Eigen Neutrosophic Z- Set and Neutrosophic Z- Relation

Neutrosophic Sets and Systems, Vol. 97, 2026 University of New Mexico P. Sheeba Maybell, M.M. Shanmugapriya, Eigen Neutrosophic ZSet and Neutrosophic ZRelation Eigen Neutrosophic ZSet and Neutrosophic ZRelation P. Sheeba Maybell1*, M.M. Shanmugapriya2 1 Department of Mathematics, Karpagam Academy of Higher Education, Coimbatore, Tamil Nadu, India; [email protected] 2 Dept. of Mathematics, Karpagam Academy of Higher Education, Coimbatore, Tamil Nadu, India; [email protected] * Correspondence: Seeba Maybell, Email: [email protected] Abstract: This paper introduces an innovative framework for computing the Greatest Eigen Neutrosophic Z-set and the Least Eigen Neutrosophic Z-set using the composition operators, namely max-min-min and min-max-max. The proposed Eigen Neutrosophic Z-set, along with the Neutrosophic Z-relation, remains constant across different computational perspectives. This study addresses the limitation of existing neutrosophic and fuzzy models that fail to effectively capture eigen-based relationships under uncertainty by introducing the Eigen Neutrosophic Z-set framework for more consistent and interpretable decision analysis. Furthermore, Neutrosophic Z-matrices are developed, and their properties are examined in relation to Neutrosophic Z-relations. In this paper several similarity relations among Neutrosophic Z-matrices are presented, along with discussions on their permutations and the invertibility characteristics. Two distinct algorithms are formulated to establish the Greatest Eigen Neutrosophic Z-set and the Least Eigen Neutrosophic Z-set, accompanied by a numerical example. Additionally, a practical application is provided to demonstrate the enhancement of score value while addressing both effectiveness and uncertainty for future advancements of hotel management decision-making systems. Keywords: Neutrosophic Z-set, Neutrosophic Z-relation, Neutrosophic Z-Matrices, Eigen Neutrosophic Z-set, Composition operators, Decision-making uncertainty modelling. 1. Introduction Zadeh [ 1] proposed a notion namely Z-number, which is an ordered pair of fuzzy numbers 𝑍= (𝑉,𝑅) in 2011.The reliability and restriction of fuzzy is mainly focused in Z-number [2]. Smarandache [3] introduced another concept of imprecise data called Neutrosophic data which deals with complicating aspects to process imprecision, vagueness, and uncertainty in data. Sanjib Mondal et.al., [4] developed similarity relations for Intuitionistic fuzzy matrices. Neutrosophic set was later developed to Quadri partitioned neutrosophic soft set, fuzzy neutrosophic soft matrices and fuzzy Quadri partitioned neutrosophic soft matrix [5, 6, 7] which was more useful in decision making. Neutrosophic qualities and neutrosophic metrics to assess trustworthiness are united in the neutrosophic z-number set technique proposed by Shigui Du et al. [8] as a generalization of the znumbers and the neutrosophic set. The three ordered pairs of neutrosophic numbers along with their reliability measures in indeterminate and inconsistent situations can be resolved by the suggested neutrosophic z-number set [9]. In z-numbers and their set, the multi criteria decision-making technique (MCDM) is readily embraced [10, 11, 12] later on MCDM developed to neutrosophic znumbers. A fuzzy relations eigen fuzzy set was presented by Sanchez [13]. He provided three main algorithms to find the Greatest Eigen Fuzzy Set (GEFS) linked with fuzzy relations using max-min Neutrosophic Sets and Systems, Vol. 97, 2026 688 P. Sheeba Maybell, M.M. Shanmugapriya, Eigen Neutrosophic ZSet and Neutrosophic ZRelation composition so that 𝑅∘A=A. Eigen fuzzy sets have been successfully used in a number of real-world applications in decision-making, genetic algorithms, image analysis, and medicine. Guleria and Bajaj [14] later proposed eigen spherical fuzzy sets and applications. Using spherical fuzzy set, they have produced an astounding achievement by identifying two distinct techniques for finding eigen spherical fuzzy sets. Further T-spherical fuzzy set for similarity measure also found for decisionmaking [15]. Harikrishnan et al. [16] in their work min-max compositions for neutrosophic fuzzy matrices was demonstrated their application in diagnosing diseases. Their work shows that changing composition operators alters diagnostic outcomes, but they do not address eigen-based stability or Z-relations. But this shows the importance of composition operators but lacks information about eigen Z-set theory. Kamran et al. [17] examined the use of neutrosophic Z-numbers in AHP-based prioritization and Z-rough structures for ranking alternatives under uncertainty. Although Znumbers provide richer representation of uncertainty, this work does not define eigen Z-sets or algorithms for stable relation evaluation. This work has a strong Z-number background but doesn’t focus on similarity or eigen properties. Mishra & Kumar [18] investigated algebraic properties of neutrosophic matrices, including invertibility and determinants for decision-oriented systems. Their theoretical work addresses classical neutrosophic matrices but does not extend these results to neutrosophic Z-matrices or eigen computations. This work doesn’t involve the Z-matrix framework; only the foundation of matrix algebra is used for analysis. Al-Faifi et al. [19] applied neutrosophic and plithogenic models to multi-criteria decisionmaking for uncertain preference structures. While they improve decision accuracy, they rely on distance and score measures and do not consider eigen-based consistency. Saha & Abdel-Basset et al. [20] explored spectral measures such as the “energy” of neutrosophic matrices for network analysis and clustering. Their results demonstrate the usefulness of spectral neutrosophic properties, but they do not propose algorithms for greatest/least eigen Z-sets or Zrelations. The Z-set definition and the composition stability are missing. 2. Research Gap The Eigen fuzzy set was introduced by Sanchez [13], along with the concept of fuzzy relations. This method established the Greatest Eigen Fuzzy Set using the max-min composition method. Numerous researchers have applied this max-min composition for image retrieval, genetic algorithms, and in the medicinal field. Subsequently, the Eigen Spherical Fuzzy Set was introduced by Guleria and Bajaj [14]. They offered two techniques for identifying the Greatest Eigen Spherical Fuzzy Set and the Least Eigen Spherical Fuzzy Set. The Neutrosophic Z-set is a novel method used to assess uncertainty in real-life scenarios. We have proposed a new composition operator for the Neutrosophic Z-set along with its Neutrosophic Z-relation. Many researchers have extensively explored Neutrosophic Fuzzy matrices, their relations, and similarity measures. Our study is significant because we extended similarity relations for Neutrosophic fuzzy matrices to Neutrosophic Z-matrices. 3. Contribution of this proposed work • Introduction of new composition operator for neutrosophic z-set: Two distinct composition operators for neutrosophic z-sets,specifically max-min-min and min-max-max, have been developed alongside the concept of neutrosophic z-relation. These composition operators identify the Greatest Eigen Neutrosophic Z-sets (GENZS) and the Least Eigen Neutrosophic Zsets (LENZS), which are tailored to yield suitable values in situations of uncertainty. • Neutrosophic Z-relation for Neutrosophic Z-matrices: A Neutrosophic Z-matrix has been introduced together with the Neutrosophic Z-relation. Various properties of similarity relations Neutrosophic Sets and Systems, Vol. 97, 2026 689 P. Sheeba Maybell, M.M. Shanmugapriya, Eigen Neutrosophic ZSet and Neutrosophic ZRelation were examined, offering a foundational framework for Neutrosophic Z-matrices, with potential for further advancements through these matrices. • Algorithm for strategy finding: This proposed work introduced two algorithms for each composition operator within the framework of Neutrosophic Z-sets. The primary objective of these algorithms is to determine the eigen neutrosophic z-set. • Numerical Example and Application: The efficacy of the proposed method is illustrated using a numerical example. Effectiveness and uncertainty are assessed in a real-world setting. The decision-making scenario provides an in-depth comprehension regarding the methods feasibility and adaptability. Table 1, depicts the comparison of the existing works in neutrosophic Z numbers and fuzzy matrices with the proposed work Eigen Neutrosophic Z set. Table 1 Comparison of existing and proposed work Dimension Existing Works (Neutrosophic, Z-numbers, Fuzzy matrices) Proposed Work Handling of uncertainty Uses truth, indeterminacy, falsity values; Z-numbers add reliability but no eigen characterization Introduces Eigen Neutrosophic Z-set to measure stable relation values under uncertainty Composition operators Studies max–min / min–max families separately Demonstrates both max–min–min and min–max–max operators and proves eigen-set consistency across compositions Matrix framework Classical neutrosophic matrices used for similarity or scoring It defines Neutrosophic Z-matrices, explores invertibility, permutations, similarity relations Eigen-based analysis Mostly absent; spectral analysis exists but not for Zrelations Provides algorithms for Greatest and Least Eigen Neutrosophic Z-sets with numerical examples Practical decisionmaking Score or distance-based ranking Uses eigen Z-sets to enhance score and interpretability in hotel management decision-making Reproducibility Often conceptual or qualitative Delivers two algorithms, operator consistency proof, and implementation steps The paper is organized as follows: Key definitions and concepts are examined in Section 3. Neutrosophic Z-matrices, Neutrosophic Z-relations, invertibility requirements, and similarity relations are introduced in Section 4. NZM idempotent is also taken for consideration in this part. Numerous characteristics and findings pertaining to neutrosophic Z-matrices are examined. The composition operator and the Neutrosophic Z-relation are defined in Section 5. The notions of the Neutrosophic Sets and Systems, Vol. 97, 2026 690 P. Sheeba Maybell, M.M. Shanmugapriya, Eigen Neutrosophic ZSet and Neutrosophic ZRelation Least Eigen Neutrosophic Z-sets (LENZS) and the Greatest Eigen Neutrosophic Z-sets (GENZS) are presented in this study along with an algorithm. The principles of GENZS and LENZS are explained with the use of a numerical example. We show how the suggested technique can be used in practical situations in Section 6. In Section 7 the work is concluded and future research directions are discussed. 3 Preliminaries 3.1 Definition Let X be a universe set then a Neutrosophic Z-number set (NZNs) in a universe set X is defined in the following as 𝑆𝑍=(<x,T(V,R)(x),I(V,R)(x),F(V,R)(x)>𝑥∈X) (1) here T(V,R)(x)=(𝑇𝑉(x),𝑇𝑅(x)) ,I(V,R)(x)=(𝐼𝑉(x),𝐼𝑅(x)),F (V,R)(x)=(𝐹𝑉(x),𝐹𝑅(x)):X→[0,1]2 (2) are the order pairs of neutrosophic values for truthfulness, indeterminacy, and falsehood; the first component consists of the neutrosophic values in a universe set X, and the second component consists of neutrosophic reliability measures, with the rule of 0≤𝑇𝑉(x)+ 𝐼𝑉(x)+𝐹𝑉(x)≤3 and 0≤𝑇𝑅(x)+ 𝐼𝑅(x)+𝐹𝑅(x)≤3 (3) 3.2 Definition Let X be a universe set and F be a set of parameters. Consider a nonempty set 𝑆𝑍, 𝑆𝑍 ∈𝐹. Let P(X) be the collection of all neutrosophic znumber sets of X. The set (E, 𝑆𝑍 ) be termed as neutrosophic znumber sets (NZNs) over X, where 𝐸∶ 𝑆𝑍→𝑃(𝑋). Consider S as neutrosophic z-matrices (NZMs) over X instead of (E, 𝑆𝑍). 3.3 Definition Let 𝑆𝐴 be a 𝑁𝑍𝑀𝑚×𝑛 and 𝑆𝐵 be a 𝑁𝑍𝑀𝑛×𝑝 then the composition of 𝑆𝐴 and 𝑆𝐵 is defined as 𝑆𝐴∘𝑆𝐵=(<(∑(𝑇𝑉𝑖𝑘 𝐴∧ 𝑇𝑉𝑘𝑗 𝐵) 𝑛 𝑖=1 ,(∑(𝑇𝑅𝑖𝑘 𝐴∧𝑇𝑅𝑘𝑗 𝐵)),(∏(𝐼𝑉𝑖𝑘 𝐴∨ 𝐼𝑉𝑘𝑗 𝐵)) 𝑛 𝑖=1 , 𝑛 𝑖=1 ( ∏(𝐼𝑅𝑖𝑘 𝐴 ∨𝐼𝑅𝑘𝑗 𝐵)) 𝑛 𝑖=1 ,( ∏(𝐹𝑉𝑖𝑘 𝐴 ∨𝐹𝑉𝑘𝑗 𝐵)),( 𝑛 𝑖=1 ∏(𝐹𝑅𝑖𝑘 𝐴∨𝐹𝑅𝑘𝑗 𝐵)) 𝑛 𝑖=1 >) (4) Equivalently it can be written as 𝑆𝐴∘𝑆𝐵=(<(⋃(𝑇𝑉𝑖𝑘 𝐴∧ 𝑇𝑉𝑘𝑗 𝐵)), 𝑛 𝑖=1 (⋃(𝑇𝑅𝑖𝑘 𝐴∧𝑇𝑅𝑘𝑗 𝐵)) 𝑛 𝑖=1 ,(⋀(𝐼𝑉𝑖𝑘 𝐴∨ 𝐼𝑉𝑘𝑗 𝐵)), 𝑛 𝑖=1 (⋀(𝐼𝑅𝑖𝑘 𝐴∨𝐼𝑅𝑘𝑗 𝐵)), 𝑛 𝑖=1 (⋀(𝐹𝑉𝑖𝑘 𝐴 ∨𝐹𝑉𝑘𝑗 𝐵)), 𝑛 𝑖=1 (⋀(𝐹𝑅𝑖𝑘 𝐴∨𝐹𝑅𝑘𝑗 𝐵))>) 𝑛 𝑖=1 (5) Neutrosophic Sets and Systems, Vol. 97, 2026 691 P. Sheeba Maybell, M.M. Shanmugapriya, Eigen Neutrosophic ZSet and Neutrosophic ZRelation If the number of 𝑆𝐴 columns equal the number of rows 𝑆𝐵, then the product is defined. This multiplication procedure is called as max-min composition operator. Consequently, 𝑆𝐴∘𝑆𝐵 and are considered conformable for multiplication, rather than using 𝑆𝐴∘𝑆𝐵 it is denoted as 𝑆𝐴𝑆𝐵, where ∑(𝑇𝑉𝑖𝑘 𝐴∧ 𝑇𝑉𝑘𝑗 𝐵) 𝑛 𝑖=1 means maxmin operation and ∏(𝐼𝑉𝑖𝑘 𝐴∨ 𝐼𝑉𝑘𝑗 𝐵)) 𝑛 𝑖=1 means min-max operation. 4 Neutrosophic Z-relation 4.1 Definition Let 𝐻(𝐴,𝐴) be an Neutrosophic Z-relation (NZR) on a set 𝑆𝐴. Let 𝑇𝑉,𝑅:𝑆𝐴 ⟶[0,1]2, 𝐼𝑉,𝑅:𝑆𝐴 ⟶ [0,1]2,𝑎𝑛𝑑 𝐹𝑉,𝑅:𝑆𝐴 ⟶[0,1]2are the three membership function and 𝑀𝐻 be the corresponding Neutrosophic ZMatrices (NZM) in relation H. 4.2 Definition The relation 𝐻(𝐴,𝐴)is reflexive if the diagonal entries of 𝑀𝐻 is [<(1,1),(0,0),(0,0)>]where 𝑇(𝑉,𝑅)𝐻(𝑥,𝑥)=(1,1), 𝐼(𝑉,𝑅)𝐻(𝑥,𝑥)=(0,0) and 𝐹(𝑉,𝑅)𝐻(𝑥,𝑥)=(0,0) for all 𝑥 ∈ 𝑆𝐴 . 4.3 Definition The relation 𝐻(𝐴,𝐴)is symmetric if 𝑀𝐻=𝑀𝐻 𝑇where 𝑀𝐻 𝑇is the transpose of 𝑀𝐻 such that 𝑇(𝑉,𝑅)𝐻(𝑥,𝑦)=𝑇(𝑉,𝑅)𝐻(𝑦,𝑥), 𝐼(𝑉,𝑅)𝐻(𝑥,𝑦)=𝐼(𝑉,𝑅)𝐻(𝑦,𝑥) and 𝐹(𝑉,𝑅)𝐻(𝑥,𝑦)=𝐹(𝑉,𝑅)𝐻(𝑦,𝑥) for all 𝑥,𝑦 ∈ 𝑆𝐴 . 4.4 Definition The relation 𝐻(𝐴,𝐴) is transitive if 𝑀𝐻≥𝑀𝐻 2 i.e., 𝑇(𝑉,𝑅)𝐻(𝑥,𝑧)≥max (min ((𝑇(𝑉,𝑅)𝐻(𝑦,𝑥),𝑇(𝑉,𝑅)𝐻(𝑦,𝑧))), 𝐼(𝑉,𝑅)𝐻(𝑥,𝑧)≤min (max ((𝐼(𝑉,𝑅)𝐻(𝑦,𝑥),𝐼(𝑉,𝑅)𝐻(𝑦,𝑧))) and 𝐹(𝑉,𝑅)𝐻(𝑥,𝑧)≤min (max((𝐹(𝑉,𝑅)𝐻(𝑦,𝑥),𝐹(𝑉,𝑅)𝐻(𝑦,𝑧))) for all pair(𝑥,𝑧)∈𝑆𝐴×𝑆𝐴. 4.5 Definition Let 𝐻(𝐴,𝐴) relation is reflexive, symmetric and transitive then 𝐻(𝐴,𝐴) relation is called as similarity relation. 4.6 Proposition For any 𝑆𝐴∈𝑁𝑍𝑀𝑛×𝑛, 𝑆𝐴 is reflexive if 𝑆𝐴≥𝐼𝑛 . proof Sinc 𝑆𝐴≥𝐼𝑛, then matrix entries which is diagonal of 𝑆𝐴 are [<(1,1),(0,0),(0,0)>].  𝑆𝐴 is a reflexive matrix. Hence the proof. 4.7 Definition For an 𝑆𝐴∈𝑁𝑍𝑀𝑛×𝑛 , we define Neutrosophic Sets and Systems, Vol. 97, 2026 692 P. Sheeba Maybell, M.M. Shanmugapriya, Eigen Neutrosophic ZSet and Neutrosophic ZRelation • 𝑆𝐴 is Reflexive if 𝑆𝐴≥𝐼𝑛 • 𝑆𝐴 is Weekly reflexive if 𝑆𝐴≥𝑆𝐴 • 𝑆𝐴 is Symmetric 𝑆𝐴=𝑆𝐴𝑇 • 𝑆𝐴 is Idempotent 𝑆𝐴=𝑆𝐴2 • 𝑆𝐴 is Transitive 𝑆𝐴2≤𝑆𝐴 4.8 Proposition Let 𝑆𝐴∈𝑁𝑍𝑀𝑛×𝑛 be a reflexive of NZM. Then I. 𝑆𝐴𝑇 is reflexive NZM, where𝑆𝐴𝑇 is transpose of 𝑆𝐴. II. 𝑆𝐴𝐾 is reflexive NZM for positive integer k. III. 𝑆𝐴𝑆𝐵 ≥ 𝑆𝐵 for 𝑆𝐵∈𝑁𝑍𝑀𝑛×𝑛 IV. 𝑆𝐵𝑆𝐴 ≥ 𝑆𝐵 for 𝑆𝐵∈𝑁𝑍𝑀𝑛×𝑛 V. 𝑆𝐴𝑆𝐵 and 𝑆𝐵𝑆𝐴 are reflexive NZMs if 𝑆𝐵 is reflexive VI. 𝑆𝐴𝑆𝐴𝑇 and 𝑆𝐴𝑇𝑆𝐴 are reflexive NZMs. Proof: I. Since 𝑆𝐴 has reflexive properties only when its diagonal entries are [<(1,1),(0,0),(0,0)>]. Hence 𝑆𝐴𝑇 is reflexive. II. Since 𝑆𝐴 is reflexive, 𝑆𝐴≥𝐼𝑛 then 𝑆𝐴2≥𝑆𝐴≥𝐼𝑛 (multiplying on both sides). Proceeding for (k-1) times we get 𝑆𝐴𝑘≥ 𝑆𝐴𝑘−1≥⋯…..≥𝑆𝐴2≥𝑆𝐴 ≥𝐼𝑛. The result holds for any scalar k. then 𝑆𝐴𝑘 is reflexive. III. 𝑆𝐴≥𝐼𝑛 then, 𝑆𝐴𝑆𝐵≥𝐼𝑛𝑆𝐵 ⇒𝑆𝐴𝑆𝐵≥𝑆𝐵 IV. 𝑆𝐴≥𝐼𝑛 then, 𝑆𝐵𝑆𝐴≥𝐼𝑛𝑆𝐵 ⇒𝑆𝐵𝑆𝐴≥𝑆𝐵 V. Since 𝑆𝐵 is reflexive 𝑆𝐵≥𝐼𝑛 then 𝑆𝐴𝑆𝐵≥𝑆𝐵≥ 𝐼𝑛 and 𝑆𝐵𝑆𝐴≥𝑆𝐵≥ 𝐼𝑛 from (III) and (IV). Hence 𝑆𝐴𝑆𝐵 and 𝑆𝐵𝑆𝐴 are reflexive. VI. Using (I) in (V) replace 𝑆𝐴𝑇 in the place of 𝑆𝐵 we derive the desire result. Hence the proof 4.9 Proposition If 𝑆𝐴∈𝑁𝑍𝑀𝑛×𝑛 be transitive and also it is reflexive then 𝑆𝐴 is idempotent. Proof: It is known that 𝑆𝐴 is reflexive, 𝑆𝐴≥𝐼𝑛 𝑆𝐴2≥𝑆𝐴≥𝐼𝑛 (6) Also, 𝑆𝐴 is transitive 𝑆𝐴2≤𝑆𝐴 (7) Combining (6) & (7) 𝑆𝐴2= 𝑆𝐴 Neutrosophic Sets and Systems, Vol. 97, 2026 693 P. Sheeba Maybell, M.M. Shanmugapriya, Eigen Neutrosophic ZSet and Neutrosophic ZRelation Hence 𝑆𝐴 is idempotent Note: Converse is not true. Example Let 𝑆𝐴 = [<(0.8,0.3),(0.3,0.4),(0.4,0.4)> <(0.8,0.3),(0.3,0.4),(0.4,0.4)> <(0.5,0.3),(0.3,0.4),(0.4,0.4)> <(0.8,0.3),(0.3,0.4),(0.4,0.4)>] I2 Hence 𝑆𝐴 is not reflexive, But 𝑆𝐴2=𝑆𝐴𝑆𝐴 (max-min) = [<(0.8,0.3),(0.3,0.4),(0.4,0.4)> <(0.8,0.3),(0.3,0.4),(0.4,0.4)> <(0.5,0.3),(0.3,0.4),(0.4,0.4)> <(0.8,0.3),(0.3,0.4),(0.4,0.4)>] = 𝑆𝐴 𝑆𝐴 is idempotent but not reflexive 4.10 Proposition If 𝑆𝐴 and 𝑆𝐵 are two symmetric NZMs of order n x n such that 𝑆𝐴𝑆𝐵= 𝑆𝐵𝑆𝐴, then 𝑆𝐴𝑆𝐵 is symmetric NZM. It can be proved easily Note: If 𝑆𝐴 is symmetric in 𝑁𝑍𝑀𝑛×𝑛 then 𝑆𝐴𝐾 is also symmetric for any scalar k. 4.11 Proposition Let 𝑆𝐴 , 𝑆𝐵 ∈𝑁𝑍𝑀𝑛×𝑛 is transitive, such that 𝑆𝐴𝑆𝐵= 𝑆𝐵𝑆𝐴 , then 𝑆𝐴 𝑆𝐵 will be transitive. Proof We know 𝑆𝐴 and 𝑆𝐵 both transitive 𝑆𝐴2≤𝑆𝐴 and 𝑆𝐵2≤𝑆𝐵. Now (𝑆𝐴𝑆𝐵)2=(𝑆𝐴𝑆𝐵)(𝑆𝐴𝑆𝐵) =𝑆𝐴(𝑆𝐵𝑆𝐴)𝑆𝐵 =𝑆𝐴(𝑆𝐴𝑆𝐵)𝑆𝐵 =(𝑆𝐴𝑆𝐴)(𝑆𝐵𝑆𝐵) =𝑆𝐴2𝑆𝐵2 ⇒(𝑆𝐴𝑆𝐵)2≤𝑆𝐴𝑆𝐵 hence 𝑆𝐴𝑆𝐵 is transitive. Note: If 𝑆𝐴 is transitive in 𝑁𝑍𝑀𝑛×𝑛 then 𝑆𝐴𝑘 is also transitive for any scalar k. 4.12 Proposition: Neutrosophic Sets and Systems, Vol. 97, 2026 694 P. Sheeba Maybell, M.M. Shanmugapriya, Eigen Neutrosophic ZSet and Neutrosophic ZRelation If 𝑆𝐴=[𝑆𝐴𝑝𝑞]=[<(𝑇𝑉𝑝𝑞 𝐴,𝑇𝑅𝑝𝑞 𝐴),(𝐼𝑉𝑝𝑞 𝐴,𝐼𝑅𝑝𝑞 𝐴),(𝐹𝑉𝑝𝑞 𝐴,𝐹𝑅𝑝𝑞 𝐴)>] ∈𝑁𝑍𝑀𝑛×𝑛 is symmetric and transitive then 𝑆𝐴𝑝𝑞 ≤ 𝑆𝐴𝑝𝑝 for p,q ∈ { 1,2, 3………n}. Proof Let 𝑆𝐴 is symmetric, 𝑆𝐴𝑝𝑞 = 𝑆𝐴𝑞𝑝for all p, q ∈ {1,2, 3………n} Also, since 𝑆𝐴 is transitive 𝑆𝐴2≤𝑆𝐴⟹𝑆𝐴≥ 𝑆𝐴2 Thus 𝑆𝐴𝑝𝑞 ≥ 𝑚𝑎𝑥⏟ 𝑟 [min(𝑆𝐴𝑝𝑟,𝑆𝐴𝑟𝑗)] for p= q and r ∈ {1,2, 3………n} 𝑆𝐴𝑝𝑞 ≥ 𝑚𝑎𝑥⏟ 𝑟 [min(𝑆𝐴𝑝𝑟,𝑆𝐴𝑟𝑞)] for p= q and r ∈ {1,2, 3………n} ≥ min(𝑆𝐴𝑝𝑟,𝑆𝐴𝑟𝑞) for r=q and each p 𝑆𝐴𝑝𝑝 ≥ 𝑆𝐴𝑝𝑞 (since 𝑆𝐴𝑝𝑞 = 𝑆𝐴𝑞𝑝) Hence proved. 4.13 Definition Let 𝑆𝐴 ∈𝑁𝑍𝑀𝑛 and 𝑆𝐵 is said to be invertible if and only if there exist 𝑆𝐵 ∈𝑁𝑍𝑀𝑛 such that 𝑆𝐴𝑆𝐵= 𝑆𝐵𝑆𝐴=𝐼𝑛 . 4.14 Definition An 𝑆𝐴 ∈𝑁𝑍𝑀𝑛 is called Neutrosophic Z-Permutation matrix (NZPM) if both row and column contains exactly one entry I and all other entries are  . 4.15 Proposition If 𝑆𝐴 be a 𝑁𝑍𝑀𝑛 of an NZPM then 𝑆𝐴𝑆𝐴𝑇= 𝑆𝐴𝑇𝑆𝐴=𝐼𝑛 Proof: 𝑆𝐴=(<(𝑇𝑉𝑖𝑗 𝐴,𝑇𝑅𝑖𝑗 𝐴),(𝐼𝑉𝑖𝑗 𝐴,𝐼𝑅𝑖𝑗 𝐴),(𝐹𝑉𝑖𝑗 𝐴,𝐹𝑅𝑖𝑗 𝐴)>) Then 𝑆𝐴𝑇=(<(𝑇𝑉𝑗𝑖 𝐴,𝑇𝑅𝑗𝑖 𝐴),(𝐼𝑉𝑗𝑖 𝐴,𝐼𝑅𝑗𝑖 𝐴),(𝐹𝑉𝑗𝑖 𝐴,𝐹𝑅𝑗𝑖 𝐴)>) now, i, jth entries of 𝑆𝐴𝑆𝐴𝑇 is ∑𝑆𝐴𝑖𝑘𝑆𝐵𝑘𝑗 𝑛 𝑘=1 = ∑𝑆𝐴𝑖𝑘𝑆𝐴𝑘𝑗 𝑛 𝑘=1 ={ 𝑖𝑓 𝑖≠𝑗 𝐼 𝑖𝑓 𝑖=𝑗 Neutrosophic Sets and Systems, Vol. 97, 2026 695 P. Sheeba Maybell, M.M. Shanmugapriya, Eigen Neutrosophic ZSet and Neutrosophic ZRelation (since 𝑆𝐴𝑆𝐵 is NZPM, ∑𝑆𝐴𝑖𝑘𝑆𝐴𝑖𝑘 𝑛 𝑘=1 =𝐼 ) Hence 𝑆𝐴𝑆𝐴𝑇 is an 𝐼𝑛. converse can be proved easily. Hence the proof. 4.19 Proposition Let 𝑆𝐴 be a NZMn, 𝑆𝐴 is invertible if and only if 𝑆𝐴 is an NZPM. Proof: First part: 𝑆𝐴𝑆𝐴𝑇= 𝑆𝐴𝑇𝑆𝐴=𝐼𝑛 (by previous proposition) hence 𝑆𝐴 is invertible and 𝑆𝐴𝑇 is the inverse of 𝑆𝐴 (i.e) 𝑆𝐴−=𝑆𝐴𝑇 Second part: Let 𝑆𝐴 be invertible and 𝑆𝐵 be the inverse of 𝑆𝐴. Thus 𝑆𝐴𝑆𝐵= 𝑆𝐵𝑆𝐴=𝐼𝑛 follows that ∑𝑆𝐴𝑝𝑟𝑆𝐵𝑟𝑞 𝑛 𝑘=1 = ∑𝑆𝐵𝑝𝑟𝑆𝐴𝑟𝑞 𝑛 𝑘=1 = for p≠q ∑𝑆𝐴𝑝𝑟𝑆𝐵𝑟𝑞 𝑛 𝑘=1 = ∑𝑆𝐵𝑝𝑟𝑆𝐴𝑟𝑝 𝑛 𝑘=1 =𝐼 𝑆𝐴𝑝𝑞𝑆𝐵𝑞𝑟=𝑆𝐵𝑝𝑟𝑆𝐴𝑟𝑞 = for p≠q and r {1,2,…..n} (8) 𝑆𝐴𝑝𝑟𝑆𝐵𝑟𝑝 =𝑆𝐵𝑝𝑟𝑆𝐴𝑟𝑝 =𝐼 for atleast one r {1,2,…..n} and for each p {1,2,…..n} (9) From (8) 𝑆𝐴𝑝𝑟 = or 𝑆𝐵𝑟𝑞 = or both 𝑆𝐴𝑝𝑟 =𝑆𝐵𝑟𝑞 = for p≠q and r {1,2,…..n} (10) and 𝑆𝐵𝑝𝑟 = or 𝑆𝐴𝑟𝑞 = or both 𝑆𝐵𝑝𝑟 =𝑆𝐴𝑟𝑞 = for p≠q and r {1,2,…..n} (11) Also, using (11) 𝑆𝐴𝑝𝑟 =𝑆𝐵𝑟𝑝 =I and 𝑆𝐴𝑟𝑝 =𝑆𝐵𝑝𝑟 =I for atleast one r {1,2,…..n} and for each p {1,2,…..n} (12) Let the results of (12) equation exists for k=p (say) that is 𝑆𝐴𝑝𝑘 =𝑆𝐵𝑘𝑝=I = [<(1,1),(0,0),(0,0)>] Then from (10), we get 𝑆𝐵𝑝𝑗 = =[<(0,0),(1,1),(1,1)>] for all i≠ j and Neutrosophic Sets and Systems, Vol. 97, 2026 702 P. Sheeba Maybell, M.M. Shanmugapriya, Eigen Neutrosophic ZSet and Neutrosophic ZRelation 𝑄3 =[<(0.8,0.7),(0.4,0.3),(0.2,0.1)> <(0.8,0.7),(0.4,0.3),(0.2,0.1)> <(0.8,0.7),(0.4,0.3),(0.2,0.1)>] Now, Q3=Q2 then Q2 is the desired GENZ set Next, algorithm I to Calculate LENZS calculate Q1 ′ 𝑄1′ =[<(0.6,0.5),(0.6,0.4),(0.3,0.1)> <(0.5,0.6),(0.5,0.3),(0.4,0.3)> <(0.5,0.6),(0.6 ,0.7),(0.3,0.4)>] Next step for n=1, 𝑄2′= 𝑄1′ ∙𝐻1 𝑄2′ =[<(0.6,0.6),(0.5,0.3),(0.3,0.1)> <(0.5,0.6),(0.5,0.3),(0.3,0.3)> <(0.5,0.6),(0.6 ,0.4),(0.3,0.1)>] Now, find 𝑄3′= 𝑄2′ ∙𝐻1 𝑄3′ =[<(0.6,0.6),(0.5,0.3),(0.3,0.1)> <(0.5,0.6),(0.5,0.3),(0.3,0.3)> <(0.5,0.6),(0.6 ,0.4),(0.3,0.1)>] Now, 𝑄3′= 𝑄2′ then 𝑄2′ is the acquired LENZS. Then using (15), (16) and (17) find 𝑆𝐴𝑚𝑎𝑥 and 𝑆𝐴𝑚in. The effectiveness (E) and Uncertainty(U) can be found using average of max, min value and difference of max, min value divided by 2. Table 2: Result of Effectiveness and Uncertainty Table 2 depicts effectiveness E1 is higher and uncertainty U1 is lower which makes the ambiance is good. The highest uncertainty of U3 gives the feedback of monetary value can be considered in future. 7. Conclusion In this work, neutrosophic z-relation along with neutrosophic zmatrices and their certain connected properties and models are introduced. The algorithms for calculating two types of eigen neutrosophic z-set with analogous were shown. At last, a utilization of neutrosophic zset in choice strategy problem using eigen neutrosophic z-set were found. As, an extension of this work in future, Neutrosophic Z-relations and Z-matrices may be generalized to higher-order structures such as Neutrosophic Z-tensors, enabling the modeling of multi-dimensional and highly uncertain data. Parameters 𝑆𝐴𝑚𝑎𝑥 0.8066 0.8066 0.8066 𝑆𝐴𝑚in 0.7266 0.6866 0.6766 E 0.7666 0.7463 0.7416 U 0.04 0.06 0.065 Neutrosophic Sets and Systems, Vol. 97, 2026 703 P. 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