EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6729 ISSN 1307-5543 – ejpam.com Published by New York Business Global Property HyperGraphs and Property SuperHyperGraphs for Data Analysis Takaaki Fujita1,∗, Florentin Smarandache2 1 Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan 2 University of New Mexico, Gallup Campus, NM 87301, USA Abstract. Graph theory provides a rigorous mathematical foundation for modeling relationships by representing entities as vertices and their interactions as edges [1, 2]. Hypergraphs generalize this paradigm by allowing hyperedges to connect arbitrary subsets of vertices [3], and SuperHyper- Graphs extend it further via iterated powerset constructions that capture hierarchical, multi-layer linkages among edges [4, 5]. These enriched models support applications across biology, social networks, signal processing, and knowledge representation. Property Graphs are directed multi- graphs in which vertices and edges carry key–value properties and edges additionally bear labels, enabling schema-flexible modeling of heterogeneous, real-world data(cf.[6–8]). In this paper, we show how to elevate Property Graphs to the settings of HyperGraphs and SuperHyperGraphs by introducing formal definitions for Property HyperGraphs and Property SuperHyperGraphs and presenting preliminary theoretical results that demonstrate their expressive power. 2020 Mathematics Subject Classifications: 05C65 Key Words and Phrases: Property graphs, superhypergraphs, hypergraphs, property superhy- pergraphs, property hypergraphs 1. Introduction Classical graphs model binary relations by representing entities as vertices and their pairwise connections as edges [1, 9]. Hypergraphs extend this notion by allowing each hyperedge to join any nonempty collection of vertices, thereby capturing higher‑order interactions [10–12]. However, even hypergraphs cannot naturally express nested or hi- erarchical groupings. To remedy this, SuperHyperGraphs were introduced: by iteratively applying the powerset operation to the vertex set, one obtains multi‑layered networks that encode nested relationships among vertex clusters [13–15]. This construction is versatile enough to subsume a wide range of graph models—including HyperGraphs and MultiHy- perGraphs—under a single unifying framework. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6729 Email addresses: Takaaki.fujita060@gmail.com (T. Fujita), smarand@unm.edu (F. Smarandache) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 2 of 36 Directed variants of these structures—such as Directed Graphs [16], Directed Hyper- Graphs [17–19], and Directed SuperHyperGraphs[20]—are also well-known and have been extensively studied in numerous research papers. Data modeling is the process of defining and organizing data structures, relationships, and constraints to support storage, retrieval, and analysis [21–23]. A Property Graph is a directed multigraph in which both vertices and edges carry arbitrary key–value attributes, and each edge is labeled. This schema‑flexible model enables the representation of rich relational data [6–8]. Although research on HyperGraphs, SuperHyperGraphs, and Property Graphs is well established, the specific combinations—Property HyperGraphs and Property SuperHyper- Graphs—have received little attention. In this paper, we fill this gap by providing formal definitions for Property HyperGraphs and Property SuperHyperGraphs and by present- ing preliminary theoretical results that demonstrate their expressive power. The author believes that one of the advantages of these frameworks is that Property Graphs can also be applied to more hierarchical concepts. This subsection explains the structure of the paper. Section 2 provides an overview of SuperHyperGraphs, HyperGraphs, and Property Graphs. Section 3 examines Property HyperGraphs, discussing their applications and key characteristics. Section 4 investigates Property SuperHyperGraphs, presenting applications and structural properties. Section 5 offers concluding remarks and outlines possible directions for future research. 2. Preliminaries Throughout this paper, we adopt a consistent vocabulary and notation. Unless oth- erwise noted, all graphs considered are finite. For further background on less familiar operations or concepts, the interested reader is referred to the cited literature. 2.1. SuperHyperGraphs A finite hypergraph generalizes the classical graph model by permitting hyperedges that connect any non‐empty subset of vertices [10, 24, 25]. Building on this concept, a finite SuperHyperGraph is obtained by iteratively applying the powerset operator, thereby creating nested hierarchies of vertex and edge sets that encode multi‐layered relation- ships [26, 27]. Such structures have demonstrated utility in areas ranging from molecular design and complex‐network analysis to advanced signal‐processing pipelines [28–30]. Un- less stated otherwise, the integer n in Pn(·) or in an n-SuperHyperGraph is assumed to be non‐negative. Definition 1 (Base Set). A base set S is the initial universe of discourse: S = {x | x belongs to the context at hand}. Every element that appears in P(S) or in any iterated powerset Pn(S) must of course lie in S. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 3 of 36 Definition 2 (Powerset). (cf.[31, 32]) For a set S, the powerset P(S) is the family of all subsets of S: P(S) = {A ⊆ S }. This collection includes both S itself and the empty set ∅. Definition 3 (Hypergraph). [3, 33] A hypergraph is an ordered pair H = (V,E) where • V is a finite vertex set, and • E is a finite family of non-empty subsets of V ; the members of E are called hyper- edges. Hypergraphs naturally represent interactions that involve more than two participants. Example 1 (Hypergraph Model for Market-Basket Analysis). Market-basket analysis seeks to uncover groups of products that are purchased together. A compact way to represent the entire transaction log is to view it as a hypergraph in the sense of Definition 3. Vertex set. Let V = {Bread,Butter,Milk,Eggs,Coffee,Cheese } be the collection of six distinct items sold in a small grocery store. Observed transactions (hyperedges). During one afternoon the cash register records four baskets: T1 = {Bread,Butter,Milk}, T2 = {Coffee,Milk}, T3 = {Bread,Eggs,Cheese}, T4 = {Butter,Eggs,Milk}. Each transaction is a non-empty subset of V and is therefore admissible as a hyperedge. Resulting hypergraph. Define E = {T1, T2, T3, T4 }. The pair H = (V,E) is a hypergraph whose hyperedges correspond one-to-one with the observed baskets. Mining tasks enabled by this representation • Support counting: the degree of a vertex equals the number of baskets that contain the corresponding item, providing its purchase frequency. • Frequent-itemset discovery: any subset of vertices that appears together in a sufficient number of hyperedges constitutes a frequent pattern. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 4 of 36 • Hyperedge clustering: grouping baskets that share many items can reveal customer segments with similar buying habits. • Edge contraction for category analysis: by merging vertices into higher-level product categories (e.g. dairy, bakery), the same hypergraph can be coarsened without revisiting the raw data. The hypergraph model thus captures all transactions simultaneously while retaining enough structure to support the full spectrum of basket-mining algorithms, from simple frequency counts to sophisticated community detection among both items and customers. Definition 4 (n-th Powerset). [34, 35] Let X be a set. The first powerset is P1(X) = P(X). For n ≥ 1 we define Pn+1(X) = P ( Pn(X) ) . When the empty set is excluded one writes P∗ n(X) = Pn(X) \ {∅}. Example 2 (n-th Powerset in Market-Basket Data Mining). Suppose a supermarket tracks customer purchases over a single week and observes the following five items: X = {Bread,Milk,Eggs,Butter,Coffee}. P1(X) = P(X): candidate itemsets. Every non-empty subset of X is a candidate itemset. For instance, {Bread,Butter} or {Milk,Eggs,Coffee}. Standard algorithms such as Apriori scan the transaction log to determine which of these subsets occur frequently. P2(X) = P ( P1(X) ) : clusters of itemsets. The second powerset groups itemsets into itemset clusters. An analyst may, for example, collect all frequent two-item combinations that involve Coffee: CCoffee = { {Coffee,Milk}, {Coffee,Bread}, {Coffee,Eggs} } ∈ P2(X). P3(X) = P ( P2(X) ) : meta-clusters. The third powerset organises clusters of itemsets into meta-clusters. One might place all itemset clusters whose underlying products form a typical breakfast assortment into a single meta-cluster: MBreakfast = { CCoffee, CMilk, CEggs } ∈ P3(X). Interpretation. • Level 1 (P1(X)) supports traditional frequent-itemset mining. • Level 2 (P2(X)) enables discovery of correlated patterns, such as sets of itemsets that often appear together across many market segments. • Level 3 (P3(X)) facilitates higher-order reasoning, e.g. comparing entire pattern families between different seasons or geographical regions. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 5 of 36 In practical pipelines, each higher powerset serves as the search space for progressively more abstract data-mining tasks: association-rule discovery at level 1, pattern clustering at level 2, and meta-pattern comparison or visual analytics at level 3 and beyond. Definition 5 (n-SuperHyperGraph). (cf. [13]) Fix a finite, nonempty base set V0 and define the iterated powerset by P0(V0) := V0, Pk+1(V0) := P ( Pk(V0) ) (k ∈ N). For an integer n ≥ 0, an n-SuperHyperGraph on V0 is a pair SHG(n) = (V,E) such that V ⊆ Pn(V0) and E ⊆ P(V ) \ {∅}. Elements of V are called n-supervertices and elements of E are n-superedges. (In partic- ular, each n-superedge is a nonempty subset of V .) Table 1 provides a concise overview of Graphs, Hypergraphs, and n-SuperHyperGraphs. In particular, SuperHyperGraphs are expected to offer an intuitive way to represent hier- archical network structures that commonly arise in real-world systems. Table 1: Concise overview of Graph, Hypergraph, and n-SuperHyperGraph. Model Vertices Edges Notes Graph V (finite) E ⊆ V × V or E = {{u, v}} Each edge links exactly two vertices; captures pairwise relations. Hypergraph V (finite) E ⊆ P(V ) \ {∅} Hyperedges connect arbi- trary nonempty subsets of vertices. n-SuperHyperGraph V ⊆ Pn(V0) E ⊆ P(V ) \ {∅} Vertices lie at level n (it- erated powerset); edges link level-n vertices in hierarchi- cal groupings. Several concrete examples of SuperHyperGraphs are presented below. Example 3 (A 2-SuperHyperGraph for market-basket data mining). We show that a toy transaction dataset naturally forms a 2-SuperHyperGraph SHG(2) = (V,E) in the sense of Definition 5. Step 0 (base set of items). Let V0 = {A,B,C,D} = {bread, milk, eggs, butter}. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 6 of 36 Step 1 (transactions as 1-supervertices). Transactions are subsets of V0, hence elements of P1(V0) = P(V0). Fix four transactions T1 = {A,B}, T2 = {B,C,D}, T3 = {A,C,D}, T4 = {A,B,D} ∈ P1(V0). Step 2 (pattern groups as 2-supervertices). Sets of transactions lie in P2(V0) = P ( P(V0) ) . Define G1 = {T1, T4}, G2 = {T2, T3}, G3 = {T2, T3, T4}, and set V := {G1, G2, G3} ⊆ P2(V0). Interpretation: G1 collects baskets with {A,B}; G2 those with {C,D}; G3 those with D plus at least one additional item. Step 3 (2-superedges via overlap of pattern groups). Define a symmetric adja- cency on V by {Gi, Gj} ∈ E ⇐⇒ i 6= j and Gi ∩Gj 6= ∅, and take E = { {G1, G3}, {G2, G3} } ⊆ P(V ) \ {∅}. Explicitly, G1 ∩G3 = {T4} 6= ∅ and G2 ∩G3 = {T2, T3} 6= ∅, while G1 ∩G2 = ∅. Each Ti ⊆ V0 ⇒ Ti ∈ P1(V0). Therefore Gj ⊆ {T1, T2, T3, T4} ⇒ Gj ∈ P2(V0). Thus V ⊆ P2(V0). Moreover, every listed superedge is a nonempty subset of V , so E ⊆ P(V ) \ {∅}. Hence SHG(2) = (V,E) is a valid 2-SuperHyperGraph. Use in pattern mining. Vertices Gj encode candidate frequent pattern groups; edges record higher-order overlaps among these groups. Level-wise algorithms (e.g. Apriori) may operate on SHG(2) to prune infrequent groups and derive rules such as {A,B} ⇒ D when supported by the underlying transactions (e.g. via the shared transaction T4). Example 4 (A 3-SuperHyperGraph for multi-tier database development). We model columns, tables, logical databases, and services at successive powerset levels. Step 0 (V0: atomic columns). V0 = {UserID,Name,Email,OrderID,OrderDate,ProductID,Quantity,Price}. Step 1 (P1(V0): tables as 1-supervertices). TUsers := {UserID,Name,Email}, TOrders := {OrderID,UserID,OrderDate}, TOrderItems := {OrderID,ProductID,Quantity,Price}, ∈ P1(V0). T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 7 of 36 Let V1 := {TUsers, TOrders, TOrderItems} ⊆ P1(V0). Define 1-superedges E1 := { {TUsers, TOrders}, {TOrders, TOrderItems} } ⊆ P(V1) \ {∅}, encoding the foreign-key relations Orders.UserID → Users.UserID and OrderItems.OrderID → Orders.OrderID. Step 2 (P2(V0): logical databases as 2-supervertices). EComDB := {TUsers, TOrders, TOrderItems}, AnalyticsDB := {TOrders, TOrderItems}, so V2 := {EComDB,AnalyticsDB} ⊆ P2(V0). Define a 2-superedge E2 := { {EComDB,AnalyticsDB} } ⊆ P(V2) \ {∅}, representing an ETL replication from operational to analytical storage. Step 3 (P3(V0): services as 3-supervertices). TransactionSvc := {EComDB}, ReportingSvc := {AnalyticsDB}, hence V3 := {TransactionSvc,ReportingSvc} ⊆ P3(V0). Define a 3-superedge E3 := { {TransactionSvc,ReportingSvc} } ⊆ P(V3) \ {∅}, capturing an event stream by which the transaction service publishes updates consumed by the reporting service. Verification (membership checks). By construction, Vk ⊆ Pk(V0) and Ek ⊆ P(Vk) \ {∅} for k = 1, 2, 3. Thus SHG(3) = (V3, E3) satisfies Definition 5. Interpretation. Level 0 lists columns; level 1 models relational schemas and key con- straints; level 2 groups tables into logical databases; level 3 groups databases into services and specifies inter-service dependencies as superedges. 2.2. Property Graph A Property Graph is a directed multigraph in which both vertices and edges may carry arbitrary key–value attributes, and where edges are additionally assigned labels. This structure enables schema-flexible modeling of relational data [6–8, 36]. Below, we provide a concise formal definition together with illustrative examples. Definition 6 (Property Graph). (cf.[6–8, 36]) Fix three (possibly infinite) sets Σ (edge–label alphabet), T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 8 of 36 K (property keys), S (property values). A property graph† is a septuple G = ( V, E, s, t, λ, µ, ⊥ ) whose components satisfy the following conditions: (a) V is a finite (or at most countable) set whose elements are called vertices (or nodes). (b) E is a finite (or at most countable) set whose elements are called edges. Distinct edges may share the same endpoints, so (V,E) is a multigraph. (c) s, t : E → V are the source and target functions. For e ∈ E we write s(e) e−→ t(e). (d) λ : E → Σ assigns a label (drawn from the alphabet Σ) to every edge. (e) µ : (V ∪ E)×K −→ S ∪ {⊥} is the property map. For an entity x ∈ V ∪ E and a key k ∈ K, the value µ(x, k) is either a member of S or the distinguished symbol ⊥ indicating that x has no value for key k. (f) The symbol ⊥ /∈ S is fixed once and for all and is not considered a valid property value. We write keyset(x) := { k ∈ K | µ(x, k) 6= ⊥}, val(x, k) := µ(x, k) (k ∈ keyset(x)), and call 〈k, µ(x, k)〉 an attribute of x. Remark 1. (i) Allowing E to be a multiset (or, equivalently, introducing edge identifiers) lets two vertices be joined by arbitrarily many edges—even with identical labels and attributes. (ii) If λ is constant (all edges share one label) and µ ≡ ⊥, Definition 6 collapses to the usual concept of a directed multigraph. (iii) Many graph-database operations (e.g. Gremlin traversals) can be formalised as func- tions T : P(V )→P(V ); see Yamaguchi et al. for a standard treatment. Example 5 (Property Graph for a Streaming-Platform Dataset). We illustrate Defini- tion 6 with a concrete, small-scale model of an on-line movie-streaming service. Label alphabet, keys, and value domain. Σ = {FOLLOWS, RATED, HAS_GENRE}, K = {type, name, age, city, title, year, rating, date}, S = N ∪ R ∪ Strings. †Equivalent to “directed, edge-labelled, attributed multigraph” in the graph-database literature. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 9 of 36 Vertices. The vertex set V = {p1, p2,m1,m2, g1} is partitioned by the attribute type: µ(pi, type) = “Person” (i = 1, 2), µ(mj , type) = “Movie” (j = 1, 2), µ(g1, type) = “Genre”. Selected vertex attributes are µ(p1, name) = “Alice” µ(p2, name) = “Bob” µ(p1, age) = 27 µ(p2, age) = 25 µ(p1, city) = “Tokyo” µ(p2, city) = “Kyoto” µ(m1, title) = “Inception” µ(m1, year) = 2010 µ(m2, title) = “Interstellar” µ(m2, year) = 2014 µ(g1, name) = “Sci-Fi”. Edges, endpoints, labels. E = {e1, e2, e3, e4, e5}, s(e1) = p1, t(e1) = p2, λ(e1) = FOLLOWS, s(e2) = p1, t(e2) = m1, λ(e2) = RATED, s(e3) = p1, t(e3) = m2, λ(e3) = RATED, s(e4) = m1, t(e4) = g1, λ(e4) = HAS_GENRE, s(e5) = m2, t(e5) = g1, λ(e5) = HAS_GENRE. Edge attributes. µ(e1, date) = “2025-05-12”, µ(e2, rating) = 5, µ(e2, date) = “2025-05-13”, µ(e3, rating) = 4, µ(e3, date) = “2025-05-14”. All other µ(x, k) not listed are set to the distinguished value ⊥. The septuple G = (V,E, s, t, λ, µ,⊥) thus obtained satisfies every clause of Defini- tion 6. It captures users (persons), movies, and genres as vertices; user-to-user FOLLOWS relationships, user ratings of movies, and movie-to-genre links as labelled, attributed edges. Additional properties or vertex types (e.g. Director, Studio) can be incorporated seam- lessly by extending the key set K and adding new vertices and edges. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 10 of 36 3. Results and Revisits: Property HyperGraphs Property HyperGraphs generalize hypergraphs by allowing vertices and hyperedges to carry key–value properties and labels, supporting flexible modeling. Notation 1. Fix three (possibly infinite) sets Σ (hyperedge‐label alphabet), K (property keys), S (property values), and let ⊥ /∈ S be a distinguished symbol. Definition 7 (Property HyperGraph). A property hypergraph is a quadruple H = ( V, E, λ, µ ) satisfying: (a) V is a finite (or at most countable) set of vertices. (b) E is a finite family of non‐empty subsets of V , called hyperedges. (c) λ : E → Σ assigns to each hyperedge a label. (d) µ : (V ∪ E)×K → S ∪ {⊥} is the property map, where µ(x, k) = { s ∈ S, if x has property k with value s, ⊥, if no value is assigned. We write (x) := { k ∈ K | µ(x, k) 6= ⊥}, (x, k) := µ(x, k) (k ∈ (x)). Example 6 (Property HyperGraph for Patient–Symptom Dataset). We illustrate Defi- nition 7 by modelling a clinical dataset in which each patient record links the symptoms they exhibit and carries patient metadata. Label alphabet, keys, and value domain. Σ = {COVID19, Influenza, Migraine}, K = {category, ICD_code, age, gender, severity}, S = {“Constitutional”, “Respiratory”, “Neurological”} ∪ {Strings} ∪ N. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 11 of 36 Vertices (symptoms). Let V = {v1, v2, v3, v4, v5}, where µ(v1, category) = “Constitutional”, µ(v1, ICD_code) = “R50.9”, µ(v2, category) = “Respiratory”, µ(v2, ICD_code) = “R05”, µ(v3, category) = “Constitutional”, µ(v3, ICD_code) = “R53.83”, µ(v4, category) = “Neurological”, µ(v4, ICD_code) = “R51”, µ(v5, category) = “Respiratory”, µ(v5, ICD_code) = “R06.02”. Hyperedges (patient records). Define three patient hyperedges: E = {e1, e2, e3}, with e1 = {v1, v2, v3}, e2 = {v2, v3, v4}, e3 = {v1, v4, v5}. Labels and properties. Assign each record a diagnosis label and patient metadata: λ(e1) = COVID19, µ(e1, age) = 45, µ(e1, gender) = “Male”, µ(e1, severity) = “Moderate”, λ(e2) = Influenza, µ(e2, age) = 30, µ(e2, gender) = “Female”, µ(e2, severity) = “Mild”, λ(e3) = Migraine, µ(e3, age) = 25, µ(e3, gender) = “Female”, µ(e3, severity) = “Severe”. Keysets and values. For example, (v4) = {category, ICD_code}, (v4, ICD_code) = “R51”, (e2) = {age, gender, severity}, (e2, severity) = “Mild”. The quadruple H = (V,E, λ, µ) thus satisfies Definition 7, modelling a patient‐symptom dataset in which each record links multiple symptoms and carries patient attributes. Example 7 (Property HyperGraph for University Course Enrollment). We illustrate Def- inition 7 by modelling a university’s course‐enrollment system. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 12 of 36 Label alphabet, keys, and value domain. Σ = {CSE101, MATH202, HIST303}, K = {role, name, semester, credits}, S = {“Instructor”, “Student”, “TA”} ∪ {Strings} ∪ N. Vertices. Let V = { v1, v2, v3, v4, v5} where µ(v1, name) = “Dr. Smith”, µ(v1, role) = “Instructor”, µ(v2, name) = “Alice”, µ(v2, role) = “Student”, µ(v3, name) = “Bob”, µ(v3, role) = “Student”, µ(v4, name) = “Carol”, µ(v4, role) = “TA”, µ(v5, name) = “Dave”, µ(v5, role) = “Student”. Hyperedges. Define three hyperedges, one per course: E = { e1, e2, e3 } , with e1 = {v1, v2, v3, v4}, e2 = {v1, v2, v5}, e3 = {v1, v3, v5}. Labels and properties. For each course hyperedge ei we set λ(e1) = CSE101, λ(e2) = MATH202, λ(e3) = HIST303, and assign µ(e1, semester) = “Fall 2025”, µ(e1, credits) = 4, µ(e2, semester) = “Spring 2025”, µ(e2, credits) = 3, µ(e3, semester) = “Fall 2025”, µ(e3, credits) = 3. Keysets and values. For example, (v2) = {name, role}, (v2, name) = “Alice”, (e1) = {semester, credits}, (e1, credits) = 4. The quadruple H = (V,E, λ, µ) thus satisfies all clauses of Definition 7: Vertices represent people with roles, hyperedges represent courses linking instructor, students, and TAs, each course carries a label (course code) and properties (semester, credits). Example 8 (Property HyperGraph for Film Productions). We illustrate Definition 7 by modelling a film‐production scenario. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 13 of 36 Label alphabet, keys, and value domain. Σ = {TheGreatAdventure, MysteryNight}, K = {roleType, name, birthYear, nationality, releaseYear, genre, boxOffice}, S = {“Actor”, “Director”, “Producer”} ∪ Strings ∪ N. Vertices. Let V = { v1, v2, v3, v4, v5}, with properties µ(v1, name) = “Alice Johnson”, µ(v1, roleType) = “Actor”, µ(v1, birthYear) = 1985, µ(v2, name) = “Bob Lee”, µ(v2, roleType) = “Actor”, µ(v2, birthYear) = 1978, µ(v3, name) = “Carol Smith”, µ(v3, roleType) = “Director”, µ(v3, nationality) = “USA”, µ(v4, name) = “David Kumar”, µ(v4, roleType) = “Producer”, µ(v4, nationality) = “UK”, µ(v5, name) = “Eva Zhang”, µ(v5, roleType) = “Actor”, µ(v5, birthYear) = 1990. Hyperedges. Define two film hyperedges: E = { e1, e2}, where e1 = {v1, v2, v3, v4} (The Great Adventure cast/crew), e2 = {v2, v3, v5} (Mystery Night cast/crew). T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 14 of 36 Labels and properties. For each film hyperedge: λ(e1) = TheGreatAdventure, µ(e1, releaseYear) = 2024, µ(e1, genre) = “Action”, µ(e1, boxOffice) = 120000000, λ(e2) = MysteryNight, µ(e2, releaseYear) = 2023, µ(e2, genre) = “Mystery”, µ(e2, boxOffice) = 85000000. Keysets and values. For example, (v3) = {name, roleType, nationality}, (v3, roleType) = “Director”, (e1) = {releaseYear, genre, boxOffice}, (e1, genre) = “Action”. The quadruple H = (V,E, λ, µ) thus satisfies all requirements of Definition 7: vertices represent cast and crew with personal attributes; hyperedges represent films linking multiple participants, each carrying a label (film title) and properties (release year, genre, box office). Theorem 1 (Generalisation of Property Graphs and Hypergraphs). Let H = (V,E, λ, µ) be a property hypergraph over (Σ,K, S,⊥). Then: (i) If every hyperedge e ∈ E satisfies |e| = 2 and we equip e = {u, v} with an arbi- trary orientation u → v, then (V,E, s, t, λ, µ,⊥) is precisely a Property Graph as in Definition 6. (ii) If Σ = {σ0} is a singleton and µ(x, k) ≡ ⊥ for all (x, k), then H = (V,E) collapses to an ordinary Hypergraph as in Definition 3. Proof. For (i), restrict every 2‐element hyperedge e = {u, v} to a directed edge by choosing one of the two orderings (u, v) or (v, u). The label map λ : E → Σ and the property map µ coincide with those of a Property Graph. All axioms (a)–(f) of Definition 6 follow immediately. For (ii), since Σ has only one element, λ carries no additional information; and because µ ≡ ⊥, no vertex or hyperedge carries a property. Thus H = (V,E) satisfies exactly the conditions of Definition 3, concluding the proof. Theorem 2 (Vertex-induced sub-Property HyperGraph). Let H = (V,E, λ, µ) be a prop- erty hypergraph and let U ⊆ V . Define VU := U, EU := { e ∈ E | e ⊆ U }, and restrict the maps by λU := λ|EU and µU := µ|(U∪EU )×K . Then HU := (VU , EU , λU , µU ) is a property hypergraph. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 15 of 36 Proof. We check the items of Definition 7. (a) By construction VU ⊆ V . (b) Each e ∈ EU is nonempty and e ⊆ U = VU , hence EU ⊆ P(VU ) \ {∅}. (c) λU maps EU into Σ by restriction. (d) µU maps (VU ∪ EU ) × K into S ∪ {⊥} by restriction. All conditions hold. Theorem 3 (Edge-induced sub-Property HyperGraph). Let H = (V,E, λ, µ) be a property hypergraph and let F ⊆ E. Set VF := ⋃ e∈F e, EF := F, and restrict λF := λ|EF and µF := µ|(VF∪EF )×K . Then HF := (VF , EF , λF , µF ) is a property hypergraph. Proof. (a) VF ⊆ V by definition. (b) Every e ∈ EF is nonempty and satisfies e ⊆ VF , hence EF ⊆ P(VF ) \ {∅}. (c)–(d) Follow immediately from restricting λ and µ to the new domains/codomains. Theorem 4 (Uniformity and inheritance). Suppose H = (V,E, λ, µ) is such that there exists k ≥ 1 with |e| = k for all e ∈ E. Then H is k-uniform. Moreover, every subfamily F ⊆ E yields a k-uniform edge-induced sub-Property HyperGraph HF . Proof. By hypothesis, for each e ∈ E we have e ⊆ V and |e| = k, which is precisely the definition of k-uniformity. If F ⊆ E, then every e ∈ F still satisfies |e| = k, so Theorem 3 gives a k-uniform sub-Property HyperGraph. Theorem 5 (Disjoint union). Let H1 = (V1, E1, λ1, µ1) and H2 = (V2, E2, λ2, µ2) be property hypergraphs with disjoint vertex sets V1 ∩ V2 = ∅. Define V := V1 ∪ V2, E := E1 ∪ E2, and the piecewise maps λ(e) := { λ1(e), e ∈ E1, λ2(e), e ∈ E2, µ(x, k) := { µ1(x, k), x ∈ V1 ∪ E1, µ2(x, k), x ∈ V2 ∪ E2. Then H := (V,E, λ, µ) is a property hypergraph. Proof. (a) V is a set of vertices. (b) Each e ∈ Ei is nonempty and e ⊆ Vi ⊆ V , hence E ⊆ P(V ) \ {∅}. (c)–(d) Disjointness of V1, V2 makes λ and µ well-defined with the same codomains as before. Theorem 6 (Incidence representation as a Property Graph). Let H = (V,E, λ, µ) be a property hypergraph. Form a (directed) Property Graph G = (VG, EG, s, t, λG, µG,⊥) T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 16 of 36 by taking VG := V ∪ E (disjoint union of tags “vertex” and “edge”), EG := { (e, v) ∈ E × V | v ∈ e }, s(e, v) = e, t(e, v) = v, and define λG(e, v) := λ(e) ∈ Σ, µG(x, k) := { µ(x, k), x ∈ V ∪ E (= VG), ⊥, x ∈ EG. Then G is a Property Graph (in the sense of a directed, edge-labelled, attributed multigraph). Proof. By construction VG is a set and EG ⊆ VG × VG. The maps s, t are projections, hence well-defined. The edge-labelling λG uses the hyperedge label λ(e); the property map µG agrees with µ on all vertices of G (i.e. original vertices and hyperedges) and assigns the sentinel ⊥ to edges, ensuring µG : (VG ∪ EG)×K → S ∪ {⊥}. Thus all clauses of the Property Graph definition are satisfied. Theorem 7 (Forgetting labels and properties). If Σ = {σ0} is a singleton and µ ≡ ⊥, then any property hypergraph H = (V,E, λ, µ) reduces to the ordinary hypergraph (V,E). Proof. With Σ a singleton, λ carries no information beyond existence, and with µ ≡ ⊥ there are no attributes on vertices or hyperedges. The remaining data are precisely V and E ⊆ P(V ) \ {∅}, which is the definition of a hypergraph. 4. Main Result: Property SuperHyperGraphs We introduce and formalise the concept of a Property SuperHyperGraph. Property SuperHyperGraphs extend SuperHyperGraphs with labels and key–value attributes on supervertices and superedges, supporting hierarchical attributed modeling. Definition 8 (Iterated powerset (Recall)). Let V0 be a finite, nonempty base set. Define P0(V0) := V0, Pk+1(V0) := P ( Pk(V0) ) (k ∈ N). For n ∈ N, the elements of Pn(V0) are called level-n carriers (or n-carriers). Definition 9 (Property n-SuperHyperGraph). Fix alphabets/sets Σ (edge-label alphabet), K (property keys), S (property values), and a distinguished symbol ⊥ /∈ S. For n ∈ N, a property n-SuperHyperGraph (over V0) is a quadruple H(n) = ( V (n), E(n), λ, µ ) consisting of: T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 17 of 36 (i) a vertex set V (n) ⊆ Pn(V0); (ii) a finite family of nonempty superedges E(n) ⊆ P ( V (n) ) \ {∅}; (iii) an edge-labelling map λ : E(n) → Σ; (iv) a property map µ : ( D(n) ×K ) −→ S ∪ {⊥}, D(n) := ( n⋃ k=0 Pk(V0) ) ∪ E(n), interpreted by µ(x, k) = s ∈ S when object x carries key k with value s, and µ(x, k) = ⊥ otherwise. We set (x) := { k ∈ K | µ(x, k) 6= ⊥}, (x, k) := µ(x, k) (k ∈ (x)). Elements of V (n) are n-supervertices; elements of E(n) are n-superedges. Example 9 (Property 3-SuperHyperGraph for an Image-Classification Dataset). We or- ganise a hierarchical dataset with images (level 0), species classes (level 1), genera (level 2), and dataset splits (level 3). Level 0 (images). Let V0 = { i1, i2, i3, i4, i5, i6 }. Assign base-level properties (allowed since V0 ⊆ D(3)): µ(ij , resolution) = “1024×768”, µ(ij , date) = “2025-06-01” (j = 1, . . . , 6). Level 1 (species; 1-supervertices). CCat = {i1, i2}, CDog = {i3, i4}, CBird = {i5, i6} ∈ P1(V0). Set V (1) = {CCat, CDog, CBird} and µ(CCat, species) = “Felis catus”, µ(CCat, count) = 2, µ(CDog, species) = “Canis lupus”, µ(CDog, count) = 2, µ(CBird, species) = “Passer domesticus”, µ(CBird, count) = 2. Level 2 (genera; 2-supervertices). GFelidae = {CCat}, GCanidae = {CDog}, GPasseridae = {CBird} ∈ P2(V0), with V (2) = {GFelidae, GCanidae, GPasseridae} and µ(GFelidae, genus) = “Felis”, µ(GFelidae, numSpecies) = 1, µ(GCanidae, genus) = “Canis”, µ(GCanidae, numSpecies) = 1, µ(GPasseridae, genus) = “Passer”, µ(GPasseridae, numSpecies) = 1. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 18 of 36 Level 3 (splits; 3-supervertices). STrain = {GFelidae, GCanidae}, STest = {GPasseridae} ∈ P3(V0), with V (3) = {STrain, STest} and µ(STrain, split) = “Train”, µ(STrain, fraction) = 0.8, µ(STest, split) = “Test”, µ(STest, fraction) = 0.2. 3-superedges. Let eFull = {STrain, STest} ∈ P ( V (3) ) \ {∅}, E(3) = { eFull }. With Σ = {ImageNetSubset}, λ(eFull) = ImageNetSubset, µ(eFull, version) = “v1.0”. Keysets. For instance, (CDog) = {species, count} and (CDog, species) = “Canis lupus”. Thus H(3) = (V (3), E(3), λ, µ) satisfies Definition 9. Example 10 (Property 3-SuperHyperGraph for Multi-Tier Database Systems). We model columns (level 0), tables (1), schemas (2), and clusters (3). Level 0 (columns). V0 = {user_id, username, email, order_id, order_date, product_id, quantity, price}. Assign µ(c, name) = c and µ(c, dataType) ∈ {INT, VARCHAR, DATE, DECIMAL}. Level 1 (tables; 1-supervertices). TUsers = {user_id, username, email}, TOrders = {order_id, user_id, order_date}, TItems = {order_id, product_id, quantity, price}, V (1) = {TUsers, TOrders, TItems}, with µ(TUsers, tableName) = “Users”, µ(TUsers, rowCount) = 120000, µ(TOrders, tableName) = “Orders”, µ(TOrders, rowCount) = 450000, µ(TItems, tableName) = “Items”, µ(TItems, rowCount) = 950000. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 19 of 36 Level 2 (schemas; 2-supervertices). SSales = {TUsers, TOrders, TItems}, V (2) = {SSales}, µ(SSales, schemaName) = “SalesDB”, µ(SSales, version) = “v1.2”. Level 3 (clusters; 3-supervertices). CPrimary = {SSales}, CReplica = {SSales}, V (3) = {CPrimary, CReplica}, with µ(CPrimary, clusterRole) = “Primary”, µ(CPrimary, region) = “us-east-1”, µ(CReplica, clusterRole) = “Replica”, µ(CReplica, region) = “us-west-2”. 3-superedges. Let eEnterprise = {CPrimary, CReplica}, E(3) = {eEnterprise}, Σ = {EnterpriseDB}, and set λ(eEnterprise) = EnterpriseDB, µ(eEnterprise, admin) = “DBA Team”, µ(eEnterprise, uptime_SLA) = 99.99. Then H(3) = (V (3), E(3), λ, µ) is a property 3-SHG. Example 11 (A 3-level Property SuperHyperGraph for Urban Transit). We model an urban transit system with stops (level 0), routes (level 1), lines (level 2), and networks (level 3). Level 0 (base stops). Let V0 = { s1, s2, s3, s4, s5, s6 }. Assign stop properties (allowed since V0 ⊆ D(3)): µ(s1, zone) = “1”, µ(s1, lat) = 35.69, µ(s1, lon) = 139.70, µ(s2, zone) = “1”, . . . Level 1 (routes as sets of stops; 1-supervertices). Define RA = {s1, s2, s3}, RB = {s3, s4, s5}, RC = {s5, s6} ∈ P1(V0). Set V (1) = {RA, RB, RC} and assign µ(RA, routeID) = “A”, µ(RA, headwayMin) = 6, µ(RB, routeID) = “B”, µ(RB, headwayMin) = 8, µ(RC , routeID) = “C”, µ(RC , headwayMin) = 12. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 20 of 36 Level 2 (lines as sets of routes; 2-supervertices). Let LRed = {RA, RB}, LBlue = {RC} ∈ P2(V0). Set V (2) = {LRed, LBlue} with µ(LRed, color) = “Red”, µ(LRed, operator) = “MetroCo”, µ(LBlue, color) = “Blue”, µ(LBlue, operator) = “MetroCo”. Level 3 (networks as sets of lines; 3-supervertices). Define NCity = {LRed, LBlue}, NRegional = {LBlue} ∈ P3(V0). Set V (3) = {NCity, NRegional} and assign µ(NCity, name) = “City Network”, µ(NCity, rev) = “2025Q3”, µ(NRegional, name) = “Regional Link”. 3-superedges and labels. Let the label alphabet Σ = {FareAgreement, Interchange}. Define e1 = {NCity, NRegional}, e2 = {NCity}, and E(3) = {e1, e2} ⊆ P ( V (3) ) \ {∅}. Set λ(e1) = FareAgreement, µ(e1, year) = 2025, λ(e2) = Interchange, µ(e2, hub) = “Central”. Verification. Each route is a subset of V0 ⇒ V (1) ⊆ P1(V0). Each line is a set of routes ⇒ V (2) ⊆ P2(V0). Each network is a set of lines ⇒ V (3) ⊆ P3(V0). Finally, e1, e2 are nonempty subsets of V (3). Thus H(3) = ( V (3), E(3), λ, µ ) is a property 3-SuperHyperGraph. Example 12 (A 3-level Property SuperHyperGraph for Academic Collections). We model academic artifacts with papers (level 0), venues (level 1), fields (level 2), and programs/con- sortia (level 3). T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 21 of 36 Level 0 (papers). Let V0 = { p1, p2, p3, p4, p5 }. Assign basic properties: µ(p1, title) = “A Study on X”, µ(p1, year) = 2023, µ(p2, title) = “Learning Y”, µ(p2, year) = 2024, . . . Level 1 (venues; 1-supervertices). Group papers by publication venue: VConfA = {p1, p2}, VConfB = {p3, p4}, VJournalC = {p5} ∈ P1(V0). Set V (1) = {VConfA, VConfB, VJournalC} and assign µ(VConfA, abbr) = “ConfA”, µ(VConfA, ISSN) = “1111-1111”, µ(VConfB, abbr) = “ConfB”, µ(VConfB, ISSN) = “2222-2222”, µ(VJournalC, abbr) = “JrnC”, µ(VJournalC, ISSN) = “3333-3333”. Level 2 (fields; 2-supervertices). Group venues by field: FML = {VConfA, VConfB}, FTheory = {VJournalC} ∈ P2(V0), with V (2) = {FML, FTheory} and µ(FML, fieldName) = “Machine Learning”, µ(FML, hIndex) = 120, µ(FTheory, fieldName) = “Theory”, µ(FTheory, hIndex) = 85. Level 3 (programs/consortia; 3-supervertices). PAIAlliance = {FML}, PSciCouncil = {FTheory} ∈ P3(V0), and V (3) = {PAIAlliance, PSciCouncil} with µ(PAIAlliance, program) = “AI Alliance”, µ(PSciCouncil, program) = “Science Council”. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 22 of 36 3-superedges and labels. Let Σ = {Consortium, DataSharing}. Define econs = {PAIAlliance, PSciCouncil}, eshare = {PAIAlliance}, E(3) = {econs, eshare} ⊆ P ( V (3) ) \ {∅}. Set λ(econs) = Consortium, µ(econs, agreementYear) = 2025, λ(eshare) = DataSharing, µ(eshare, scope) = “anonymized-metadata”. Verification. Each venue is a subset of V0 ⇒ V (1) ⊆ P1(V0). Each field is a set of venues ⇒ V (2) ⊆ P2(V0). Each program is a set of fields ⇒ V (3) ⊆ P3(V0). The listed edges are nonempty subsets of V (3). Hence H(3) = ( V (3), E(3), λ, µ ) is a property 3-SuperHyperGraph. Theorem 8 (Generalisation). Let H(n) = (V (n), E(n), λ, µ) be a property n-SuperHyperGraph over the base V0. Then: (i) If n = 1, every e ∈ E(1) has |e| = 2, and an orientation is chosen on each such pair, then H(1) induces a Property Graph in the sense of Definition 6. (ii) If n = 1, Σ = {σ0} is a singleton and µ ≡ ⊥, then H(1) reduces to the classical hypergraph ( V (1), E(1) ) . (iii) If Σ = {σ0} and µ ≡ ⊥ (for arbitrary n), then H(n) reduces to the ordinary n- SuperHyperGraph ( V (n), E(n) ) . Proof. (1) Since E(1) ⊆ P(V (1)) and |e| = 2 for each e ∈ E(1), every edge is a 2–element subset {u, v} ⊆ V (1). Fix, once and for all, a choice of orientation for each unordered pair: θ : E(1) −→ V (1) × V (1), θ({u, v}) = (u, v) or (v, u). Define the directed edge set EG := { θ(e) | e ∈ E(1) } ⊆ V (1) × V (1). Set VG := V (1) and define source/target maps by s(u, v) = u, t(u, v) = v. Let π : EG → E(1) be the forgetful map π(u, v) = {u, v}. Define the label and property maps by λG(a) := λ ( π(a) ) ∈ Σ, µG(x, k) := { µ(x, k), x ∈ VG = V (1), µ ( π(x), k ) , x ∈ EG. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 23 of 36 Then G = (VG, EG, s, t, λG, µG,⊥) satisfies all clauses of Definition 6: VG is the vertex set, EG ⊆ VG × VG, s, t are well-defined, λG labels directed edges via their underlying undirected pairs, and µG is a property map with codomain S ∪ {⊥} (using the original µ on vertices and pulling back along π on edges). Hence H(1) yields a Property Graph. (2) If n = 1, Σ = {σ0} and µ ≡ ⊥, then labels carry no information and no vertex/edge has a nontrivial property. The structure H(1) reduces to its underlying pair (V (1), E(1)), with E(1) ⊆ P(V (1)) \ {∅}, which is precisely the definition of a hypergraph. (3) The same argument applies verbatim for general n: with a trivial label alphabet and µ ≡ ⊥, only V (n) ⊆ Pn(V0) and E(n) ⊆ P(V (n)) \ {∅} remain. This is exactly an n-SuperHyperGraph. Theorem 9 (Induced sub-n-SuperHyperGraph). Let H(n) = (V (n), E(n), λ, µ) be a prop- erty n-SuperHyperGraph over the base V0, and let U0 ⊆ V0. Define V (n) U := {x ∈ V (n) | x ⊆ Pn−1(U0) }, E (n) U := { e ∈ E(n) | e ⊆ V (n) U }. Let D (n) U := ( n⋃ r=0 Pr(U0) ) ∪ E (n) U , and set H(n) U := ( V (n) U , E (n) U , λ| E (n) U , µ| D (n) U ×K ) . Then H (n) U is a property n-SuperHyperGraph over U0. Proof. We verify each clause of the definition. (i) Vertex condition. We first note the monotonicity of iterated powersets: if A ⊆ B, then Pr(A) ⊆ Pr(B) for all r ≥ 0. This is proved by induction on r: for r = 0 it is A ⊆ B; if Pr(A) ⊆ Pr(B), then Pr+1(A) = P ( Pr(A) ) ⊆ P ( Pr(B) ) = Pr+1(B). Applying this with A = U0 ⊆ B = V0 and r = n − 1, we get Pn−1(U0) ⊆ Pn−1(V0). Hence every x ∈ V (n) U lies in Pn(U0) (because x is by definition a subset of Pn−1(U0)), so V (n) U ⊆ Pn(U0). (ii) Edge condition. By construction, every e ∈ E (n) U satisfies e ⊆ V (n) U and e 6= ∅ (because E(n) consisted of nonempty subsets of V (n), and we only removed vertices). Thus E (n) U ⊆ P ( V (n) U ) \ {∅}. (iii) Label map. The restriction λ| E (n) U : E (n) U → Σ is well-defined with the same codomain Σ. (iv) Property map. Since D (n) U ⊆ (⋃n r=0 Pr(V0) ) ∪ E(n) by monotonicity and by the definition of E(n) U , the restricted map µ| D (n) U ×K : D (n) U ×K → S ∪ {⊥} is well-defined with the same codomain. All axioms are satisfied, hence H (n) U is a property n-SuperHyperGraph over U0. Example 13 (Induced sub-n-SuperHyperGraph: a concrete n = 2 instance). Fix Σ = {Link}, K = {tag}, S = N ∪ Strings, ⊥ /∈ S, T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 24 of 36 and let the base be V0 = {a, b, c, d}. Define 1-level carriers (subsets of V0) A1 = {a, b}, A2 = {b, c}, A3 = {c, d} ∈ P1(V0), and 2-supervertices G1 = {A1, A2}, G2 = {A2, A3} ∈ P2(V0). Set V (2) = {G1, G2} ⊆ P2(V0) and E(2) = { e }, e = {G1, G2} ∈ P(V (2)) \ {∅}. Label and properties: λ(e) = Link, µ(G1, tag) = “north”, µ(G2, tag) = “south”, µ(e, tag) = “overlap”. Thus H(2) = (V (2), E(2), λ, µ) is a property 2-SHG over V0. Now take U0 = {a, b, c} ⊆ V0. Then P1(U0) = P(U0), and we check membership: A1, A2 ⊆ U0 ⇒ G1 ⊆ P1(U0), A3 * U0 ⇒ G2 * P1(U0). Hence V (2) U = {G1}, E (2) U = ∅, and with D (2) U = P0(U0) ∪ P1(U0) ∪ P2(U0) ∪ E (2) U we obtain the induced substructure H (2) U = ( V (2) U , E (2) U , λ| E (2) U , µ| D (2) U ×K ) , which is (by Theorem 9) a property 2-SuperHyperGraph over the base U0. Theorem 10 (Flattening projection). Let H(n) = (V (n), E(n), λ, µ) be a property n-SuperHyperGraph with n ≥ 1. Define the vertex-flattening map ϕV : Pn(V0) −→ Pn−1(V0), ϕV (X) = ⋃ x∈X x, and its action on edges by image ϕE : P ( V (n) ) \ {∅} −→ P ( ϕV [V (n)] ) \ {∅}, ϕE(e) = { ϕV (v) | v ∈ e }. Assume the following compatibility on fibres: (labels on edges) ϕE(e) = ϕE(e ′) =⇒ λ(e) = λ(e′), T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 25 of 36 (properties on vertices) ϕV (v) = ϕV (v ′) =⇒ µ(v, k) = µ(v′, k) ∀k ∈ K, (properties on edges) ϕE(e) = ϕE(e ′) =⇒ µ(e, k) = µ(e′, k) ∀k ∈ K. Then with V ′ := ϕV [V (n)] ⊆ Pn−1(V0), E′ := { ϕE(e) | e ∈ E(n) } ⊆ P(V ′) \ {∅}, and with λ′(ϕE(e)) := λ(e), µ′(z′, k) := { µ(v, k), z′ = ϕV (v) ∈ V ′, µ(e, k), z′ = ϕE(e) ∈ E′, the quadruple H ′ = (V ′, E′, λ′, µ′) is a property (n− 1)-SuperHyperGraph over V0. Proof. (i) Vertex condition. By definition of ϕV , we have ϕV (X) ∈ Pn−1(V0) for every X ∈ Pn(V0). Hence V ′ = ϕV [V (n)] ⊆ Pn−1(V0). (ii) Edge condition. Let e ∈ E(n). Since e ⊆ V (n) and e 6= ∅, its image ϕE(e) = {ϕV (v) | v ∈ e} is a nonempty subset of V ′. Thus E′ ⊆ P(V ′) \ {∅}. (iii) Labels. If ϕE(e) = ϕE(e ′), the label-compatibility assumption gives λ(e) = λ(e′), so λ′ is well-defined on E′ and λ′ : E′ → Σ. (iv) Properties. If ϕV (v) = ϕV (v ′) then µ(v, k) = µ(v′, k) for all k ∈ K, and if ϕE(e) = ϕE(e ′) then µ(e, k) = µ(e′, k) for all k ∈ K. Hence µ′ is well-defined on (V ′∪E′)×K with codomain S ∪ {⊥}. All clauses of the definition of a property (n− 1)-SuperHyperGraph are satisfied. Example 14 (Flattening projection: a concrete 2 → 1 instance). Let the base set be V0 = {a, b, c}. Consider the property 2-SuperHyperGraph H(2) = (V (2), E(2), λ, µ) with V (2) = { v1 = { {a, b}, {b, c} } , v2 = { {a}, {a, c} }} ⊆ P2(V0), E(2) = { e = {v1, v2} } ⊆ P(V (2)) \ {∅}. Let the label alphabet be Σ = {α} and set λ(e) = α. Fix keys K = {tag, weight} and a value domain S containing strings and integers; take ⊥ /∈ S. Define the property map by µ(v1, tag) = “T1”, µ(v2, tag) = “T2”, µ(e, weight) = 3, and µ(·, ·) = ⊥ for all other key–object pairs. The vertex–flattening map ϕV : P2(V0) → P1(V0) yields ϕV (v1) = {a, b} ∪ {b, c} = {a, b, c}, ϕV (v2) = {a} ∪ {a, c} = {a, c}. The induced edge map ϕE gives ϕE(e) = { ϕV (v1), ϕV (v2) } = { {a, b, c}, {a, c} } . T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 26 of 36 Hence the flattened structure H ′ = (V ′, E′, λ′, µ′) is a property 1-SuperHyperGraph with V ′ = ϕV [V (2)] = { {a, b, c}, {a, c} } ⊆ P1(V0), E′ = { ϕE(e) } . By the theorem, we set λ′(ϕE(e) ) = λ(e) = α, µ′({a, b, c}, tag ) = µ(v1, tag) = “T1”, µ′({a, c}, tag ) = µ(v2, tag) = “T2”, µ′(ϕE(e), weight ) = µ(e, weight) = 3, and µ′(·, ·) = ⊥ otherwise. Since no two distinct vertices (resp. edges) of H(2) share the same ϕV -image (resp. ϕE-image), the fibre–compatibility conditions hold trivially, and Theorem 10 applies. Theorem 11 (Iterated flattening). With notation as above, for 1 ≤ k ≤ n define ϕ (k) V := ϕV ◦ · · · ◦ ϕV︸ ︷︷ ︸ k times : Pn(V0) → Pn−k(V0), and extend to edges by ϕ (k) E (e) := { ϕ (k) V (v) | v ∈ e }. If the label/property compatibilities hold at each intermediate stage, then H(n−k) := ( ϕ (k) V [V (n)], {ϕ(k) E (e) | e ∈ E(n)}, λ(k), µ(k) ) is a property (n− k)-SuperHyperGraph. Proof. Fix n ∈ N and 1 ≤ k ≤ n. Recall the vertex–flattening map ϕV : Pr(V0) → Pr−1(V0), X 7→ ⋃ x∈X x for r ≥ 1, and its extension to (nonempty) edge-sets ϕE : P(V (r))\ {∅} → P ( ϕV [V (r)] ) \ {∅}, ϕE(e) := { ϕV (v) | v ∈ e }. For k ≥ 1, define the k-fold compositions ϕ (k) V := ϕV ◦ · · · ◦ ϕV︸ ︷︷ ︸ k times : Pn(V0) −→ Pn−k(V0), ϕ (k) E (e) := { ϕ (k) V (v) | v ∈ e }. We prove by induction on k that H(n−k) := ( V (n−k), E(n−k), λ(k), µ(k) ) := ( ϕ (k) V [V (n)], {ϕ(k) E (e) | e ∈ E(n)}, λ(k), µ(k) ) is a property (n− k)-SuperHyperGraph, provided the label/property compatibilities hold at each intermediate stage. Base case k = 1. Set V ′ := ϕV [V (n)] ⊆ Pn−1(V0), E′ := { ϕE(e) | e ∈ E(n) } ⊆ P(V ′) \ {∅}. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 27 of 36 The inclusion V ′ ⊆ Pn−1(V0) is immediate from the codomain of ϕV . Nonemptiness of each ϕE(e) holds because e 6= ∅ and ϕV is total. By the assumed edge-label compatibility ϕE(e) = ϕE(e ′) =⇒ λ(e) = λ(e′), the map λ′ : E′ → Σ, λ′(ϕE(e)) := λ(e) is well-defined. By the assumed property compatibility (for both vertices and edges) ϕV (v) = ϕV (v ′) =⇒ µ(v, k) = µ(v′, k), ϕE(e) = ϕE(e ′) =⇒ µ(e, k) = µ(e′, k), the map µ′ : (V ′ ∪ E′)×K → S ∪ {⊥} defined by µ′(ϕV (v), k) := µ(v, k), µ′(ϕE(e), k) := µ(e, k) is well-defined. Consequently (V ′, E′, λ′, µ′) is a property (n− 1)-SuperHyperGraph. Induction step. Assume the claim holds for some k with 1 ≤ k < n: that is, H(n−k) = ( V (n−k), E(n−k), λ(k), µ(k) ) = ( ϕ (k) V [V (n)], {ϕ(k) E (e) | e ∈ E(n)}, λ(k), µ(k) ) is a property (n− k)-SuperHyperGraph. By the hypothesis “label/property compatibilities hold at each intermediate stage”, we have, for the (k+1)-st stage, the compatibilities on H(n−k): ϕV (x) = ϕV (y) =⇒ µ(k)(x, ·) = µ(k)(y, ·), ϕE(a) = ϕE(b) =⇒ λ(k)(a) = λ(k)(b), µ(k)(a, ·) = µ(k)(b, ·). Applying the base case to H(n−k) yields a property (n− k − 1)-SuperHyperGraph Ĥ = ( ϕV [V (n−k)], {ϕE(e ?) | e? ∈ E(n−k)}, λ̂, µ̂ ) . Using V (n−k) = ϕ (k) V [V (n)] and E(n−k) = {ϕ(k) E (e) | e ∈ E(n)}, we compute ϕV [V (n−k)] = ϕV [ ϕ (k) V [V (n)] ] = ϕ (k+1) V [V (n)], and for each e ∈ E(n), ϕE ( ϕ (k) E (e) ) = { ϕV (w) ∣∣ w ∈ {ϕ(k) V (v) | v ∈ e} } = { ϕ (k+1) V (v) ∣∣ v ∈ e } = ϕ (k+1) E (e). Hence Ĥ coincides with( ϕ (k+1) V [V (n)], {ϕ(k+1) E (e) | e ∈ E(n)}, λ(k+1), µ(k+1) ) , where λ(k+1) and µ(k+1) are the induced maps defined by λ(k+1) ( ϕ (k+1) E (e) ) := λ(e), T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 28 of 36 µ(k+1) ( ϕ (k+1) V (v), k0 ) := µ(v, k0), µ(k+1) ( ϕ (k+1) E (e), k0 ) := µ(e, k0), which are well-defined by the stated compatibilities at each stage. Thus the statement holds for k + 1. By induction on k, the structure H(n−k) = ( ϕ (k) V [V (n)], {ϕ(k) E (e) | e ∈ E(n)}, λ(k), µ(k) ) is a property (n− k)-SuperHyperGraph for all 1 ≤ k ≤ n. Example 15 (Iterated flattening: a concrete 3→1 instance with k = 2). Let the base set be V0 = {x, y}. Build a property 3-SuperHyperGraph H(3) = (V (3), E(3), λ, µ) as follows. Level–1 carriers are {x}, {y}, {x, y}. Define two level–2 carriers U1 = { {x}, {y} } , U2 = { {x, y} } , and two level–3 supervertices w1 = {U1}, w2 = {U2}. Set V (3) = {w1, w2} ⊆ P3(V0), E(3) = { e = {w1, w2} } . Take Σ = {σ} with λ(e) = σ, and let the property map be trivial: µ(·, ·) ≡ ⊥. (This choice satisfies the fibre–compatibility conditions automatically.) First flattening (k = 1). The map ϕV : P3(V0) → P2(V0) gives ϕV (w1) = U1 = { {x}, {y} } , ϕV (w2) = U2 = { {x, y} } . Thus V (2) (1) = ϕV [V (3)] = {U1, U2}, E (2) (1) = { ϕE(e) = {U1, U2} } . With λ(1) ( {U1, U2} ) = σ and µ(1) ≡ ⊥, Theorem 10 yields a property 2-SuperHyperGraph H (2) (1) . Second flattening (k = 2). Apply ϕV again, now ϕV : P2(V0) → P1(V0): ϕV (U1) = {x} ∪ {y} = {x, y}, ϕV (U2) = {x, y}. Hence both level–2 vertices collapse to the same level–1 carrier, and V (1) (2) = { {x, y} } , E (1) (2) = { ϕ (2) E (e) = {{x, y}} } . T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 29 of 36 Define λ(2) ( {{x, y}} ) = σ and keep µ(2) ≡ ⊥. The fibre–compatibility assumptions are satisfied (both labels constant and properties trivial), so by Theorem 11 with k = 2 we obtain a property 1-SuperHyperGraph H(1) = ( V (1) (2) , E (1) (2) , λ (2), µ(2) ) = ( {{x, y}}, { {{x, y}} } , σ, ⊥ ) . Theorem 12 (Uniformity). Let H(n) = (V (n), E(n), λ, µ) be a property n-SuperHyperGraph. Assume there exists an integer k ≥ 1 such that |e| = k for every e ∈ E(n). Then H(n) is k-uniform, i.e., every level-n superedge links exactly k n-supervertices. Moreover, any subfamily E′(n) ⊆ E(n) inherits k-uniformity. Proof. By hypothesis, for each e ∈ E(n) we have e ⊆ V (n) and |e| = k. By the usual definition, this is precisely the statement that H(n) is k-uniform. Now let E′(n) ⊆ E(n). For any e ∈ E′(n) we still have e ⊆ V (n) and (because e ∈ E(n)) |e| = k. Hence the restriction (V (n), E′(n), λ|E′(n) , µ|(V (n)∪E′(n))×K) is again k-uniform. No further conditions are required. Theorem 13 (Disjoint union). Let H(n) i = (V (n) i , E (n) i , λi, µi) be property n-SuperHyperGraphs over pairwise disjoint base sets V (i) 0 for i = 1, 2 (so V (1) 0 ∩ V (2) 0 = ∅). To avoid accidental identifications (e.g. of ∅) across levels, form tagged copies ι (r) i : Pr(V (i) 0 ) −→ Pr ( V (1) 0 ∪ V (2) 0 ) × {i}, x 7→ (x, i) (0 ≤ r ≤ n), and define V (n) = ι (n) 1 [ V (n) 1 ] ∪ ι (n) 2 [ V (n) 2 ] , E(n) = { ι (n) 1 [e] | e ∈ E (n) 1 } ∪ { ι (n) 2 [e] | e ∈ E (n) 2 } . Set the label and property maps by λ ( ι (n) i [e] ) := λi(e), µ ( (ι (r) i (x), i), k ) := µi(x, k), for e ∈ E (n) i , x ∈ Pr(V (i) 0 ), 0 ≤ r ≤ n. Then ( V (n), E(n), λ, µ ) is a property n- SuperHyperGraph over the disjoint base V (1) 0 ∪ V (2) 0 (identifying (v, i) with v since the bases are disjoint). Proof. We verify the clauses of the definition. (Vertices) Since V (n) i ⊆ Pn(V (i) 0 ) for each i, we have ι (n) i [V (n) i ] ⊆ Pn(V (1) 0 ∪V (2) 0 )×{i}. Hence V (n) ⊆ Pn(V (1) 0 ∪ V (2) 0 ) × {1, 2}, which (via the obvious identification) sits inside Pn(V (1) 0 ∪ V (2) 0 ). (Edges) For each e ∈ E (n) i we know e ⊆ V (n) i and e 6= ∅. Therefore ι(n)i [e] ⊆ ι (n) i [V (n) i ] ⊆ V (n), and ι (n) i [e] 6= ∅. Thus every element of E(n) is a nonempty subset of V (n). (Labels) If ι (n) i [e] = ι (n) j [e′], then necessarily i = j and e = e′ (by tagging), so the piecewise definition λ(ι (n) i [e]) = λi(e) is well-defined and maps into Σ. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 30 of 36 (Properties) For any x ∈ ⋃n r=0 Pr(V (i) 0 ) or x ∈ E (n) i , the tag ensures that (ι (r) i (x), i) determines the source structure uniquely, so µ ( (ι (r) i (x), i), k ) := µi(x, k) is well-defined with codomain S ∪ {⊥}. All axioms are therefore satisfied, so (V (n), E(n), λ, µ) is a property n-SuperHyperGraph. If one prefers literal unions without tags, the same construction applies because the base sets are disjoint; tagging is used only to preclude pathological coincidences such as ∅. Example 16 (Disjoint union: a concrete n = 1 instance). Fix common alphabets Σ = {α, β}, K = {info}, S = Strings, ⊥ /∈ S. First component H (1) 1 . Base V (1) 0 = {x1, x2} (disjoint from the other base). Take V (1) 1 = { A = {x1}, B = {x1, x2} } ⊆ P1 ( V (1) 0 ) , E (1) 1 = {e1}, e1 = {A,B}, λ1(e1) = α, µ1(A, info) = “single”, µ1(B, info) = “pair”, µ1(e1, info) = “includes”. Second component H (1) 2 . Base V (2) 0 = {y1, y2, y3} with V (1) 0 ∩ V (2) 0 = ∅. Let V (1) 2 = { C = {y1, y2}, D = {y2, y3} } ⊆ P1 ( V (2) 0 ) , E (1) 2 = {e2}, e2 = {C,D}, λ2(e2) = β, µ2(C, info) = “left”, µ2(D, info) = “right”, µ2(e2, info) = “adjacent”. Tagged disjoint union. Using the injections ι (1) i : P1(V (i) 0 ) → P1(V (1) 0 ∪ V (2) 0 ) × {i}, form V (1) = ι (1) 1 [V (1) 1 ] ∪ ι (1) 2 [V (1) 2 ], E(1) = {ι(1)1 [e1], ι (1) 2 [e2]}. Define λ ( ι (1) 1 [e1] ) = α, λ ( ι (1) 2 [e2] ) = β, and for x ∈ V (1) j ∪ E (1) j , µ ( (ι (1) j (x), j), info ) := µj(x, info) (j = 1, 2). Then H(1) = ( V (1), E(1), λ, µ ) is a property 1-SuperHyperGraph (Property HyperGraph) over the disjoint base V (1) 0 ∪V (2) 0 by Theorem 13. If one suppresses tags, disjointness of the bases still prevents accidental identifications. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 31 of 36 Theorem 14 (Line graph as a Property Graph). Let H(n) = (V (n), E(n), λ, µ) be a property n-SuperHyperGraph. Define its line graph G by VG := E(n), EG := { (e1, e2) ∈ VG × VG ∣∣ e1 6= e2, e1 ∩ e2 6= ∅ } . Let s(e1, e2) = e1 and t(e1, e2) = e2, set λG(e1, e2) := λ(e1) ∈ Σ, and define a property map µG(x, k) := { µ(x, k), x ∈ VG(= E(n)), ⊥, x ∈ EG. Then G = (VG, EG, s, t, λG, µG,⊥) is a Property Graph. Proof. We check each item of the Property Graph definition. (1) Vertices and edges. By construction VG = E(n). Every element of EG is an ordered pair of distinct superedges that intersect, so EG ⊆ VG × VG. (2) Source/target maps. The maps s, t : EG → VG defined by s(e1, e2) = e1 and t(e1, e2) = e2 are well-defined by the previous item. (3) Edge labels. For (e1, e2) ∈ EG we set λG(e1, e2) = λ(e1) ∈ Σ. This is a bona fide edge-labelling (variants such as (λ(e1), λ(e2)) ∈ Σ× Σ are also admissible). (4) Properties. On vertices VG = E(n) we reuse µ; on edges we assign the sentinel ⊥. Thus µG : (VG∪EG)×K → S∪{⊥} has the correct codomain, and ⊥ /∈ S by assumption. All requirements are met; hence G is a Property Graph. 5. Conclusion and Outlook This study has demonstrated how the expressive power of Property Graphs can be lifted to the richer settings of HyperGraphs and SuperHyperGraphs. We provided rigor- ous, uniform definitions for these generalizations and presented initial results that under- score their modeling potential. For clarity and ease of reference, Table 2 summarizes the concepts introduced and defined in this paper. Future work may explore further extensions based on advanced uncertainty formalisms—including Fuzzy Sets[37], Vague Sets[38], Intuitionistic Fuzzy Sets[39], Para- consistent Set[40], Soft Sets[41, 42], Picture Fuzzy Set[43, 44], Rough Sets[45, 46], Neu- trosophic Sets (and their quadri-partitioned variants) [47, 48], Hesitant Fuzzy Sets [49], and Plithogenic Sets [50, 51]. Each of these set-theoretic paradigms already possesses a graph-theoretic interpretation, and incorporating them into the Property HyperGraph and Property SuperHyperGraph frameworks promises a fertile direction for research. Also an exciting future direction would be to extend Property SuperHyperGraphs over complex algebraic structures such as Gaussian integers, Eisenstein integers, Quaternion integers, and even Octonion algebras (cf.[52, 53]). These extensions could yield novel frameworks for secure data modeling, multi-dimensional coding theory, and advanced cryptographic schemes, exploiting the non-commutative, associative, or non-associative properties of such algebras. T. Fujita, F. Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6729 32 of 36 Table 2: Concise overview of Property Graph, Property Hypergraph, and Property n-SuperHyperGraph. Model Labels Properties Notes Property Graph λ : E → Σ µ : (V ∪ E) × K → S ∪ {⊥} Directed multigraph G = (V,E); ver- tices/edges carry at- tributes; edges are labelled. Property Hyper- graph λ : E → Σ µ : (V ∪ E) × K → S ∪ {⊥} Hypergraph H = (V,E); hyperedges connect arbi- trary vertex subsets; at- tributes and labels al- lowed. Property n- SuperHyperGraph λ : E(n) → Σ µ : (D(n)×K) → S∪{⊥} H(n) = (V (n), E(n), λ, µ); supervertices at level n, edges between them; supports hierarchical, attributed structures. Funding This work was carried out without any external financial support. Acknowledgements We are grateful to all colleagues and reviewers whose insights and encouragement have strengthened this work. We also thank the authors of the cited literature for laying the groundwork that enabled our study, and acknowledge the institutions that provided the resources and infrastructure essential for its completion. Author’s Contributions Conceptualization, Takaaki Fujita; Investigation, Takaaki Fujita; Methodology, Takaaki Fujita; Writing – original draft, Takaaki Fujita; Writing – review & editing, All authors. Data Availability This is a theoretical study and did not produce any data. We invite future researchers to perform empirical validations of the concepts presented herein. Ethical Approval No human participants or animal subjects were involved in this research; ethical ap- proval was therefore not required. T. Fujita, F. Smarandache / Eur. J. Pure Appl. 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