Structural Theory of Interval-Valued Neutrosophic Ẑ-Ideals in Ẑ-algebraic Structures

Neutrosophic Sets and Systems, Vol. 97, 2026 University of New Mexico Shanmugapriya K P , Hemavathi P ,Vinod Kumar R, Saeid Jafari,Structural Theory of Intervalvalued Neutrosophic Ẑ-Ideals in Ẑ-algebraic Structures Structural Theory of Interval-Valued Neutrosophic Ẑ-Ideals in Ẑ-algebraic Structures Shanmugapriya K P 1, Hemavathi P 2,* ,Vinod Kumar R3, Saeid Jafari 4 1 Department of Mathematics, Saveetha Institute if Medical and Technical Sciences (SIMATS), Saveetha School of Engineering, Thandalam; 602 105, [email protected] 2 Department of Mathematics, Saveetha Institute if Medical and Technical Sciences (SIMATS), Saveetha School of Engineering, SIMATS; [email protected] 3 Department of Mathematics, Rajalakshmi Engineering College (Autonomous), Thandalam, Chennai, 602 105, India.; [email protected] 4 College of Vedtsjaelland South and Mathematical and Physical Science Foundation, Sidevej 5, 4200 Slagelse, Denmark; [email protected] * [email protected] Abstract: This paper presents a structural theory of Interval-Valued Neutrosophic (IVN) Ẑ-ideals in Ẑ-algebraic structures to address an ongoing problem of representing uncertainty and indeterminacy. The available fuzzy and Neutrosophic ideals provide useful methods, but they lack the expressive power to portray simultaneous fluctuations in truth, falsehood, and indeterminacy. To deal with this, presented an IVN membership functions, in which each component is represented as an interval rather than a single value, resulting in a more flexible and realistic representation of uncertain phenomena. The work defines IVN Ẑ-ideals and discusses their structural, closure, and include relationships. It has been demonstrated step by step to enhance transparency in result development and are accompanied by illustrated examples to show how they differ from classical fuzzy and Neutrosophic ideals. A comparative investigation reveals that IVN-based techniques outperform current techniques in representing complex uncertainty and indeterminacy. Though the proposed framework offers an important generalisation, it has limits in terms of computational complexity and scalability. Additional study may be done regarding algorithmic optimisation, its application to large-scale algebraic systems, decision-making, and artificial intelligence. In summary, the study strengthens theoretical principles of algebraic models with uncertainty and leads towards more advanced applications. Keywords: Fuzzy set, Fuzzy Ẑ-ideal, Neutrosophic Fuzzy Ẑ-ideal, IVN Ẑideal. 1. Introduction In the current era of algebraic analysis, soft computing techniques like fuzzy set theory and neutrosophic set theory play a pivotal role in resolving uncertainty and vagueness in complex algebraic structures. Among these developments, the study of Ẑ-algebras has opened up new dimensions in algebraic frameworks, giving rise to various types of ideal structures under fuzzy and Neutrosophic Sets and Systems, Vol. 97, 2026 440 Shanmugapriya K P , Hemavathi P ,Vinod Kumar R, Saeid Jafari,Structural Theory of Intervalvalued Neutrosophic Ẑ-Ideals in Ẑ-algebraic Structures neutrosophic environments. The introduction of fuzzy set theory by Zadeh [18] in 1965 was a significant milestone that allowed representation of data with gradations rather than absolute membership, which led to applications across diverse fields. The progress of fuzzy ideals in Ẑ-algebras has been well established by Sowmiya and Jeyalakshmi through their exploration of fuzzy algebraic structures [14], fuzzy ideals [15], and homomorphisms in fuzzy contexts [16]. These investigations laid the foundation for deeper understanding of how fuzzy mappings interact with Ẑ-algebraic operations. Hemavathi et al. [4] made significant contributions to the theory of IV fuzzy β-subalgebras, emphasizing the usefulness of interval information in fuzzy environments. This motivated researchers to explore further generalizations of fuzzy sets, particularly Neutrosophic and IVN sets. Smarandache [12,13] developed the concept of neutrosophy, introducing neutrosophic sets to model incomplete, indeterminate, and inconsistent information, thus generalizing classical and intuitionistic fuzzy frameworks. This new mathematical paradigm found its utility in various algebraic structures. Wang et al. [17] offered rigorous treatment of IVN sets, which further enhanced the modeling capabilities for uncertain environments in logic and computation. The concept of Neutrosophic ideals, especially in the context of Ẑ-algebras, gained attention through works such as those by Muralikrishna and Manokaran [7], and Nagaiah et al. [8], who analyzed Neutrosophic algebra from a foundational perspective. In recent contributions, Shanmugapriya and Hemavathi [10,11] focused specifically on Neutrosophic fuzzy sets and IVN structures in Ẑ-algebras. These studies introduce new ideal types and substructures that exhibit algebraic consistency while accommodating indeterminacy. The evolution of neutrosophic structures progressed toward INK-algebras and their associated ideal systems. Al Omeri et al. [1] introduced the notion of translation in Neutrosophic INK-algebras, emphasizing the utility of these transformations in extending existing ideal theories. The work of Rajakumari et al. [6] on IVN INK ideals via INK-algebra provided a detailed framework that synthesized the ideas of fuzzy, Neutrosophic, and IV sets under the algebraic structures. Other vital studies include that of Chandramouleeswaran et al. [3], discussed the structural aspects of Ẑ-algebras, and Jun et al. [5], who explored cubic subalgebras in broader algebraic settings. These papers provided theoretical scaffolding that influenced the conceptualization of new fuzzy and Neutrosophic ideals. Satyanarayana and Baji [2,9] also contributed to the understanding of IVN ideals in d-algebras, which conceptually parallels the IVN ideals in Ẑ-algebras. Overall, these advancements have led to a rich algebraic landscape where various generalized ideals such as fuzzy, neutrosophic, and IVN ideals can be studied within Ẑ-algebraic and INKalgebraic frameworks. This paper builds upon these contributions, focusing on a structural approach to analyzing IVN Ẑ-ideals, expanding the algebraic structures and fostering further development in the domain. 1.1 The structure of the paper is outlined below: Section-1: This section provides an introduction to the concept of IVN sets and their significance in extending classical Ẑ-algebra. It outlines the motivation, objectives, and scope of the study, emphasizing the need for handling uncertainty and indeterminacy in algebraic structures. Section-2: Presents the essential preliminaries and background definitions related to Ẑ-algebra and IVN sets. It includes basic notations, operations, and structural properties relevant to the proposed work. Section-3: Introduces the concept of IVN Ẑ-ideal in Ẑ-algebra. The section defines this new class of ideals, discusses its algebraic properties, and illustrates them through examples. Section-4: Explores the theme of IVN Ẑ-ideals under homomorphisms of Ẑ-algebras. Key theorems are established to demonstrate how these ideals are preserved or transformed under structure-preserving mappings. Neutrosophic Sets and Systems, Vol. 97, 2026 441 Shanmugapriya K P , Hemavathi P ,Vinod Kumar R, Saeid Jafari,Structural Theory of Intervalvalued Neutrosophic Ẑ-Ideals in Ẑ-algebraic Structures Section-5: The paper concludes with a summary of the main contributions and suggests directions for future research in the field of fuzzy and neutrosophic algebraic systems. 1.2. Research Gaps: 1.2.1 While Ẑ-ideals, fuzzy Ẑ-ideals, and Interval-Valued fuzzy Ẑ-ideals have been widely studied, Neutrosophic extensions of Ẑ-ideals, particularly IVN Ẑ-ideals, have yet to be thoroughly investigated. 1.2.2 Current fuzzy and interval-valued fuzzy models cannot express indeterminacy as well as truth and false, making them insufficient for describing complicated uncertainty in algebraic systems. 1.2.3 Comparative studies on classical fuzzy ideals, fuzzy Ẑ-ideals, Interval-Valued fuzzy Ẑ-ideals, and Neutrosophic-related ideals in the context of Z-algebra have offered limited insight into the relative advantages and disadvantages of each of these techniques. 1.3. Objective of the Study: 1.3.1 To describe their structural characteristics of IVN Ẑ-ideals to Ẑ-algebraic structures. Provide detailed demonstrations and examples that clarify the theoretical framework. 1.3.2 Conduct a comparative analysis with the currently available fuzzy, neutrosophic, and neutrosophic BCK-ideals to demonstrate the benefit of the IVN approach. 1.3.3 To address limits and research limitations, it is crucial to have a framework in place that allows for practical extension. 1.4 The study makes the following key contributions: 1.4.1 New Framework: It gives Interval-Valued Neutrosophic (IVN) membership functions for Ẑideals while also generalising conventional fuzzy and Neutrosophic theories. 1.4.2 Structural Theory: Theorems, proofs, and fundamental definitions of IVN Ẑ-ideals are developed systematically. 1.4.3 Comparison perspective: The results of this study show inconsistencies between the fuzzy, Neutrosophic, Neutrosophic BCK-ideals and IVN Ẑ-ideals and provide a more accurate description of their strengths and weaknesses. 1.4.4 Theoretical Illustrations: Illustrations based on the Cayley table are presented to demonstrate the certainty of the offered definitions and concepts that are theoretical. 1.4.5 Application Pathway: The structure is theoretical, but it may be expanded in the future to include decision-making, artificial intelligence, and other applications that require thorough modelling of uncertainty and indeterminacy. 2. Preliminaries The particular notations are used in this research as follows X specified as 𝔐, x denoted as ℰ, y specified as 𝛿 and Y represented as 𝔙. In the following section, discussed about the necessary definitions that are involved for this research article. Definition 2.1.[18] Let 𝜁 be Fuzzy set in the non-empty set 𝔐. The set 𝜁 = {ℰ : 𝜇𝜁(ℰ) / ℰ ∈ 𝔐 }for all ℰ ∈𝔐. Definition 2.2.[4] An IVN Fuzzy set on 𝔚 is to be defined on 𝔚 , 𝜁 ={ℰ, 𝜇𝜁 (ℰ)/ ℰ ∈ 𝔚 }, briefly denoted by, 𝜇𝜁(ℰ)= [𝜇𝜁𝐿(ℰ),𝜇𝜁𝑈(𝜖)], where 𝜇𝜁𝐿(ℰ) & 𝜇𝜁𝑈(ℰ) are the two fuzzy sets in 𝔚 such that 𝜇𝜁𝐿(ℰ) Neutrosophic Sets and Systems, Vol. 97, 2026 442 Shanmugapriya K P , Hemavathi P ,Vinod Kumar R, Saeid Jafari,Structural Theory of Intervalvalued Neutrosophic Ẑ-Ideals in Ẑ-algebraic Structures ≤ 𝜇𝜁𝑈(ℰ) for all ℰ∈ 𝔚. Let 𝜇𝜁(ℰ) = [𝜇𝜁𝐿(ℰ), 𝜇𝜁𝑈(ℰ)] ∀ ℰ∈ 𝔚, and let 𝔻[0,1] denotes the family of all closed sub-intervals of [0,1]. If 𝜇𝜁𝐿(ℰ) = 𝜇𝜁𝑈(ℰ) =𝔠, where 0≤𝔠≤1, then there exists 𝜇𝜁(ℰ) = [𝔠, 𝔠] = 𝔠. For the convenience, ℰ belongs to 𝔻 [0,1] ∀ ℰ ∈ 𝔚, ∴ The IVN fuzzy set is given by 𝜁 ={ ℰ, 𝜇𝜁 (ℰ)/ ℰ ∈ 𝔚}, where 𝜇𝜁: 𝔚 → 𝔻 [0,1]. Now, Define a refined minimum (briefly rmin) of two elements in 𝔻[0,1]. Define the symbols "≤ ", "≥" & "=". In case, if two elements are in 𝔻 [0,1], then it will be expressed by 𝔻1:=[ 𝛼1,𝛽1], 𝔻2:=[ 𝛼2,𝛽2] ∈ 𝔻 [0,1]. Then, rmin (𝔻1,𝔻2) = [min{ 𝛼1,𝛼2}, min {𝛽1,𝛽2}], 𝔻1≥𝔻2 iff 𝛼1≥𝛼2, 𝛽1≥𝛽2. Similarly, there exist 𝔻1≤𝔻2 & 𝔻1=𝔻2. Definition 2.3.[3] A Ẑ-algebra (𝔐,⁕,0) is defined as a structure, where 𝔐 is a non-empty set, with constant 0 and ⁕ is a binary operation on 𝔐, satisfying the following conditions i) ℰ ⁕ 0 = 0 ii) 0 ⁕ ℰ = ℰ iii) ℰ ⁕ ℰ = ℰ iv) ℰ ⁕ 𝛿 = 𝛿 ⁕ ℰ, when ℰ ≠ 0 and 𝛿≠ 0 ∀ ℰ,𝛿∈ 𝔐. Example 2.4. Consider a Cayley’s table, let {𝔐 = 0, a1,a2,a3,a4}, with binary operation * and constant 0 * 0 a1 a2 a3 a4 0 0 a1 a2 a3 a4 a1 0 a1 a2 a1 a3 a2 0 a4 a2 a1 a3 a3 0 a2 a1 a3 a4 a4 0 a3 a3 a4 a4 Definition 2.5.[3] Let 𝕊 be a non-empty subset of Ẑ-algebra of 𝔚, Then 𝕊 is defined to be a Ẑsubalgebra of 𝔚, if (ℰ ⁕ 𝛿) ∈ 𝕊, ∀ ℰ,𝛿 ∈ 𝕊. Definition 2.6.[14] Let (𝔚, ⁕,0) be Ẑ– algebra. An IVN fuzzy set 𝜇𝜁 is represented as an IVN fuzzy Ẑsubalgebra of a Ẑ– algebra 𝔚, if 𝜇𝜁 (ℰ ⁕ 𝛿) ≥ rmin {𝜇𝜁(ℰ), 𝜇𝜁(𝛿)} ∀ ℰ,𝛿 ∈ 𝔚. Definition 2.7.[16] Let (𝔐,⁕,0) & (𝔜, ⁕′,0′) be two Ẑ-algebras. Then, the mapping 𝕙: (𝔐,⁕,0) ⟶(𝔜, ⁕′,0′) is known as Ẑhomomorphism of Ẑ-algebras, if 𝕙(ℰ ⁕ 𝛿) = 𝕙(ℰ) ⁕′ 𝕙(𝛿) ∀ ℰ,𝛿∈ 𝔐. Neutrosophic Sets and Systems, Vol. 97, 2026 443 Shanmugapriya K P , Hemavathi P ,Vinod Kumar R, Saeid Jafari,Structural Theory of Intervalvalued Neutrosophic Ẑ-Ideals in Ẑ-algebraic Structures Definition 2.8. Consider a Ẑ-homomorphism 𝕙 mapping the Ẑ-algebra (𝔐,⁕,0) onto the another Ẑalgebra (𝔜, ⁕′,0′). Under this mapping, the following properties are satisfied: i) 𝕙 is called as a monomorphism of Ẑ-algebras, if it is injective (one to one). ii) 𝕙 is called an epimorphism of Ẑ-algebras if it is surjective (onto). Definition 2.9.[14] Consider a Ẑ-algebra (𝔐,⁕,0). A fuzzy set 𝜁 on 𝔐, characterized by the membership_function 𝜇𝜁, is called a fuzzy Ẑ-subalgebra of 𝔐, if for all ℰ,𝛿∈ 𝔐, the following condition is satisfied 𝜇𝜁(ℰ ⁕ 𝛿) ≥ min{ 𝜇𝜁(ℰ), 𝜇𝜁(𝛿)}. Definition 2.10.[14] Consider a fuzzy set 𝜁 defined on 𝔐. For a fixed 𝔱 ∈ [0,1], the set U (𝜁, 𝔱) = {ℰ∈ 𝔐 | 𝜇𝜁(ℰ) ≥ 𝔱} is called the upper-level subset of 𝜁, also known as the upper-level cut, upper 𝔱-level subset. Definition 2.11.[15] Let 𝔐 be Ẑ-algebra and ℑ ⊆ 𝔐. Then, ℑ is known as Ẑ-ideal of 𝔐, if the following conditions hold, ∀ ℰ,𝛿∈ 𝔐: i) 0 ∈ ℑ ii) ℰ ⁕ 𝛿 ∈ℑand 𝛿 ∈ ℑ implies ℰ ∈ ℑ Definition 2.12.[15] Let ( 𝔐,⁕,0) be a Ẑ-algebra. The fuzzy set 𝜁 on 𝔐, defined by the membership_function 𝜇𝜁, is called as fuzzy Ẑ-ideal of the Ẑ-algebra for every ℰ,𝛿 in 𝔐, if it holds the following properties, i) 𝜇𝜁(0) ≥ 𝜇𝜁(ℰ) ii) 𝜇𝜁(ℰ) ≥ min {𝜇𝜁(ℰ⁕𝛿), 𝜇𝜁(𝛿)} Definition 2.13.[10] Let the neutrosophic set 𝜁= { ℰ: T𝜁(ℰ), I𝜁(ℰ), F𝜁(ℰ)/ ℰ ∈ 𝔐 }, if it holds the following conditions, then it is said to be Neutrosophic fuzzy in Ẑ-algebra, i) T𝜁( ℰ ⁕ 𝛿) ≥ min{T𝜁(ℰ), T𝜁(𝛿)} ii) I𝜁(ℰ ⁕ 𝛿) ≥ min{I𝜁(ℰ), I𝜁(𝛿)} iii) F𝜁(ℰ ⁕ 𝛿) ≤ max{F𝜁(ℰ), F𝜁(𝛿)} 3. IVN Ẑ - ideal in Ẑ - algebra This section explores IVN Ẑ-ideals in Ẑ-algebras, merging IVN sets with algebraic frameworks. Definition 3.1. An IVN Ẑ-algebra 𝕸 is called an IVN Ẑideal of 𝕸 , if it satisfies i) 𝜻𝑻(0) ≥ 𝜻𝑻 (𝓔), 𝜻𝑻(𝓔) ≥ rmin {𝜻𝑻(𝓔⁕𝜹),𝜻𝑻(𝜹)} ii) 𝜻𝑰(0) ≥ 𝜻𝑰 (𝓔), 𝜻𝑰( 𝓔) ≥ rmin {𝜻𝑰(𝓔⁕𝜹),𝜻𝑰(𝜹)} iii) 𝜻𝑭(0) ≤ 𝜻𝑭 (𝓔), 𝜻𝑭( 𝓔) ≤ rmax {𝜻𝑭(𝓔⁕𝜹),𝜻𝑭(𝜹)} ∀ 𝓔 , 𝜹 ∈ 𝕸. Example 3.2. Refer to Example 2.4 for illustration 𝜻𝑻,𝑰,𝑭={[𝟎.𝟒,𝟎.𝟓] 𝓔=𝟎,𝐰𝐡𝐞𝐧 𝓔 ≠ 𝟎 𝐚𝐧𝐝 𝜹≠ 𝟎 [𝟎.𝟐,𝟎.𝟔] 𝓔= 𝐚𝟏,𝐚𝟑 [𝟎.𝟏,𝟎.𝟑] 𝓔= 𝐚𝟐,𝐚𝟒 Hence, the above example satisfies the condition of IVN Ẑ-ideal of Ẑ-algebra. Theorem 3.3. An IVN Ẑideal 𝜁 = [𝜁𝑇,𝜁𝐼,𝜁𝐹] in 𝔐 is an IVN Ẑideal of 𝔐 iff 𝜁𝑇𝐿,𝜁𝑇𝑈, 𝜁𝐼𝐿, 𝜁𝐼𝑈, 𝜁𝐹𝐿 and 𝜁𝐹𝑈 are Neutrosophic Ẑideal of 𝔐. Neutrosophic Sets and Systems, Vol. 97, 2026 444 Shanmugapriya K P , Hemavathi P ,Vinod Kumar R, Saeid Jafari,Structural Theory of Intervalvalued Neutrosophic Ẑ-Ideals in Ẑ-algebraic Structures Proof : Let 𝜁𝑇𝐿,𝜁𝑇𝑈, 𝜁𝐼𝐿, 𝜁𝐼𝑈, 𝜁𝐹𝐿 and 𝜁𝐹𝑈 are Neutrosophic Ẑideal of 𝔐 & ℰ , 𝛿 ∈ 𝔐. i) 𝜁𝑇 𝐿(0) ≥ 𝜁𝑇𝐿(ℰ) & 𝜁𝑇𝑈(0) ≥ 𝜁𝑇𝑈(ℰ) Therefore, 𝜁𝑇(0) ≥ 𝜁𝑇(ℰ) 𝜁𝑇(ℰ) = [ 𝜁𝑇𝐿(ℰ),𝜁𝑇𝑈(ℰ)] ≥ min {𝜁𝑇𝐿(ℰ⁕𝛿), 𝜁𝑇𝐿(𝛿)}, min {𝜁𝑇𝑈(ℰ⁕𝛿), 𝜁𝑇𝑈(𝛿)} ≥ rmin {[𝜁𝑇𝐿(ℰ⁕𝛿),𝜁𝑇𝑈(ℰ⁕𝛿)], [𝜁𝑇𝐿(𝛿),𝜁𝑇𝑈(𝛿)]} = rmin {𝜁𝑇( ℰ⁕𝛿), 𝜁𝑇(𝛿)}. ii) Similarly, 𝜁𝐼(ℰ) = rmin {𝜁𝐼( ℰ⁕𝛿), 𝜁𝐼(𝛿)}. iii) 𝜁𝐹𝐿(0) ≤ 𝜁𝐹𝐿(ℰ) & 𝜁𝐹𝑈(0) ≤ 𝜁𝐹𝑈(ℰ) Therefore, 𝜁𝐹(0) ≤ 𝜁𝑇(ℰ) 𝜁𝐹(ℰ) = [ 𝜁𝐹𝐿(ℰ),𝜁𝑇𝑈(ℰ)] ≤ max {𝜁𝐹𝐿(ℰ⁕𝛿), 𝜁𝐹𝐿(𝛿)}, max {𝜁𝐹𝑈(ℰ⁕𝛿), 𝜁𝐹𝑈(𝛿)} ≤ rmax {[𝜁𝐹𝐿(ℰ⁕𝛿),𝜁𝐹𝑈(ℰ⁕𝛿)], [𝜁𝐹𝐿(𝛿),𝜁𝐹𝑈(𝛿)]} = rmax {𝜁𝐹( ℰ⁕𝛿), 𝜁𝐹(𝛿)}. Hence, 𝜁𝑇𝐿,𝜁𝑇𝑈, 𝜁𝐼𝐿, 𝜁𝐼𝑈, 𝜁𝐹𝐿 and 𝜁𝐹𝑈 are IVN Ẑideals of 𝔐. Theorem 3.4. Every Ẑideal of 𝔐 can be realized as an IVN level set of IVN Ẑideal of 𝔐. Proof: Let 𝔙 be a Ẑideal of 𝔐, and let 𝜁 be an IVN set of 𝔐, defined by 𝜁𝑇(ℰ) = {[ℰ1,ℰ2],ℰ ∈ 𝔙 [1,1],ℰ ∉ 𝔙 where [ℰ1,ℰ2] ∈ [0,1] & ℰ1 >ℰ2 𝜁𝐼(ℰ) = {[ℰ1,ℰ2],ℰ ∈ 𝔙 [1,1],ℰ ∉ 𝔙 where [ℰ1,ℰ2] ∈ [0,1] & ℰ1 >ℰ2 𝜁𝐹(ℰ) = {[ℰ1,ℰ2],ℰ ∈ 𝔙 [0,0],ℰ ∉ 𝔙 where [ℰ1,ℰ2] ∈ [0,1] & ℰ1<ℰ2 It is clear that 𝔙(𝜁 ; [ℰ1,ℰ2] = 𝔙. Let 𝜁𝑇,𝐼,𝐹 be an IVN Ẑideal of 𝔐. Let ℰ , 𝛿 ∈ 𝔐. If (ℰ⁕𝛿) ∈ 𝔙 & 𝛿 ∈ 𝔙, then ℰ ∈ 𝔙 i) 𝜁𝑇(ℰ) = 𝜁𝑇(𝛿) = [ℰ1,ℰ2]. Thus, 𝜁𝑇(ℰ) ≥ rmin{𝜁𝑇(ℰ⁕𝛿), 𝜁𝑇(𝛿)}. If (ℰ⁕𝛿) ∉ 𝔙 & 𝛿 ∉ 𝔙, then 𝜁𝑇(𝛿) = 𝜁𝑇(ℰ⁕𝛿) = [1,1]. Thus, 𝜁𝑇(ℰ) ≥ [1,1] = rmin {𝜁𝑇(ℰ⁕𝛿), 𝜁𝑇(𝛿)}. If (ℰ⁕𝛿) ∈ 𝔙 & 𝛿 ∉ 𝔙, then 𝜁𝑇(ℰ⁕𝛿) = [ℰ1,ℰ2], 𝜁𝑇(𝛿) =[1,1]. Thus, rmin {𝜁𝑇(ℰ⁕𝛿), 𝜁𝑇(𝛿)} = [1,1]. Hence 𝜁𝑇(ℰ) ≥ rmin {𝜁𝑇(ℰ⁕𝛿), 𝜁𝑇(𝛿)}. ii) 𝜁𝐼(ℰ) =𝜁𝐼(𝛿) = [ℰ1,ℰ2]. Thus, 𝜁𝐼(ℰ) ≥ rmin{𝜁𝐼(ℰ⁕𝛿), 𝜁𝐼(𝛿)}.If (ℰ⁕𝛿) ∉ 𝔙 & 𝛿 ∉ 𝔙, then 𝜁𝐼(𝛿) = 𝜁𝐼(ℰ⁕𝛿) = [1,1]. Thus, 𝜁𝐼(ℰ) ≥ [1,1] = rmin {𝜁𝐼(ℰ⁕𝛿), 𝜁𝑇(𝛿)}. If (ℰ⁕𝛿) ∈ 𝔙 & 𝛿 ∉ 𝔙, then 𝜁𝐼(ℰ⁕𝛿) = [ℰ1,ℰ2], 𝜁𝐼(𝛿) =[1,1]. Thus, rmin {𝜁𝐼(ℰ⁕𝛿), 𝜁𝐼(𝛿)} = [1,1]. Hence 𝜁𝐼(ℰ) ≥ rmin {𝜁𝐼(ℰ⁕𝛿), 𝜁𝐼(𝛿)}. iii) 𝜁𝐹(ℰ) = 𝜁𝐹(𝛿) = [ℰ1,ℰ2]. Thus, 𝜁𝐹(ℰ) ≤ rmax{𝜁𝐹(ℰ⁕𝛿), 𝜁𝐹(𝛿)}. If (ℰ⁕𝛿) ∉ 𝔙 & 𝛿 ∉ 𝔙, then 𝜁𝐹(𝛿) = 𝜁𝐹(ℰ⁕𝛿) = [0,0]. Thus, 𝜁𝐹(ℰ) ≤ [0,0] = rmax {𝜁𝐹(ℰ⁕𝛿), 𝜁𝐹(𝛿)}. If (ℰ⁕𝛿) ∈ 𝔙 & 𝛿 ∉ 𝔙, then 𝜁𝐹(ℰ⁕𝛿) = [ℰ1,ℰ2], 𝜁𝐹(𝛿) =[0,0]. Thus, rmax {𝜁𝐹(ℰ⁕𝛿), 𝜁𝐹(𝛿)} = [0,0]. Hence 𝜁𝐹(ℰ) ≤ rmax {𝜁𝐹(ℰ⁕𝛿), 𝜁𝐹(𝛿)}. Neutrosophic Sets and Systems, Vol. 97, 2026 445 Shanmugapriya K P , Hemavathi P ,Vinod Kumar R, Saeid Jafari,Structural Theory of Intervalvalued Neutrosophic Ẑ-Ideals in Ẑ-algebraic Structures Similarly, for the case (ℰ⁕𝛿) ∉ 𝔙 & 𝛿 ∈ 𝔙, we have i) 𝜁𝑇(ℰ) ≥ rmin{𝜁𝑇(ℰ⁕𝛿), 𝜁𝑇(𝛿)}, On the otherhand, since 0 ∈ 𝔙, thus 𝜁𝑇(0) = [ ℰ1,ℰ1] (i.e.) 𝜁𝑇(0) ≥ 𝜁𝑇(ℰ). ii) 𝜁𝐼(ℰ) ≥ rmin{𝜁𝑇(ℰ⁕𝛿), 𝜁𝐼(𝛿)}, On the otherhand, since 0 ∈ 𝔙, thus 𝜁𝐼(0) = [ ℰ1,ℰ1] (i.e.) 𝜁𝐼(0) ≥ 𝜁𝐼(ℰ). iii) 𝜁𝐹(ℰ) ≤ rmax{𝜁𝐹(ℰ⁕𝛿), 𝜁𝐹(𝛿)}, On the otherhand, since 0 ∈ 𝔙, thus 𝜁𝐹(0) = [ ℰ1,ℰ1] (i.e.) 𝜁𝐹(0) ≤ 𝜁𝐹(ℰ). Hence, 𝜁𝑇,𝐼,𝐹 is an IVN Ẑideal of 𝔐. Theorem 3.5. Let 𝔙 be a non-empty set of 𝔐 and let 𝜁𝑇,𝐼,𝐹 be an Interval_Valued (Int.V) Neutrosophic set on 𝔐 defined by 𝜁𝑇(ℰ) = {[ℰ1,ℰ2],ℰ ∈ 𝔙 [1,1],ℰ ∉ 𝔙 where [ℰ1,ℰ2] ∈ [0,1] & ℰ1 >ℰ2 𝜁𝐼(ℰ) = {[ℰ1,ℰ2],ℰ ∈ 𝔙 [1,1],ℰ ∉ 𝔙 where [ℰ1,ℰ2] ∈ [0,1] & ℰ1 >ℰ2 𝜁𝐹(ℰ) = {[ℰ1,ℰ2],ℰ ∈ 𝔙 [0,0],ℰ ∉ 𝔙 where [ℰ1,ℰ2] ∈ [0,1] & ℰ1<ℰ2 If 𝔐 is an IVN Ẑideal of 𝔙, then 𝔙 is also an IVN Ẑideal of 𝔐. Proof: Assume, 𝜁 is an IVN Ẑideal of 𝔐. Then, i) 𝜁𝑇(0) ≥ 𝜁𝑇 (ℰ), 𝜁𝑇( ℰ) ≥ rmin {𝜁𝑇(ℰ⁕𝛿),𝜁𝑇(𝛿)} ii) 𝜁𝐼(0) ≥ 𝜁𝐼 (ℰ), 𝜁𝐼( ℰ) ≥ rmin {𝜁𝐼(ℰ⁕𝛿),𝜁𝐼(𝛿)} iii) 𝜁𝐹(0) ≤ 𝜁𝐹 (ℰ),𝜁𝐹( ℰ) ≤ rmax {𝜁𝐹(ℰ⁕𝛿),𝜁𝐹(𝛿)} ∀ ℰ , 𝛿 ∈ 𝔐. Since 𝔙 is a non-empty subset of 𝔐, thus there exists ℰ ∈ 𝔐, such that i) 𝜁𝑇(ℰ) = [ℰ1,ℰ2], 𝜁𝑇(0) ≥ 𝜁𝑇 (ℰ) = [ℰ1,ℰ2] ii) 𝜁𝐼(ℰ) = [ℰ1,ℰ2], 𝜁𝐼(0) ≥ 𝜁𝐼 (ℰ) = [ℰ1,ℰ2] iii) 𝜁𝐹(ℰ) = [ℰ1,ℰ2], 𝜁𝐹(0) ≤ 𝜁𝐹 (ℰ) = [ℰ1,ℰ2] This shows, 0 ∈ 𝔙, if ℰ⁕𝛿 ∈ 𝔙 and 𝛿 ∈ 𝔙, then i) 𝜁𝑇(ℰ),𝜁𝑇(𝛿) ≥ [ℰ1,ℰ2], therefore 𝜁𝑇(ℰ) ≥ rmin {𝜁𝑇(ℰ⁕𝛿),𝜁𝑇(𝛿)} = rmin{[ℰ1,ℰ2], [ℰ1,ℰ2]} = [ℰ1,ℰ2] ∴ 𝜁𝑇(ℰ) = [ℰ1,ℰ2] , (i.e.) ℰ ∈ 𝔙. Neutrosophic Sets and Systems, Vol. 97, 2026 446 Shanmugapriya K P , Hemavathi P ,Vinod Kumar R, Saeid Jafari,Structural Theory of Intervalvalued Neutrosophic Ẑ-Ideals in Ẑ-algebraic Structures ii) Similarly, 𝜁𝐼(ℰ) = [ℰ1,ℰ2] iii) 𝜁𝐹(ℰ) ≤ 𝜁𝐹(ℰ) = [ℰ1,ℰ2], therefore 𝜁𝐹(ℰ) ≤ rmax {𝜁𝐹(ℰ⁕𝛿),𝜁𝐹(𝛿)} = rmax{[ℰ1,ℰ2], [ℰ1,ℰ2]} = [ℰ1,ℰ2] ∴ 𝜁𝐹(ℰ) = [ℰ1,ℰ2], (i.e.) ℰ ∈ 𝔙. Hence 𝔙 is an IVN Ẑideal of 𝔐. Definition 3.6. Let 𝜁1 = {(ℰ, 𝜁𝑇1(ℰ),𝜁𝐼1(ℰ),𝜁𝐹1(ℰ)), ℰ∈𝔐} and 𝜁2 = {(ℰ,𝜁𝑇2(ℰ),𝜁𝐼2(ℰ),𝜁𝐹2(ℰ)), ℰ∈𝔐} are two IVN sets on Ẑ-algebra 𝔐, define the IVN set (𝜁1∩𝜁2) = {𝜁𝑇1∩𝑇2,𝜁𝑇1∩𝑇2,𝜁𝑇1∩𝑇2) on 𝔐, by i) 𝜁𝑇1∩𝑇2(ℰ) = rmin {𝜁𝑇1(ℰ),𝜁𝑇2(ℰ)}ℰ∈𝔐, then 𝜁𝑇1∩𝑇2 is referred as the intersection of 𝜁𝑇1 and 𝜁𝑇2. ii) 𝜁𝐼1∩𝐼2(ℰ) = rmin {𝜁𝐼1(ℰ),𝜁𝐼2(ℰ)}ℰ∈𝔐, then 𝜁𝐼1∩𝐼2 is referred as the intersection of 𝜁𝐼1 and 𝜁𝐼2. iii) 𝜁𝐹1∩𝐹2(ℰ) = rmax {𝜁𝐹1(ℰ),𝜁𝐹2(ℰ)}ℰ∈𝔐, then 𝜁𝐹1∩𝐹2 is referred as the intersection of 𝜁𝐹1 and 𝜁𝐹2. Theorem 3.7. Let 𝜁1 and 𝜁2 be two IVN Ẑ-ideals of Ẑ-algebra 𝔐. Then 𝜁1∩𝜁2 is an IVN Ẑ-ideal of 𝔐. Proof: Assume 𝜁1 and 𝜁2 are two IVN Ẑ-ideals of Ẑ-algebra 𝔐, then for all ℰ,𝛿 ∈𝔐 i) 𝜁𝑇1(0) ≥ 𝜁𝑇1(ℰ), 𝜁𝑇1(ℰ) ≥rmin{𝜁𝑇1(ℰ⁕𝛿), 𝜁𝑇1(𝛿)}, 𝜁𝑇2(0) ≥ 𝜁𝑇2(ℰ), 𝜁𝑇2(ℰ) ≥rmin{𝜁𝑇2(ℰ⁕𝛿), 𝜁𝑇2(𝛿)} Hence, 𝜁𝑇1∩𝑇2(0) = rmin{𝜁𝑇1(0),𝜁𝑇2(0)} ≥rmin{𝜁𝑇1(ℰ),𝜁𝑇2(ℰ)} = 𝜁𝑇1∩𝑇2(ℰ). 𝜁𝑇1∩𝑇2(ℰ) = rmin {𝜁𝑇1(ℰ),𝜁𝑇2(ℰ)} ≥rmin{rmin{𝜁𝑇1(ℰ⁕𝛿), 𝜁𝑇1(𝛿)},rmin{𝜁𝑇2(ℰ⁕𝛿), 𝜁𝑇2(𝛿)}} Neutrosophic Sets and Systems, Vol. 97, 2026 447 Shanmugapriya K P , Hemavathi P ,Vinod Kumar R, Saeid Jafari,Structural Theory of Intervalvalued Neutrosophic Ẑ-Ideals in Ẑ-algebraic Structures = rmin{rmin{𝜁𝑇1(ℰ⁕𝛿), 𝜁𝑇2(ℰ⁕𝛿)}rmin{𝜁𝑇1(𝛿),𝜁𝑇2(𝛿)}} = rmin{𝜁𝑇1∩𝑇2(ℰ⁕𝛿), 𝜁𝑇1∩𝑇2(𝛿)}. ii) Similarly,𝜁𝐼1∩𝐼2(0)≥𝜁𝐼1∩𝐼2(ℰ), 𝜁𝐼1∩𝐼2(ℰ)≥ rmin{𝜁𝐼1∩𝐼2(ℰ⁕𝛿), 𝜁𝐼1∩𝐼2(𝛿)}. iii) 𝜁𝐹1(0) ≤ 𝜁𝐹1(ℰ), 𝜁𝐹1(ℰ) ≤ rmax{𝜁𝐹1(ℰ⁕𝛿), 𝜁𝐹1(𝛿)}, 𝜁𝐹2(0) ≤ 𝜁𝐹2(ℰ), 𝜁𝐹2(ℰ) ≤ rmax{𝜁𝐹2(ℰ⁕𝛿), 𝜁𝐹2(𝛿)} Hence, 𝜁𝐹1∩𝐹2(0) = rmax{𝜁𝐹1(0),𝜁𝐹2(0)} ≤ rmax{𝜁𝐹1(ℰ),𝜁𝐹2(ℰ)} = 𝜁𝐹1∩𝐹2(ℰ) 𝜁𝐹1∩𝐹2(ℰ) = rmax {𝜁𝐹1(ℰ),𝜁𝐹2(ℰ)} ≥rmax{rmax{𝜁𝐹1(ℰ⁕𝛿), 𝜁𝐹1(𝛿), rmax{𝜁𝐹2(ℰ⁕𝛿), 𝜁𝐹2(𝛿)}} = rmax{rmax{𝜁𝐹1(ℰ⁕𝛿), 𝜁𝐹2(ℰ⁕𝛿)}rmax{𝜁𝐹1(𝛿),𝜁𝐹2(𝛿)}} = rmax{𝜁𝐹1∩𝐹2(ℰ⁕𝛿), 𝜁𝑇1∩𝑇2(𝛿)}. ∴ Intersection of 𝜁1 and 𝜁2 is also an IVN Ẑ-ideal of 𝔐. Definition 3.8. Let 𝜁1 = {(ℰ, 𝜁𝑇1(ℰ),𝜁𝐼1(ℰ),𝜁𝐹1(ℰ)), ℰ∈𝔐} and 𝜁2 = {(ℰ,𝜁𝑇2(ℰ),𝜁𝐼2(ℰ),𝜁𝐹2(ℰ)), ℰ∈𝔐} are two IVN sets on Ẑ-algebra 𝔐, define the IVN set (𝜁1∪ 𝜁2) = { 𝜁𝑇1∪𝑇2,𝜁𝑇1∪𝑇2,𝜁𝑇1∪𝑇2) on 𝔐, by i) 𝜁𝑇1∪𝑇2(ℰ) ≥rmin {𝜁𝑇1(ℰ),𝜁𝑇2(ℰ)}ℰ∈𝔐, then 𝜁𝑇1∪𝑇2 is referred as the union of 𝜁𝑇1 and 𝜁𝑇2. ii) 𝜁𝐼1∪𝐼2(ℰ) ≥rmin {𝜁𝐼1(ℰ),𝜁𝐼2(ℰ)}ℰ∈𝔐, then 𝜁𝐼1∩𝐼2 is referred as the union of 𝜁𝐼1 and 𝜁𝐼2. iii) 𝜁𝐹1∪𝐹2(ℰ) ≤rmax {𝜁𝐹1(ℰ),𝜁𝐹2(ℰ)}ℰ∈𝔐, then 𝜁𝐹1∩𝐹2 is referred as the union of 𝜁𝐹1 and 𝜁𝐹2. Theorem 3.9. Let 𝜁1 and 𝜁2 are two IVN Ẑ-ideals of Ẑ-algebra 𝔐. Then 𝜁1∪ 𝜁2 is an IVN Ẑ-ideal of 𝔐. Proof: Assume 𝜁1 and 𝜁2 be two IVN Ẑ-ideals of Ẑ-algebra 𝔐, then for all ℰ,𝛿 ∈𝔐 i) 𝜁𝑇1(0) ≥ 𝜁𝑇1(ℰ), 𝜁𝑇1(ℰ) ≥rmin{𝜁𝑇1(ℰ⁕𝛿), 𝜁𝑇1(𝛿)}, 𝜁𝑇2(0) ≥ 𝜁𝑇2(ℰ), 𝜁𝑇2(ℰ) ≥rmin{𝜁𝑇2(ℰ⁕𝛿), 𝜁𝑇2(𝛿)} Hence, 𝜁𝑇1∪𝑇2(0) ≥ rmin{𝜁𝑇1(0),𝜁𝑇2(0)} ≥rmin{𝜁𝑇1(ℰ),𝜁𝑇2(ℰ)} Neutrosophic Sets and Systems, Vol. 97, 2026 454 Shanmugapriya K P , Hemavathi P ,Vinod Kumar R, Saeid Jafari,Structural Theory of Intervalvalued Neutrosophic Ẑ-Ideals in Ẑ-algebraic Structures general applications, such as decision-making, artificial intelligence, and information sciences, have not yet been explored. 6.4 Lack of Algorithm: There are no efficient algorithms for computing and verifying IVN Z-ideals, despite knowing their structural features. 7. Conclusions In this paper, we introduced and investigated the concept of IVN Fuzzy Ẑ-ideals within the framework of Ẑ-algebraic structures. By extending classical fuzzy ideal theory through the lens of Neutrosophic logic and IV representation, and successfully formulated new structural definitions and explored their fundamental properties. The presented approach offers a more refined and flexible mathematical tool to handle higher degrees of uncertainty, indeterminacy, and imprecision that often arise in complex algebraic systems. Through illustrative examples and theorems, the role of these ideals in preserving the underlying algebraic operations and structural consistencies was thoroughly analyzed. Our results contribute significantly to the enrichment of fuzzy algebra and open new directions in Neutrosophic algebraic studies. 8. Future Research Directions 8.1 Algorithm Optimization: Discover computational strategies for simplifying and enhancing scale. Specifically, the IVN framework must be integrated with the multi-criteria decision-making and optimisation problem, in which uncertainty plays a significant role. 8.2 Hybrid Models: Combine IVN principles with other uncertainty models, such as rough sets and Intuitionistic Fuzzy sets. 8.3 Applied to artificial intelligence: Research into how IVN Z-ideals can be used in machine learning and intelligent systems, particularly when the data is inadequate, ambiguous, or inconsistent. 8.4 Analytical evaluation: Does the proposed theory handle practical challenges, such as risk assessment, network reliability, and medical diagnostics, demonstrating its applicability? Acknowledgement The authors would like to thank the reviewers who contributed such useful comments and constructive ideas. References [1] Al-Omeri, W. F., Kaviyarasu, M., & Rajeshwari, M. Translation of Neutrosophic INK-Algebra. Neutrosophic Sets and Systems, 66, 119–135, (2024). [2] Bavanari Satyanarayana, & Baji, S. Interval-Valued Neutrosophic N-d-Ideal in d-Algebra. Neutrosophic Optimization and Intelligent Systems, 3, 57–70, (2024). Neutrosophic Sets and Systems, Vol. 97, 2026 455 Shanmugapriya K P , Hemavathi P ,Vinod Kumar R, Saeid Jafari,Structural Theory of Intervalvalued Neutrosophic Ẑ-Ideals in Ẑ-algebraic Structures [3] Chandramouleeswaran, M., Muralikrishna, P., Sujatha, K., & Sabarinathan, S. A note on Ẑ-algebra. Italian Journal of Pure and Applied Mathematics, 38, 707–714, (2017). [4] Hemavathi, P., Muralikrishna, P., & Palanivel, P. A note on interval-valued fuzzy β-subalgebras. Global Journal of Pure and Applied Mathematics, 11(4), 2553–2560, (2015). [5] Jun, Y. B., Kim, C. S., & Yang, K. O. Cubic subalgebras and ideals of BCK/BCI-algebras. Far East Journal of Mathematical Sciences, 44(2), 239–250, (2010). [6] Rajakumari, R., Balasubramanian, K. R., & Vadivel, A. Interval Valued Neutrosophic INK-Ideal Via INK-Algebra. International Journal of Neutrosophic Science, 24(1), 35–50, (2024). [7] Muralikrishna, P., & Manokaran, S. MBJ-Neutrosophic β-Ideal of β-Algebra. Neutrosophic Sets and Systems, 35, (2020). [8] Nagaiah, T., Bhaskar, L., Narasimha Swamy, P., & Broumi, S. A study on neutrosophic algebra. Neutrosophic Sets and Systems, 50,(2022). [9] Satyanarayana, B., & Baji, S. Interval-Valued Neutrosophic N-d-Ideal in d-Algebra. Neutrosophic Optimization and Intelligent Systems, 3, 57–70, (2024). [10] Shanmugapriya, K. P., & Hemavathi, P. A Novel Concept of Neutrosophic Fuzzy Sets in Ẑ-Algebra. In Recent Developments in Algebra and Analysis (Trends in Mathematics), pp. 47–56, (2024). [11] Shanmugapriya, K. P., & Hemavathi, P. A New Framework of Interval-Valued Neutrosophic in Ẑ-Algebra. Neutrosophic Sets and Systems, 60(1), (2024). [12] Smarandache, F. Neutrosophy: Neutrosophic Probability, Set and Logic. American Research Press, (1999). [13] Smarandache, F. Neutrosophic set—a generalization of the intuitionistic fuzzy set. International Journal of Pure and Applied Mathematics, (2004). [14] Sowmiya, S., & Jeyalakshmi, P. Fuzzy algebraic structures in Ẑ-algebras. World Journal of Engineering Research and Technology, 5(4), 74–88, (2019). [15] Sowmiya, S., & Jeyalakshmi, P. On fuzzy Ẑ-ideals in Ẑ-algebras. Global Journal of Pure and Applied Mathematics, 15(4), 505–516, (2019). [16] Sowmiya, S., & Jeyalakshmi, P. Ẑ-Homomorphism and Cartesian product on fuzzy α-translations and fuzzy β-multiplications of Ẑ-algebras. AIP Conference Proceedings, 226, (2020). [17] Wang, H., Smarandache, F., Zhang, Y.-Q., & Sunderraman, R. Interval Neutrosophic Sets and Logic: Theory and Applications in Computing, (2005). [18] Zadeh, L. A. Fuzzy sets. Information and Control, 8(3), 338–353, (1965). Received: May 6, 2025. Accepted: Oct 4, 2025