An Introduction to Single-Valued Neutrosophic Primal Theory

: This article explores the interconnections among the single-valued neutrosophic grill, single-valued neutrosophic primal and their stratification, uncovering their fundamental characteristics and correlated findings. By introducing the notion of a single-valued neutrosophic primal, a broader framework including the fuzzy primal and intuitionistic fuzzy primal is established. Additionally, the concept of a single-valued neutrosophic open local function for a single-valued neutrosophic topological space is presented. We introduce an operator based on a single-valued neutrosophic primal, illustrating that the single-valued neutrosophic primal topology is finer than the single-valued neutrosophic topology. Lastly, the concept of single-valued neutrosophic open compatibility between the single-valued neutrosophic primal and single-valued neutrosophic topologies is introduced, along with the establishment of several equivalent conditions related to this notion.


Introduction
Topology, a highly versatile field of mathematics [1], finds extensive application across both the scientific and social science domains, prompting the emergence of numerous innovative concepts within its standard frameworks.Kuratowski [2] examined the notion of ideals derived from filters, which can be seen as dual to filters.The notion of the fuzzy grill was given by [3].Chattopadhyay and Thron [4] utilized grills to establish various topics, including closure spaces, while Thron [5] defined proximity structures within grills.Roy et al. [6] introduced novel definitions related to grills, with Roy and Mukherjee [7][8][9] subsequently exploring diverse topological properties associated with grills.Numerous applications stemming from these studies are documented in various works [10][11][12][13][14][15][16].Given the dual nature of primal concerning grills, we draw inspiration from Jankovi ć and Hamlett [17] to introduce a new topology based on ideal structures.
The concept of neutrosophic sets, introduced as a generalization of intuitionistic fuzzy sets, was initially proposed in [18].Salama et al. [19] and Wang et al. [20] have extensively investigated neutrosophic sets and their single-valued neutrosophic, abbreviated as svn , counterparts.Numerous applications stemming from these studies are documented in various works [21][22][23][24].Stratified single-valued soft topogenous structures have been studied by Alsharari et al. [25].Saber et al. have conducted extensive research on single-valued neutrosophic soft uniform spaces, single-valued neutrosophic ideals, abbreviated as svnis, and the connectedness and stratification of single-valued neutrosophic topological spaces expanded with an ideal [26][27][28].The neutrosophic compound orthogonal neural network (NCONN), for the first time, contained the NsN weight values, NsN input and output, and hidden layer neutrosophic neuron functions; to approximate neutrosophic functions, NsN data have been studied by Ye et al. [29].Shao et al. [30] introduced the concept of the probabilistic single-valued (interval) neutrosophic hesitant fuzzy set, extensively investigating the operational relations of PINHFS and the comparison method of probabilistic interval neutrosophic hesitant fuzzy numbers (PINHFNs).Rıdvan et al. [31] examined the notion of the neutrosophic subsethood measure for single-valued neutrosophic sets.The neutrosophic fuzzy set and its application in decision-making was defined by Das et al. [32].
The objective of this paper is to explore the inter-relations between the single-valued neutrosophic grill (svn-grill) and single-valued neutrosophic primal (svn-primal), along with their stratification, while showcasing some of their inherent properties.Additionally, we investigate the quantum behaviors within a novel structure denoted as Ξ ⋆ r (T τπσ , P τπσ ), as defined in Definition 11.Furthermore, we introduce and analyze both the svn primal and its associated topology.We also derive several preservation properties and characterizations regarding svn-primal open compatibility.
We begin with the definition of a neutrosophic set as follows: Definition 1 ([18]).Let £ be a non-empty set.An n-set on £ is defined as representing the degree of membership where (τ Ξ (z)), the degree of indeterminacy (π Ξ (z)) and degree of nonmembership (σ Ξ (z)); ∀ z ∈ £ to the set Ξ.
We now discuss the concept of the svn-set, which is a more specific type of neutrosophic set.Definition 2 ([20]).Let £ be a space of points (objects) with a generic element in £ denoted by z.
In this case, σ Ξ , π Ξ and τ Ξ are called the falsity membership function, indeterminancy membership function and truth membership function, respectively.
To better understand the properties of svn-sets, we will now discuss the complement of an svn-set.Definition 3 ([20]).Let Ξ = {⟨x, τ Ξ (z), π Ξ (z), σ Ξ (z)⟩}e : x ∈ £} be an svn-set on £.The complement of the set Ξ (Ξ c ) is defined as follows: The following definition provides more insight into the relationships between svn-sets, introducing the notions of subsets, equality and special sets.
In the context of svn-sets, we introduce definitions related to the intersection and union of svn-sets.Further, we discuss the concept of a single-valued neutrosophic topological space (svnts) and the properties it entails.
The following theorem establishes an operator that satisfies specific conditions, which further clarifies the properties of svnts.

Stratified Single-Valued Neutrosophic Grill with Single-Valued Neutrosophic Primal
In this section, we explores the interconnections between single-valued neutrosophic grills (svn-grills ) and single-valued neutrosophic primals (svn-primals), along with their stratification.We also present a new structure within the context of single-valued neutrosophic topology, referred to as an svn-primal.This novel structure is the dual counterpart to the svn-grill.
We start with the following definition: For the sake of brevity and clarity, we will occasionally denote (G τ , G π , G σ ) as G τπσ .A single-valued neutrosophic grill topological space (svngts) is defined as the triple (£, T τπσ , G τπσ ).
Proof.First, we will prove that On the other hand, Then, we have This is a contradiction.Thus, (G 3 ) holds.
(G st ) Assume that there exist Ξ ∈ ξ £ and α ∈ ξ such that This is a contradiction.Thus, G st holds.Hence, Finally, consider (G ⋆τ , G ⋆π , G ⋆σ ), a stratified svn-grill on £ which is finer than (G τ , G π , G σ ).And we will show that G τ On the other hand, (G ⋆τ , G ⋆π , G ⋆σ ) is stratified; then, we have Likewise, we can establish through a similar line of reasoning that In order to better understand the notion of svn-primal mappings, let us provide some context for the following definition.Consider a non-empty set £ and a mapping P τ , P π , P σ : ζ £ → ζ.We will now introduce certain conditions that, when satisfied, characterize the mapping as an svn-primal on £.Definition 8. Let £ be a non-empty set.A mapping P τ , P π , P σ : ζ £ → ζ is said to be svn-primal on £, if it meets the following conditions: Sometimes, we will write P τπσ for (P τ , P π , P σ ).
Proposition 1.Let {P τπσ j } j∈J be a collection of svn-primals on £.Then, their union i∈J P τπσ j is also an svn-primal on £.
Proof.Directly from Definition 7.

Single-Valued Neutrosophic Primal Open Local Function in Šostak Sense
In this section, we investigate the concept of svn-primal open local functions within the context of Šostak's sense.Our primary focus is to explore the properties of these functions and their relationship with svnts and neutrosophic primals.Through a series of definitions, theorems and discussions, we aim to provide a comprehensive understanding of this unique concept and its implications in the domain of neutrosophic topology.The introductory results presented here lay the foundation for further exploration of this fascinating topic, shedding light on the details of svn structures in the sense of Šostak.Definition 9. Let n, m, v ∈ ξ 0 and n + m + v ≤ 3. A single-valued neutrosophic point (svn-point) y n,m,v is an svn-set in ξ £ for each z ∈ Θ, defined by We indicate the set of all svn-points in £ as (svn-point (£)).
For each y n,m,v ∈ svn-point(£) and Θ ∈ ξ ω we shall write y n,m,v quasi-coincident with Ξ, denoted by y n,m,v qΞ, if For all Θ, Ξ ∈ ξ £ we shall write Ξ q θ to mean that Ξ is quasi-coincident with Θ if there exists z ∈ £ such that Definition 10.Let (£, T τπσ ) be an svnts, for all Ξ ∈ ξ £ , y n,m,v ∈svn-point (£) and r ∈ ξ 0 .Then, Ξ is said to be an r-open Q T τπσ -neighborhood (r-OQN) of y n,m,v , defined as follows Lemma 1.An svn-point y n.m.v ∈ C T τπσ (Θ, r) iff every r-OQN of y n,n,v is quasi-coincident with Ξ. Definition 11.Let (£, T τπσ , P τπσ ) be an svnpts, for each r ∈ ξ 0 and Ξ ∈ ξ £ .Then, the singlevalued neutrosophic primal open local function Ξ ⋆ r (T τπσ , P τπσ ) of Ξ is the union of all svn-points y n,m,v such that if Θ ∈ Q T τπσ (y n,m,v , r) and In this article, we will write Ξ ⋆ r for Ξ ⋆ r (T τπσ , P τπσ ) without any ambiguity.
We will now prove this relationship Since r , r).So, there exists at least one x ∈ £ with A ∈ Q T τπσ (y n,m,v , r) such that r).This is also true for A. So, there is at ).Since A is an arbitrary and A ∈ Q T τπσ (y n,m,v , r), then This contradicts Equation (4).Thus, Ξ ⋆ r ≥ C T τπσ (Ξ ⋆ r , r). (4) Using (3) we obtain that (Ξ r) such that for each x ∈ £ and for some P τ (Π 1 ) ≥ r, P π (Π 1 ) ≤ 1 − r, P σ (Π 1 ) ≤ 1 − r we have, Similarly, there exists A 2 ∈ Q T τπσ (y n,m,v , r) such that for each x ∈ £ and for some P r) and by (P 3 ), we obtain 7) and ( 8) are obvious.

Conclusions
In this paper, first, we investigated the complex area of stratification of svn grills and determined some of their fundamental features.The links between svn grills and svn-primals were investigated.We also introduced and explored the concept of svn-primal open local functions in the context of Šostak's sense.By extending the notions of svn sets and related topological structures, we have presented a novel approach to understanding the properties and relationships within this unique framework.
Our investigation began by defining svn-sets and their corresponding notions.Building upon these fundamental definitions, we introduced svnts and explored various operations and properties within these spaces, such as the neutrosophic closure and interior.A central contribution of this work has been the introduction and exploration of svnprimals and their associated operators.We have provided several equivalent conditions characterizing the compatibility of svnts with neutrosophic primals.Additionally, we discussed the properties of primal open local functions, their relationship with svnts, and the induced operators.
In conclusion, the results presented in this paper contribute to the growing field of neutrosophic topology by offering a deeper understanding of svn structures in Šostak's sense.The properties and correlations investigated here establish the framework for further studies in this field, opening options for future investigations and applications of these innovative notions in various fields.
In terms of related and future research directions, it is of great interest to investigate the connections between our findings and the advancements in the field of neural networks (Gu and Sheng [36]; Gu et al. [37]; Deng et al. [38]; Gu et al. [39]).Furthermore, the possible connections to multidimensional systems and signal processing need further investigation, as illustrated by Wang et al. [40] and Xiong et al. [41].
By bridging the gap between svn structures and related domains, we might promote collaboration across disciplines and discover new applications for our findings.This not only enriches the discipline of neutrosophic topology, but also benefits the larger scientific community by providing new insights and fresh approaches to difficult problems.
For forthcoming papers The theory can be extended in the following normal methods.1-Basic concepts of neutrosophic metric topological spaces can be studied using the notion of svn-primal present in this article; 2-Examine the connected, separation axioms and soft closure spaces in the context of svn-primals.