNS}$. Then, for any sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ in $\mathfrak{X}$, and for each $\mathfrak{t}\ge 0$, ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ is convex.

Proof. 

Let $\mathfrak{x},\mathfrak{y}\in {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ and $\rho \in (0,1)$ be given. Then, $\exists \varrho \in (0,1)$ such that $(1-\varrho )★(1-\varrho )>1-\rho$, $\varrho \circ \varrho <\rho$ and $\varrho ⊛\varrho <\rho$. The requirement is to show that $\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}\in {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$, assuming any $\alpha \in [0,1]$. When $\alpha =0$ or 1, the outcome is evident.

Consider $\alpha \in (0,1)$. Given $\varsigma >0$, define

$\mathfrak{U}=\left\{\mathfrak{j}\in \mathbb{N}:\mu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2\alpha }}\right)\le 1-\varrho \mathrm{or}\nu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2\alpha }}\right)\ge \varrho ,\omega \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2\alpha }}\right)\ge \varrho \right\}$ and

$\mathfrak{H}=\left\{\begin{array}{c}\mathfrak{j}\in \mathbb{N}:\mu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2(1-\alpha )}}\right)\le 1-\varrho \mathrm{or}\\ \nu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2(1-\alpha )}}\right)\ge \varrho ,\omega \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2(1-\alpha )}}\right)\ge \varrho \end{array}\right\}$.

Evidently, for each $ϵ>0$,

$\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mathfrak{j}\in \mathfrak{U}\right\}\right|\ge ϵ\right\}\in \mathrm{I}$ and

$\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mathfrak{j}\in \mathfrak{H}\right\}\right|\ge ϵ\right\}\in \mathrm{I}$.

Therefore,

$\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mathfrak{j}\in \mathfrak{U}\cup \mathfrak{H}\right\}\right|\ge ϵ\right\}\in \mathrm{I}$.

In order to have $0<1-\mathfrak{b}<ϵ$, select $0<\mathfrak{b}<1$. Then, the set

$\mathfrak{D}=\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mathfrak{j}\in \mathfrak{U}\cup \mathfrak{H}\right\}\right|\ge 1-\mathfrak{b}\right\}\in \mathrm{I}.$

Hence, for $m\in {\mathfrak{D}}^c$,

${\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mathfrak{j}\in \mathfrak{U}\cup \mathfrak{H}\right\}\right|<1-\mathfrak{b}$

implies

${\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mathfrak{j}\in {\mathfrak{U}}^c\cap {\mathfrak{H}}^c\right\}\right|\ge 1-(1-\mathfrak{b}$) = $\mathfrak{b}$.

Now, take $\mathfrak{j}\in {\mathfrak{U}}^c\cap {\mathfrak{H}}^c$. Then,

$$\begin{array}{cc}\hfill \mu \left({\mathfrak{x}}_{\mathfrak{j}}-[\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}],\varsigma +\mathfrak{t}\right)& =\mu \left((1-\alpha )({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y})+\alpha ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x}),(1-\alpha )\mathfrak{t}+\alpha \mathfrak{t}+\varsigma \right)\hfill \\ \hfill & \ge \mu \left((1-\alpha )({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y}),(1-\alpha )\mathfrak{t}+{\displaystyle \frac{\varsigma }{2}}\right)★\mu \left(\alpha ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x}),\alpha \mathfrak{t}+{\displaystyle \frac{\varsigma }{2}}\right)\hfill \\ \hfill & =\mu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2(1-\alpha )}}\right)★\mu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2\alpha }}\right)\hfill \\ \hfill & >(1-\varrho )★(1-\varrho )\hfill \\ \hfill & >1-\rho \hfill \\ \hfill \nu \left({\mathfrak{x}}_{\mathfrak{j}}-[\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}],\varsigma +\mathfrak{t}\right)& =\nu \left((1-\alpha )({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y})+\alpha ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x}),(1-\alpha )\mathfrak{t}+\alpha \mathfrak{t}+\varsigma \right)\hfill \\ \hfill & \le \nu \left((1-\alpha )({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y}),(1-\alpha )\mathfrak{t}+{\displaystyle \frac{\varsigma }{2}}\right)\circ \nu \left(\alpha ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x}),\alpha \mathfrak{t}+{\displaystyle \frac{\varsigma }{2}}\right)\hfill \\ \hfill & =\nu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2(1-\alpha )}}\right)\circ \nu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2\alpha }}\right)\hfill \\ \hfill & <\varrho \circ \varrho <\rho \mathrm{and}\hfill \\ \hfill \omega \left({\mathfrak{x}}_{\mathfrak{j}}-[\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}],\varsigma +\mathfrak{t}\right)& =\omega \left((1-\alpha )({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y})+\alpha ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x}),(1-\alpha )\mathfrak{t}+\alpha \mathfrak{t}+\varsigma \right)\hfill \\ \hfill & \le \omega \left((1-\alpha )({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y}),(1-\alpha )\mathfrak{t}+{\displaystyle \frac{\varsigma }{2}}\right)⊛\omega \left(\alpha ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x}),\alpha \mathfrak{t}+{\displaystyle \frac{\varsigma }{2}}\right)\hfill \\ \hfill & =\omega \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2(1-\alpha )}}\right)⊛\omega \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\mathfrak{t}+{\displaystyle \frac{\varsigma }{2\alpha }}\right)\hfill \\ \hfill & <\varrho ⊛\varrho <\rho .\hfill \end{array}$$

As a result, we have

${\mathfrak{U}}^c\cap {\mathfrak{H}}^c\subseteq \left\{\begin{array}{c}\mathfrak{j}\in \mathbb{N}:\mu ({\mathfrak{x}}_{\mathfrak{j}}-[\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}],\varsigma +\mathfrak{t})>1-\rho ,\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-[\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}],\varsigma +\mathfrak{t})<\rho \mathrm{and}\omega ({\mathfrak{x}}_{\mathfrak{j}}-[\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}],\varsigma +\mathfrak{t})<\rho \end{array}\right\}$.

Hence, for $m\in {\mathfrak{D}}^c$,

$$\begin{array}{cc}\hfill \mathfrak{b}& \le {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mathfrak{j}\in {\mathfrak{U}}^c\cap {\mathfrak{H}}^c\right\}\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mu ({\mathfrak{x}}_{\mathfrak{j}}-[\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}],\varsigma +\mathfrak{t})>1-\rho ,\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-[\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}],\varsigma +\mathfrak{t})<\rho \mathrm{and}\\ \omega ({\mathfrak{x}}_{\mathfrak{j}}-[\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}],\varsigma +\mathfrak{t})<\rho \end{array}\right\}\right|\hfill \\ \hfill & \Rightarrow {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mu ({\mathfrak{x}}_{\mathfrak{j}}-[\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}],\varsigma +\mathfrak{t})\le 1-\rho \mathrm{or}\\ \nu ({\mathit{x}}_{\mathfrak{j}}-[\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}],\varsigma +\mathfrak{t})\ge \rho ,\\ \omega ({\mathit{x}}_{\mathfrak{j}}-[\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}],\varsigma +\mathfrak{t})\ge \rho \end{array}\right\}\right|\hfill \\ \hfill & <1-\mathfrak{b}<ϵ.\hfill \end{array}$$

Therefore,

${\mathfrak{D}}^c\subseteq \left\{\begin{array}{c}\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mu ({\mathfrak{x}}_{\mathfrak{j}}-[\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}],\varsigma +\mathfrak{t})\le 1-\rho \mathrm{or}\right.\right.\\ \left.\left.\nu ({\mathfrak{x}}_{\mathfrak{j}}-[\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}],\varsigma +\mathfrak{t})\ge \rho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-[\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}],\varsigma +\mathfrak{t})\ge \rho \right\}\right|<ϵ\end{array}\right\}\in \mathfrak{F}\left(\mathrm{I}\right)$.

Therefore, $\alpha \mathfrak{x}+(1-\alpha )\mathfrak{y}\in {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-LIM\left({\mathfrak{x}}_{\mathfrak{j}}\right)$. □

Evidently, when $\mathfrak{t}=0$, the limit set ${(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ reduces to a singleton, which is trivially convex.

However, for $\mathfrak{t}>0$, the limit set may consist of multiple elements. In such cases, it becomes natural to investigate the extent of this set. With this motivation, we establish the following result concerning its diameter.

Theorem 4. 

Let $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$ be an $\mathfrak{NNS}$. Then, for any $\mathfrak{t}>0$, there do not exist elements $\mathfrak{x},\mathfrak{y}\in {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ satisfying the following condition: $\mu (\mathfrak{x}-\mathfrak{y},\mathfrak{q}\mathfrak{t})\le 1-\rho \mathrm{or}\nu (\mathfrak{x}-\mathfrak{y},\mathfrak{q}\mathfrak{t})\ge \rho ,\omega (\mathfrak{x}-\mathfrak{y},\mathfrak{q}\mathfrak{t})\ge \rho$ for each $\rho \in (0,1)$, where $\mathfrak{q}>2$.

Proof. 

Given a value of $\rho$ in the interval (0,1), there exists $\varrho \in (0,1)$ such that $(1-\varrho )★(1-\varrho )>1-\rho$, $\varrho \circ \varrho <\rho$ and $\varrho ⊛\varrho <\rho$. Ideally, let $\mathfrak{x},\mathfrak{y}\in {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ such that $\mu (\mathfrak{x}-\mathfrak{y},\mathfrak{qt})\le 1-\rho \mathrm{or}\nu (\mathfrak{x}-\mathfrak{y},\mathfrak{qt})\ge \rho ,\omega (\mathfrak{x}-\mathfrak{y},\mathfrak{qt})\ge \rho$ for some $\mathfrak{q}>2$.

When $\varsigma >0$, take into consideration

$\mathfrak{R}=\left\{\mathfrak{j}\in \mathbb{N}:\mu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},{\displaystyle \frac{\varsigma }{2}}+\mathfrak{t}\right)\le 1-\varrho \mathrm{or}\nu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},{\displaystyle \frac{\varsigma }{2}}+\mathfrak{t}\right)\ge \varrho ,\omega \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},{\displaystyle \frac{\varsigma }{2}}+\mathfrak{t}\right)\ge \varrho \right\}$ and

$\mathfrak{B}=\left\{\mathfrak{j}\in \mathbb{N}:\mu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y},{\displaystyle \frac{\varsigma }{2}}+\mathfrak{t}\right)\le 1-\varrho \mathrm{or}\nu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y},{\displaystyle \frac{\varsigma }{2}}+\mathfrak{t}\right)\ge \varrho ,\omega \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y},{\displaystyle \frac{\varsigma }{2}}+\mathfrak{t}\right)\ge \varrho \right\}$.

Since $\mathfrak{x},\mathfrak{y}\in {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_r}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$, by Lemma 2 we have

${\displaystyle \mathrm{I}-\underset{\mathfrak{n}\to \infty }{lim}\frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mathfrak{j}\in \mathfrak{R}\right\}\right|=0}$ and

${\displaystyle \mathrm{I}-\underset{\mathfrak{n}\to \infty }{lim}\frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mathfrak{j}\in \mathfrak{B}\right\}\right|=0}$.

Now,

$$\begin{array}{cc}\hfill & \mathrm{I}-\underset{\mathfrak{n}\to \infty }{lim}{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mathfrak{j}\in \mathfrak{R}\cup \mathfrak{B}\right\}\right|\hfill \\ \hfill & \le \mathrm{I}-\underset{\mathfrak{n}\to \infty }{lim}{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mathfrak{j}\in \mathfrak{R}\right\}\right|\hfill \\ \hfill & +\mathrm{I}-\underset{n\to \infty }{lim}{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mathfrak{j}\in \mathfrak{B}\right\}\right|\hfill \\ \hfill & =0.\hfill \end{array}$$

Hence, for every $ϵ>0$,

$\mathfrak{L}=\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mathfrak{j}\in \mathfrak{R}\cup \mathfrak{B}\right\}\right|\ge ϵ\right\}\in \mathrm{I}$.

Let $m\in {\mathfrak{L}}^c$ and $ϵ={\displaystyle \frac{1}{4}}$. Then

${\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mathfrak{j}\in \mathfrak{R}\cup \mathfrak{B}\right\}\right|<{\displaystyle \frac{1}{4}}$

$\Rightarrow {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mathfrak{j}\in {\mathfrak{R}}^c\cap {\mathfrak{B}}^c\right\}\right|\ge {\displaystyle \frac{3}{4}}$.

As a result, we have $\mathfrak{D}=\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mathfrak{j}\in {\mathfrak{R}}^c\cap {\mathfrak{B}}^c\right\}\ne \varphi$.

For some $\varsigma >0$, use $\mathfrak{qt}=2\mathfrak{t}+\varsigma$ since $\mathfrak{q}>2$. Let $\mathfrak{j}\in \mathfrak{D}$. Next, we have the following three cases:

Firstly, if $\mu (\mathfrak{x}-\mathfrak{y},\mathfrak{qt})\le 1-\rho$, then

$$\begin{array}{cc}\hfill 1-\rho & \ge \mu \left(\mathfrak{x}-\mathfrak{y},\varsigma +2\mathfrak{t}\right)\ge \mu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},{\displaystyle \frac{\varsigma }{2}}+\mathfrak{t}\right)★\mu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y},{\displaystyle \frac{\varsigma }{2}}+\mathfrak{t}\right)>(1-\varrho )★(1-\varrho )>1-\rho ,\hfill \end{array}$$

secondly, if $\nu (\mathfrak{x}-\mathfrak{y},\mathfrak{qt})\ge \rho$, then

$$\begin{array}{cc}\hfill \rho & \le \nu \left(\mathfrak{x}-\mathfrak{y},\varsigma +2\mathfrak{t}\right)\le \nu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},{\displaystyle \frac{\varsigma }{2}}+\mathfrak{t}\right)\circ \nu \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y},{\displaystyle \frac{\varsigma }{2}}+\mathfrak{t}\right)<\varrho \circ \varrho <\rho ,\hfill \end{array}$$

and finally, if $\omega (\mathfrak{x}-\mathfrak{y},\mathfrak{qt})\ge \rho$, then

$$\begin{array}{cc}\hfill \rho & \le \omega \left(\mathfrak{x}-\mathfrak{y},\varsigma +2\mathfrak{t}\right)\le \omega \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},{\displaystyle \frac{\varsigma }{2}}+\mathfrak{t}\right)⊛\omega \left({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{y},{\displaystyle \frac{\varsigma }{2}}+r\right)<\varrho ⊛\varrho <\rho .\hfill \end{array}$$

This yields a contradiction in each of the above scenarios. Hence, all possible cases result in inconsistencies. Therefore, the proof is complete. □

Theorem 4 establishes that the diameter of the limit set ${(\mu ,\nu ,\omega )}_{I_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ does not exceed $2\mathfrak{t}$.

Theorem 5. 

Let $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$ be an $\mathfrak{NNS}$. For a sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ in $\mathfrak{X}$, if ${\displaystyle {(\mu ,\nu ,\omega )}_{\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x}}$ then ∃ $\varrho \in (0,1)$ such that $cl\left({\mathfrak{B}}_{\mathfrak{x}}^{(\mu ,\nu ,\omega )}(\mathfrak{t},\varrho )\right)\subseteq {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-LIM\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ for a certain $\mathfrak{t}>0$.

Proof. 

When $\rho$ is given, there exists $\varrho \in (0,1)$ such that $(1-\varrho )★(1-\varrho )>1-\rho ,$$\varrho \circ \varrho <\rho$, and $\varrho ⊛\varrho <\rho$. Suppose ${\displaystyle {(\mu ,\nu ,\omega )}_{\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x}}$. Then,

$\mathfrak{A}=\left\{\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\le 1-\varrho \mathrm{or}\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \varrho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \varrho \end{array}\right\}\right|\ge ϵ\right\}\in \mathrm{I}$ for each $\varsigma ,ϵ>0$. Considering $m\in {\mathfrak{A}}^c$, we obtain

${\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\le 1-\varrho \mathrm{or}\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \varrho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \varrho \end{array}\right\}\right|<ϵ$

$\Rightarrow {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )>1-\varrho ,\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )<\varrho \mathrm{and}\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )<\varrho \end{array}\right\}\right|\ge 1-ϵ$.

Now, let $\mathfrak{w}\in cl\left({\mathfrak{B}}_{\mathfrak{x}}^{(\mu ,\nu ,\omega )}(\mathfrak{t},\varrho )\right)$ for some $\mathfrak{t}>0$. Then $\mu (\mathfrak{x}-\mathfrak{w},\mathfrak{t})\ge 1-\varrho ,$$\nu (\mathfrak{x}-\mathfrak{w},\mathfrak{t})\le \varrho$ and $\omega (\mathfrak{x}-\mathfrak{w},\mathfrak{t})\le \varrho$.

Define $\mathfrak{C}=\left\{\mathfrak{j}\in \mathbb{N}:\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )>1-\varrho ,\nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )<\varrho \mathrm{and}\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )<\varrho \right\}$.

Thus, for $\mathfrak{j}\in \mathfrak{C}$, similarly to above, we have $\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{w},\varsigma +\mathfrak{t})>1-\rho ,$$\nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{w},\varsigma +\mathfrak{t})<\rho$ and $\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{w},\varsigma +\mathfrak{t})<\rho$.

Therefore, $\mathfrak{C}\subseteq \left\{\mathfrak{j}\in \mathbb{N}:\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{w},\varsigma +\mathfrak{t})>1-\rho ,\nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{w},\varsigma +\mathfrak{t})<\rho \mathrm{and}\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{w},\varsigma +\mathfrak{t})<\rho \right\}$

and hence

$1-ϵ\le {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mathfrak{j}\in \mathfrak{C}\right\}\right|$

$\le {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{w},\varsigma +\mathfrak{t})>1-\rho ,\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{w},\varsigma +\mathfrak{t})<\rho \mathrm{and}\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{w},\varsigma +\mathfrak{t})<\rho \end{array}\right\}\right|$.

This means that

${\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{w}\varsigma +\mathfrak{t})\le 1-\rho \mathrm{or}\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{w}\varsigma +\mathfrak{t})\ge \rho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{w}\varsigma +\mathfrak{t})\ge \rho \end{array}\right\}\right|<ϵ$.

Because of $m\in {\mathfrak{A}}^c$ and ${\mathfrak{A}}^c\in \mathfrak{F}\left(\mathrm{I}\right)$, we get

${\mathfrak{A}}^c\subseteq \left\{\begin{array}{c}\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{w},\varsigma +\mathfrak{t})\le 1-\rho \mathrm{or}\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{w},\varsigma +\mathfrak{t})\ge \rho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{w},\varsigma +\mathfrak{t})\ge \rho \end{array}\right\}\right|<ϵ\end{array}\right\}\in \mathfrak{F}\left(\mathfrak{I}\right).$

Therefore, $\mathfrak{w}\in {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$, and hence, $cl\left({\mathfrak{B}}_{\mathfrak{x}}^{(\mu ,\nu ,\omega )}(\mathfrak{t},\varrho )\right)\subseteq {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{{\mathfrak{st}}_{\alpha \beta }^{\gamma }}-\mathrm{LIM}\left({\mathfrak{x}}_j\right)$. □

4. Rough Ideal Cluster Points

We present and explore the concept of a ${\mathrm{I}}_{\mathfrak{t}}-{\mathrm{st}}_{\alpha \beta }^{\gamma }$–cluster point of $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ in an $\mathfrak{NNS}$ in this section.

Definition 15. 

Let $\mathfrak{NNS}$, $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$. A point $\mathfrak{z}\in \mathfrak{X}$ for a sequence $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ in $\mathfrak{X}$ and some $\mathfrak{t}\ge 0$ is called ${\mathrm{I}}_{\mathfrak{t}}-{\mathrm{st}}_{\alpha \beta }^{\gamma }$–cluster point w.r.t. $(\mu ,\nu ,\omega )$ of $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ if $ϵ,\varsigma >0$ and $\rho \in (0,1)$,

$\left\{\begin{array}{c}\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_n,{\beta }_n];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})\le 1-\rho \mathrm{or}\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})\ge \rho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})\ge \rho \end{array}\right\}\right|<ϵ\end{array}\right\}\notin \mathrm{I}$.

Let us represent the set of all ${\mathrm{I}}_{\mathfrak{t}}-{\mathrm{st}}_{\alpha \beta }^{\gamma }$–cluster points of $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ w.r.t. $(\mu ,\nu ,\omega )$ by ${\Gamma }_{(\mu ,\nu ,\omega ){\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$. We say $\mathfrak{z}$ is $\mathrm{I}-{\mathrm{st}}_{\alpha \beta }^{\gamma }$–cluster point of $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ w.r.t. $(\mu ,\nu ,\omega )$ for $\mathfrak{t}=0$, and ${\Gamma }_{(\mu ,\nu ,\omega )\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ represents the set of such cluster points.

Theorem 6. 

Let $\mathfrak{NNS}$, $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$. The set ${\Gamma }_{(\mu ,\nu ,\omega ){\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ is closed if for every $\mathfrak{t}\ge 0$ and $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ in $\mathfrak{X}$.

Proof. 

This proof’s outline is similar to that of Theorem 2’s proof. □

Lemma 3. 

For a given $\mathfrak{t}\ge 0$, $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ is a sequence in an $\mathfrak{NNS}$$(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$. Assume that $\mathfrak{z}\in {\Gamma }_{(\mu ,\nu ,\omega ){\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ and $\mu (\mathfrak{z}-\mathfrak{y},\mathfrak{t})>1-\rho$, $\nu (\mathfrak{z}-\mathfrak{y},\mathfrak{t})<\rho$ and $\omega (\mathfrak{z}-\mathfrak{y},\mathfrak{t})<\rho$ for $\rho \in (0,1)$. Then $\mathfrak{y}\in {\Gamma }_{(\mu ,\nu ,\omega ){\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$.

Proof. 

Since the outcome is obvious, the proof is not needed. □

The following theorem explores the connection between the set of ${\mathrm{I}}_{\mathfrak{t}}-{\mathrm{st}}_{\alpha \beta }^{\gamma }$–cluster points and the set of $\mathrm{I}-s{t}_{\alpha \beta }^{\gamma }$–cluster points.

Theorem 7. 

Let $\mathfrak{NNS}$, $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$ and $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ a sequence in $\mathfrak{X}$. Then for $\mathfrak{t}>0,\exists \varrho \in (0,1)$ such that ${\displaystyle {\Gamma }_{(\mu ,\nu ,\omega ){\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)=\bigcup _{\mathfrak{x}\in {\Gamma }_{(\mu ,\nu ,\omega )\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)}cl\left({\mathfrak{B}}_{\mathfrak{x}}^{(\mu ,\nu ,\omega )}(\mathfrak{t},\varrho )\right)}$.

Proof. 

For a given $\rho \in (0,1),\exists \varrho \in (0,1)$ such that $(1-\varrho )★(1-\varrho )>1-\rho ,$$\varrho \circ \varrho <\rho$ and $\varrho ⊛\varrho <\rho$. Let ${\displaystyle \mathfrak{z}\in \bigcup _{\mathfrak{x}\in {\Gamma }_{(\mu ,\nu ,\omega )\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)}cl\left({\mathfrak{B}}_{\mathfrak{x}}^{(\mu ,\nu ,\omega )}(r,\varrho )\right),\mathfrak{t}>0}$.

Then, there exists ${\displaystyle \mathfrak{x}\in {\Gamma }_{(\mu ,\nu ,\omega )\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)}$ such that $\mathfrak{z}\in cl\left({\mathfrak{B}}_{\mathfrak{x}}^{(\mu ,\nu ,\omega )}(\mathfrak{t},\varrho )\right)$, i.e., $\mu (\mathfrak{x}-\mathfrak{z},\mathfrak{t})\ge 1-\varrho ,$$\nu (\mathfrak{x}-\mathfrak{z},\mathfrak{t})\le \varrho$ and $\omega (\mathfrak{x}-\mathfrak{z},\mathfrak{t})\le \varrho$.

Considering that ${\displaystyle \mathfrak{x}\in {\Gamma }_{(\mu ,\nu ,\omega )\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)}$, where $ϵ,\varsigma >0$ is specified, the set

$\mathfrak{J}=\left\{\begin{array}{c}\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\le 1-\varrho \mathrm{or}\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \varrho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \varrho \end{array}\right\}\right|<ϵ\end{array}\right\}\notin \mathrm{I}$.

Define, ${\mathfrak{J}}_1=\left\{\mathfrak{j}\in \mathbb{N}:\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\le 1-\varrho \mathrm{or}\nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \varrho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{x},\varsigma )\ge \varrho \right\}$.

Then, similar to above, we obtain

$${\mathfrak{J}}_1^c\subseteq \left\{\mathfrak{j}\in \mathbb{N}:\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})>1-\rho ,\nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})<\rho \mathrm{and}\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})<\rho \right\}.$$

(6)

Take $m\in \mathfrak{J}$. Then,

${\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mathfrak{j}\in {\mathfrak{J}}_1\right\}\right|<ϵ$

$\Rightarrow {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mathfrak{j}\in {\mathfrak{J}}_1^c\right\}\right|\ge 1-ϵ$.

Therefore, it follows from (6) that

${\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})>1-\rho \mathrm{and}\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})<\rho \mathrm{and}\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})<\rho \end{array}\right\}\right|\ge 1-ϵ$.

$\Rightarrow {\displaystyle \frac{1}{{({\beta }_m+1-{\alpha }_m)}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_m,{\beta }_m];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})\le 1-\rho \mathrm{or}\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})\ge \rho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})\ge \rho \end{array}\right\}\right|<ϵ$.

Thus, we obtain

$\mathfrak{J}\subseteq \left\{\begin{array}{c}\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})\le 1-\rho \mathrm{or}\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})\ge \rho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})\ge \rho \end{array}\right\}\right|<ϵ\end{array}\right\}$.

Since $\mathfrak{J}\notin \mathrm{I}$, the collection

$\left\{\begin{array}{c}\mathfrak{n}\in \mathbb{N}:{\displaystyle \frac{1}{{({\beta }_{\mathfrak{n}}+1-{\alpha }_{\mathfrak{n}})}^{\gamma }}}\left|\left\{\begin{array}{c}\mathfrak{j}:\mathfrak{j}\in [{\alpha }_{\mathfrak{n}},{\beta }_{\mathfrak{n}}];\mu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})\le 1-\rho \mathrm{or}\\ \nu ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})\ge \rho ,\omega ({\mathfrak{x}}_{\mathfrak{j}}-\mathfrak{z},\varsigma +\mathfrak{t})\ge \rho \end{array}\right\}\right|<ϵ\end{array}\right\}\notin I$.

Therefore, $\mathfrak{z}\in {\Gamma }_{(\mu ,\nu ,\omega ){\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$. Hence,

$${\displaystyle {\Gamma }_{(\mu ,\nu ,\omega ){\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)\supseteq \bigcup _{\mathfrak{x}\in {\Gamma }_{(\mu ,\nu ,\omega )\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)}cl\left({\mathfrak{B}}_{\mathfrak{x}}^{(\mu ,\nu ,\omega )}(\mathfrak{t},\varrho )\right).}$$

(7)

Conversely, assume that $\mathfrak{y}\in {\Gamma }_{(\mu ,\nu ,\omega ){\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$, but ${\displaystyle \mathfrak{y}\notin \bigcup _{\mathfrak{x}\in {\Gamma }_{(\mu ,\nu ,\omega )\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)}cl\left({\mathfrak{B}}_{\mathfrak{x}}^{(\mu ,\nu ,\omega )}(r,\varrho )\right)}$. Then $\mathfrak{y}\notin cl\left({\mathfrak{B}}_{\mathfrak{x}}^{(\mu ,\nu ,\omega )}(\mathfrak{t},\varrho )\right)$ for any $\mathfrak{x}\in {\Gamma }_{(\mu ,\nu ,\omega )\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$. Therefore, $\mathfrak{y}\notin {\Gamma }_{(\mu ,\nu ,\omega ){\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ follows from Lemma 3, which defies what we assumed. Thus,

$${\displaystyle {\Gamma }_{(\mu ,\nu ,\omega ){\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}\left(\mathfrak{x}\right)\subseteq \bigcup _{\mathfrak{x}\in {\Gamma }_{(\mu ,\nu ,\omega )\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)}cl\left({\mathfrak{B}}_{\mathfrak{x}}^{(\mu ,\nu ,\omega )}(\mathfrak{t},\varrho )\right)}$$

(8)

From (7) and (8), the result follows. □

We illustrate this relationship in the corollary that follows, where we consider the collection of ${\mathrm{I}}_{\mathfrak{t}}-{\mathrm{st}}_{\alpha \beta }^{\gamma }$–cluster points and the limit set of $\mathcal{RI}$–$\alpha \beta$–$\mathcal{STC}$ of order $\gamma$.

Corollary 1. 

Let $(\mathfrak{X},\mu ,\nu ,\omega ,★,\circ ,⊛)$ be an $\mathfrak{NNS}$. For $\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ in $\mathfrak{X}$, if ${\displaystyle {(\mu ,\nu ,\omega )}_{\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}}$ exists then ${\Gamma }_{(\mu ,\nu ,\omega ){\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)\subseteq {(\mu ,\nu ,\omega )}_{{\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}-LIM\left({\mathfrak{x}}_{\mathfrak{j}}\right)$ for some $\mathfrak{t}>0$.

Proof. 

Suppose ${\displaystyle {(\mu ,\nu ,\omega )}_{\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}-\underset{\mathfrak{j}}{lim}{\mathfrak{x}}_{\mathfrak{j}}=\mathfrak{x}}$. Then ${\Gamma }_{(\mu ,\nu ,\omega )\mathrm{I}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)=\left\{\mathfrak{x}\right\}$. Hence, by Theorem 7,

$${\Gamma }_{(\mu ,\nu ,\omega ){\mathrm{I}}_{\mathfrak{t}}}^{s{t}_{\alpha \beta }^{\gamma }}\left({\mathfrak{x}}_{\mathfrak{j}}\right)=cl\left({\mathfrak{B}}_{\mathfrak{x}}^{(\mu ,\nu ,\omega )}(\mathfrak{t},\rho )\right).$$

(9)

for some $\rho \in (0,1)$ and $\mathfrak{t}>0$. The result is derived from (9) and Theorem 5. □

5. Conclusions

In this paper, we introduced the concept of rough I–$\alpha \beta$–statistical convergence of order $\gamma$ in neutrosophic normed spaces, which unifies and extends several established modes of convergence, including rough convergence, ideal convergence, and statistical convergence. By incorporating the neutrosophic framework, the proposed approach effectively handles indeterminacy and imprecision inherent in real-world data. We investigated several fundamental properties of this convergence, including the convexity and closedness of the rough statistical limit set, and established inclusion relations between different classes of cluster points and limit sets. The examples provided illustrated the novelty and practical relevance of the new convergence notion. These results not only generalize previous findings in classical normed spaces but also give rise to fresh research in neutrosophic functional analysis, particularly in the study of sequence spaces, operator theory, and approximation processes under uncertainty. Future work may explore applications of rough I–$\alpha \beta$–statistical convergence in solving differential equations, optimization problems, and modeling in data science where neutrosophic structures naturally arise.