For a given set S, its power set is the family of all subsets of S, including both the empty set and S itself. The n-th iterated power set of S is obtained by repeatedly applying the power set operation n times, beginning with S [1, 2].

Definition 1.1 (Universal Set). Let U denote a set that contains every element under discussion. Throughout, any set S is assumed to satisfy S ⊆ U.

Definition 1.2 (Base Set). A base set S is defined as a subset of U that serves as the starting point for further constructions, such as power sets and hyperstructures.

Definition 1.3 (Power Set). [3] The power set of a set S, written P(S), is given by P(S) = { X | X ⊆ S }.

Definition 1.4 (Iterated Power Set). [4, 5, 6, 7] For each integer n ≥ 1, the n-fold iterated power set of S is defined recursively as P (S) = P(S),

k+1 k

P (S) = P P (S) , k ≥ 1.

Equivalently, one may denote Pn(S) = P (S).

Definition 1.5 (Nonempty Iterated Power Set). [4, 8] The nonempty iterated power set is given by P1 (S) = P(S) \ {∅},

∗ ∗ ∗

Pk+1(S) = P Pk (S) , k ≥ 1, where, for any set X, one writes P (X) = P(X) \ {∅}.

Example 1.6 (Nonempty Iterated Power Set in Beverage Selection). Let S = {Tea, Coffee} represent two drink options. Then:

P1 (S) = P(S) \ {∅} = {Tea}, {Coffee}, {Tea, Coffee} .

The second-level nonempty iterated power set is

∗ ∗ ∗

P2 (S) = P P1 (S) = {{Tea}}, {{Coffee}}, {{Tea, Coffee}}, {{Tea}, {Coffee}}, {{Tea}, {Tea, Coffee}}, {{Coffee}, {Tea, Coffee}}, {{Tea}, {Coffee}, {Tea, Coffee}} .

For instance, {{Tea}, {Coffee}} ∈ P2 (S) can model offering both a “tea-only” service and a “coffee-only” service

∗ ∗ ∗

as distinct package options. One may continue to P3 (S) = P (P2 (S)), whose elements are nonempty collections of these service-packages.

Example 1.7 (Crop rotations; a concrete nonempty iterated power set for |S| = 3). Let the set of candidate crops be S = {Wheat, Corn, Soybean}.

Then P1 (S) = {Wheat}, {Corn}, {Soybean}, {Wheat, Corn}, {Wheat, Soybean}, {Corn, Soybean}, {Wheat, Corn, Soybean} , so |P1 (S)| = 2 − 1 = 7. The next level is P2 (S) = P P1 (S) \ {∅}, |P2 (S)| = 2 − 1 = 127.

Two explicit elements of P2 (S) are A1 = {Wheat}, {Corn, Soybean} , A2 = {Wheat, Corn}, {Wheat, Soybean}, {Wheat, Corn, Soybean} . Agricultural interpretation: elements of P1 (S) are concrete rotation blocks (e.g. {Corn, Soybean} for a two-year rotation); elements of P2 (S) are sets of such blocks (e.g. portfolios of rotations assigned to different fields or planning scenarios compared at the farm scale).

Example 1.8 (Crop rotations; a concrete nonempty iterated power set for |S| = 3). Let the set of candidate crops be S = {Wheat, Corn, Soybean}.

Then P1 (S) = {Wheat}, {Corn}, {Soybean}, {Wheat, Corn}, {Wheat, Soybean}, {Corn, Soybean}, {Wheat, Corn, Soybean} , so |P1 (S)| = 2 − 1 = 7. The next level is

∗ ∗ ∗ 7

P2 (S) = P P1 (S) \ {∅}, |P2 (S)| = 2 − 1 = 127.

Two explicit elements of P2 (S) are A1 = {Wheat}, {Corn, Soybean} , A2 = {Wheat, Corn}, {Wheat, Soybean}, {Wheat, Corn, Soybean} . Agricultural interpretation: elements of P1 (S) are concrete rotation blocks (e.g. {Corn, Soybean} for a two-year rotation); elements of P2 (S) are sets of such blocks (e.g. portfolios of rotations assigned to different fields or planning scenarios compared at the farm scale).

Example 1.9 (Field operations (management bundles); a concrete nonempty iterated power set for |S| = 4). Let S = {Tillage, Sowing, Irrigation, Fertilization}.

Then P1 (S) = {{Tillage}, {Sowing}, {Irrigation}, {Fertilization}, {Tillage, Sowing}, {Tillage, Irrigation}, {Tillage, Fertilization}, {Sowing, Irrigation}, {Sowing, Fertilization}, {Irrigation, Fertilization}, {Tillage, Sowing, Irrigation}, {Tillage, Sowing, Fertilization}, {Tillage, Irrigation, Fertilization}, {Sowing, Irrigation, Fertilization}, {Tillage, Sowing, Irrigation, Fertilization}}.

so |P1 (S)| = 2 − 1 = 15 and

∗ 15

|P2 (S)| = 2 − 1 = 32767.

Two explicit elements of P2 (S) are B1 = {Tillage, Sowing}, {Irrigation} , B2 = {Sowing, Fertilization}, {Irrigation, Fertilization}, {Tillage, Sowing, Irrigation, Fertilization} . Agricultural interpretation: members of P1 (S) are operation bundles applied together (e.g. {Tillage, Sowing} at establishment); members of P2 (S) are sets of bundles (e.g. alternative management templates to be compared across fields, seasons, or irrigation availability scenarios).

Hypergraphs generalize ordinary graphs by allowing each hyperedge to join an arbitrary nonempty subset of vertices, thereby modeling higher-order relations among elements [9, 10, 11]. A SuperHyperGraph further extends this idea by incorporating iterated powerset structures, enabling multi-layered, self-referential connections among hyperedges [12, 13, 14, 15].

Definition 1.10 (Hypergraph). [9, 16] Let V be a finite set of vertices. A hypergraph is a pair H = V, E , E ⊆ P(V ) \ {∅}, where each element of E is called a hyperedge. No restriction is imposed on the size of a hyperedge.

Example 1.11 (Agricultural Hypergraph: crop rotations across fields). Let the vertex set be candidate crops V = {Wheat, Corn, Soybean, Canola}.

Each field’s multi-year rotation induces a hyperedge equal to the set of crops appearing in that field’s plan: eField 1 = {Wheat, Soybean}, eField 2 = {Corn, Soybean, Wheat}, eField 3 = {Canola}.

Then H = (V, E) with E = {eField 1, eField 2, eField 3} ⊆ P(V ) \ {∅} is a hypergraph per Definition 1.10. Each hyperedge captures a (possibly multi-year) rotation used on a specific field.

Definition 1.12 (n-SuperHyperGraph). [12] Let V0 be a finite base set. Define the iterated powersets by P (V0) = V0,

k+1 k

P (V0) = P P (V0) , k ≥ 0.

For a fixed n ≥ 1, an n-SuperHyperGraph is a pair SHG = V, E ,

V, E ⊆ P (V0) and V ̸= ∅, E = ̸ ∅.

Elements of V are called n-supervertices and elements of E are called n-superedges.

Example 1.13 (Corporate Hierarchy as a 2-SuperHyperGraph). Let the set of employees be V0 = {A, B, C, D}, with A = Alice, B = Bob, C = Carol, and D = Dave. First-level subsets (teams) are chosen as T1 = {A, B}, T2 = {C, D}, so T1, T2 ∈ P(V0). Next, define two departments as elements of the second-level iterated power set:

D1 = { T1}, D2 = { T2}, so {D1, D2} ⊆ P (V0).

Thus we set V2 = { D1, D2} ⊆ P (V0), E2 = { D1, D2} ⊆ P (V0).

The pair F = V2, E2 is a 2-SuperHyperGraph encoding the hierarchy: employees → teams → departments → the company division. Example 1.14 (Agricultural n=1 SuperHyperGraph: management bundles and protocols). Let the base set collect crops, inputs, and practices V0 = {Wheat, Corn, Urea (N), DAP (P), Drip, Sprinkler, HerbicideA}.

Fix n = 1. Choose 1-supervertices (nonempty subsets of V0) that represent reusable management bundles:

v1 = {Wheat, Urea (N)}, v2 = {Corn, DAP (P)}, v3 = {Drip}, v4 = {HerbicideA, Sprinkler}. Set V = {v1, v2, v3, v4} ⊆ P (V0). Define superedges (each a nonempty subset of V ) to encode whole-field protocols:

e1 = {v1, v3} (Wheat + N with drip irrigation), e2 = {v2, v4} (Corn + P with herbicide under sprinkler). Then SHG = (V, E) with E = {e1, e2} ⊆ P(V ) \ {∅} is an n=1 SuperHyperGraph per Definition 1.12. Here, supervertices are management bundles (collections of base actions/resources), and superedges bundle those bundles into implementable protocols.

A food web is a directed graph representing species as nodes and predator–prey interactions as directed edges in an ecosystem[17, 18, 19, 20, 21].

Definition 1.15 (Food Web). [17, 18] Let V be a finite set whose elements represent biological species (or trophic groups) in an ecosystem. A food web is the directed graph G = (V, E), where the edge set E ⊆ V × V satisfies (u, v) ∈ E ⇐⇒ species u preys upon species v.

We require additionally that • There are no self-loops: (v, v) ̸∈ E for all v ∈ V .

• There are no parallel edges: E is a set (not a multiset).

Thus G encodes all direct predator–prey relationships among the species in the ecosystem.

Example 1.16 (Simple Pond Food Web). Consider a small pond ecosystem with five trophic levels:

V = { A, Z, S, B, D},

• A = Algae • Z = Zooplankton • S = Small Fish • B = Big Fish • D = Duck The predator–prey relationships are captured by E = { (Z, A), (S, Z), (B, S), (D, B), (D, S) }, meaning:

Z → A (zooplankton eats algae), S → Z (small fish eat zooplankton), B → S (big fish eat small fish), D → B (duck eats big fish), D → S (duck also eats small fish).

Thus the directed graph G = (V, E) illustrates a simple pond food web.

Food HyperWeb is a hypergraph whose vertices represent species and whose hyperedges correspond to each predator’s unique complete prey set.

Definition 2.1 (Base Food Web). Let V0 be a finite set of species and let G = (V0, E0) be the directed graph (the Food Web) where (u, v) ∈ E0 if and only if species u preys upon species v.

Definition 2.2 (Food HyperWeb). Let G = (V0, E0) be a Food Web. Define the Food HyperWeb H = (V1, E1) by V1 = V0, E1 = eu ⊆ V0 : eu = { v ∈ V0 : (u, v) ∈ E0}, eu ̸= ∅ .

Each hyperedge eu collects all prey of predator u.

Example 2.3 (Simple Pond Food HyperWeb). Consider the same small pond ecosystem as in the Food Web example:

V0 = { A, Z, S, B, D},

E0 = {(Z, A), (S, Z), (B, S), (D, B), (D, S)}, with species labels:

• A: Algae • Z: Zooplankton • S: Small Fish • B: Big Fish • D: Duck The Food HyperWeb H = (V1, E1) is obtained by grouping each predator’s prey into a hyperedge:

V1 = V0, E1 = { eZ, eS, eB, eD},

eZ = { A}, (zooplankton eats algae); eS = { Z}, (small fish eat zooplankton); eB = { S}, (big fish eat small fish); eD = { B, S}, (duck eats big fish and small fish).

Thus H is the hypergraph whose vertices are the five species and whose hyperedges capture each predator’s full prey set.

Example 2.4 (Marine Food HyperWeb). Consider a simple marine ecosystem with six species:

V0 = { P, Z, K, F, S, H}, • P: Phytoplankton • Z: Zooplankton • K: Krill • F: Small Fish • S: Seal • H: Shark The predator–prey relations (the Food Web G = (V0, E0)) are E0 = { (Z, P), (K, Z), (F, K), (F, Z), (S, F), (S, K), (H, S), (H, F) }.

From this we form the Food HyperWeb H = (V1, E1) with V1 = V0 and E1 = { eZ, eK, eF , eS, eH}, where each hyperedge collects all prey of a given predator:

eZ = { P}, (zooplankton eats phytoplankton); eK = { Z}, (krill eats zooplankton); eF = { K, Z}, (small fish eat krill and zooplankton); eS = { F, K}, (seal eats small fish and krill); eH = { S, F}, (shark eats seal and small fish).

Thus H is the hypergraph whose vertices are the six marine species and whose hyperedges exactly describe each predator’s full prey set.

Theorem 2.5 (Food HyperWeb is a Hypergraph). Let H = (V1, E1) be the Food HyperWeb constructed from a Food Web G = (V0, E0) as in Definition 2.1. Then E1 ⊆ P(V1) \ {∅}, V1 is finite, so H satisfies the axioms of a finite hypergraph.

Proof : By definition V1 = V0 is finite. Each hyperedge eu = { v ∈ V1 : (u, v) ∈ E0} is nonempty exactly when u preys on at least one species, hence eu ∈ P(V1) \ {∅}. Moreover, the assignment u 7→ eu is injective, so there are no duplicate hyperedges. Thus H meets the standard definition of a finite hypergraph [10]. □

Theorem 2.6 (Reconstruction of Food Web by Flattening). The original Food Web G = (V0, E0) is recovered from the Food HyperWeb H = (V1, E1) via E0 = (u, v) ∈ V1 × V1 : v ∈ eu, eu ∈ E1 .

Proof : By construction eu = {v : (u, v) ∈ E0}. Hence (u, v) ∈ E0 ⇐⇒ v ∈ eu, so the directed edges of G coincide exactly with the incidences of vertices in hyperedges of H. Therefore flattening H recovers E0. □

Food SuperHyperWeb is an n-superhypergraph whose vertices are iterated species subsets and whose hyperedges encode complete multilevel hierarchical predator–prey relationships.

Definition 2.7 (Food n-SuperHyperWeb). Let G = (V0, E0) be a finite directed graph (the Food Web), and for each u ∈ V0 write P(u) = { v ∈ V0 : (u, v) ∈ E0} for its prey set. Fix an integer n ≥ 1, and let P (V0) denote the n-fold iterated powerset of V0. Define

n (n) (n) n

Vn = P (V0), En = eu : u ∈ V0, P(u) = ̸ ∅, eu = P P(u) .

Then the pair F = Vn, En is called the Food n-SuperHyperWeb.

Example 2.8 (Forest Ecosystem as a Food 2-SuperHyperWeb). Let the species set be V0 = {G, M, O, F},

G = Grass, M = Mouse, O = Owl, F = Fox.

The Food Web G = (V0, E0) has predator–prey arcs E0 = {(M, G), (O, M), (F, O), (F, M)}.

For each u ∈ V0, its prey set is P(M) = {G}, P(O) = {M}, P(F) = {O, M}, P(G) = ∅.

Fix n = 2. Then

2 (2)

V2 = P (V0), E2 = { eu : u ∈ {M, O, F}}, with

(2) 2 2

e = P P(M) = P ({G}) = ∅, {∅}, {{G}}, {∅, {G}} ,

(2) 2

e = P ({M}) = ∅, {∅}, {{M}}, {∅, {M}} ,

(2) 2

e = P {O, M} , which has 2 = 16 elements (all subsets of P({O, M})). Thus F = V2, E2 is the Food 2-SuperHyperWeb encoding the multi-level trophic structure:

species → prey sets −→ collections of prey-sets.

Example 2.9 (Agricultural Food 2-SuperHyperWeb). Let the set of trophic species be V0 = {G, I, C, Co, H},

G = Grass, I = Insect, C = Chicken, Co = Cattle, H = Human.

The Food Web G = (V0, E0) has predator–prey arcs E0 = {(I, G), (C, I), (Co, G), (H, C), (H, Co)}.

Hence each prey set is P(I) = {G}, P(C) = {I}, P(Co) = {G}, P(H) = {C, Co}.

Fix n = 2. Then

2 (2)

V2 = P (V0), E2 = { eu : u ∈ {I, C, Co, H}}, with

(2) 2

e = P {G} = ∅, {∅}, {{G}}, {∅, {G}} , (2) 2

e = P {I} = ∅, {∅}, {{I}}, {∅, {I}} ,

(2) 2 (2)

e = P {G} (same as e ),

Co I (2) 2

e = P {C, Co} , which has 2 = 16 elements, each representing a possible “meal bundle” drawn from the basic prey sets {C, Co}. Therefore F = V2, E2 is the Food 2-SuperHyperWeb encoding both primary producer consumption and human dietary options in a two-layer superhypergraph structure.

Theorem 2.10. F = (Vn, En) is an n-SuperHyperGraph.

Proof : By construction Vn is a finite set and each hyperedge

(n) n

eu = P P(u) is a nonempty element of P (V0). Distinct predators u yield distinct hyperedges, so there are no duplicates. Hence F satisfies the definition of an n-SuperHyperGraph. □

Theorem 2.11 (Generalization of Food HyperWeb and Food Web). Define the “full flattening” of a nested set by taking the union of all its elements at every level. Since

(n) n

eu = P P(u) , flattening eu recovers exactly the prey set P(u). Therefore P(u) : u ∈ V0, P(u) = ̸ ∅ = E1 and (u, v) : v ∈ P(u) = E0, showing that F simultaneously generalizes the Food HyperWeb H = (V0, E1) and the original Food Web G = (V0, E0).

S S S Proof : Immediate from the identity (n) · · · X = P(u) and the definitions of E1 and E0. □

X∈eu