Hyperstructures and superhyperstructures are obtained by allowing algebraic operations to return sets, and then iterating this idea via iterated powersets [3]. We first fix notation for powersets and their iterations.

Definition 1 (Base set). A base set is a nonempty finite set S. All higher-level objects in this paper (subsets, iterated subsets, hyperoutputs, and superhyperoutputs) are ultimately built from elements of S.

Definition 2 (Powerset and nonempty powerset) ([4]). Let S be a set. The powerset of S is

P(S) := {A | A ⊆ S}.

The nonempty powerset of S is P (S) := P(S) \ {∅}.

Definition 3 (Iterated powerset and iterated nonempty powerset) ([3]). Let S be a set and let n ∈ N0. Define recursively

0 n+1 n

P (S) := S, P (S) := P P (S) .

Similarly, define the iterated nonempty powerset by

∗ 0 ∗ n+1 ∗ ∗ n

(P ) (S) := S, (P ) (S) := P (P ) (S) .

Definition 4 (Classical operation and classical structure [3]) . Let H be a nonempty set and let m ≥1. A classical m-ary operation on H is a function # : H → H.

A classical structure is a tuple A = (H,#1, . . . , #t) equipped with specified axioms relating the operations (e.g., associativity, distributivity, identities), depending on the intended algebraic type.

Definition 5 (Hyperoperation) ([5]). Let H be a nonempty set and let m ≥ 1. A hyperoperation (of arity m) on H is a map ◦ : H −→ P (H), so the value ◦(x1, . . . , xm) is a nonempty subset of H rather than a single element.

Definition 6 (Hyperstructure) [3]). A hyperstructure is a tuple H = (H,◦1, . . . , ◦t) where H is a nonempty set and each ◦j is a hyperoperation on H (possibly of its own arity), together with axioms appropriate to the model under consideration.

Definition 7 (Superhyperoperation of order (m, n) ([3]). Let H be a nonempty set, and fix integers m ≥ 1 and n ≥ 1. A (classical-type) (m, n)-superhyperoperation on H is a map

(m,n) m ∗ n

⊙ : H −→ (P ) (H), so each output is a nonempty element of the n-fold iterated powerset. If one allows the empty set at some level, one may instead take codomain P (H); we refer to this as the Neutrosophic-type variant.

Definition 8 (n-superhyperstructure) ([3]). Fix n ≥ 1. An n-superhyperstructure is a tuple SH = (H,⊙1, . . . , ⊙t) where H is a nonempty set and each ⊙j is a superhyperoperation

mj ∗ n

⊙j : H → (P ) (H) for some arity mj ≥ 1, together with axioms suited to the application.

Example 1 (Group-based 2-superhyperstructure). Let G = Z3 = {0, 1, 2} with addition modulo 3. Then P (G) = P(G), P (G) = P P(G) .

Define a binary operation

2 2 2

⋆ : P (G) × P (G) −→ P (G) by A ⋆ B := X ⊕ Y X ∈ A, Y ∈ B , X ⊕ Y := { x + y mod 3 | x ∈ X, y ∈ Y }.

For instance, if A = {0, 1}, {2} , B = {1} , then {0, 1} ⊕ {1} = {1, 2}, {2} ⊕ {1} = {0}, so A ⋆ B = {1, 2}, {0} ∈ P (G).

Thus (P (G), ⋆) is a concrete example of a 2-level superhyperstructure built from the group (G, +).

Remark 1. The same lifting principle extends to any level n ≥ 2: one replaces P (H) by P (H) (or (P ) (H)) and defines the lifted operation by taking pointwise images of lower-level combinations.

For reference, a comparison of Classical Structures, HyperStructures, and n-SuperHyperStructures is presented in Table 1. It is expected that n-SuperHyperStructures offer advantages such as clearly modeling hierarchical and multi-level real-world concepts.

Notion Carrier (Universe) Operation Type Codomain of an m-ary Operation Classical Structure H classical operation # : H → H HyperStructure H hyperoperation ◦ : H → P (H)

(m,n) m ∗ n

SuperHyperStructure (level n) H (m, n)-superhyperoperation ⊙ : H → (P ) (H)

SuperHyperStructures. Here P (H) = P(H)\{∅} and (P ) (H) is the n-fold iterated nonempty powerset.

A queue is a linear data structure in which elements are inserted at the rear and removed from the front, following the FIFO principle [6, 7, 8]. In this subsection, we extend this concept to hyperqueues and superhyperqueues.

Definition 9 (Queue). Let E be a nonempty set of elements. A queue over E is an abstract data type defined by:

Q = E the set of all finite sequences (lists) of elements of E. The following operations are defined on Q:

• empty : Q, the empty queue:

empty = ⟨ ⟩.

• enqueue : Q × E → Q, which adds an element at the tail:

enqueue(q,e) = q ∥ ⟨e⟩, where ∥ denotes sequence concatenation.

• dequeue : Q → Q × E, which removes and returns the head element:

(q , e), if q = ⟨e⟩ ∥ q , dequeue(q) = undefined, if q = ⟨ ⟩.

• isEmpty : Q → {true, false}, true, q = ⟨ ⟩, isEmpty(q) = false, q ̸= ⟨ ⟩.

These operations satisfy the following axioms for all q ∈ Q and e ∈ E:

(1) isEmpty(empty) = true.

(2) isEmpty(enqueue(q, e)) = false.

(3) If dequeue(q) = (q , e), then

′ ′ ′

q = ⟨e⟩ ∥ q , enqueue(q , e) ̸= q ∥ e and repeated dequeue operations remove elements in the same order they were enqueued (FIFO property).

Example 2 (Customer support ticket queue). A Customer Support Ticket Queue is a system that manages and orders customer service requests for processing, typically in a FIFO manner (cf.[9, 10, 11]). Let the set of customer support tickets be E = {T1, T2, T3}.

We illustrate a sequence of queue operations on Q = E .

Initial state:

Q0 = empty = ⟨ ⟩, isEmpty(Q0) = true.

Enqueue T1:

Q1 = enqueue(Q0, T1) = ⟨T1⟩, isEmpty(Q1) = false.

Enqueue T2:

Q2 = enqueue(Q1, T2) = ⟨T1, T2⟩.

Enqueue T3:

Q3 = enqueue(Q2, T3) = ⟨T1, T2, T3⟩.

Dequeue twice:

dequeue(Q3) = (Q2, T1), Q2 = ⟨T2, T3⟩,

′ ′ ′

dequeue(Q2) = (Q1, T2), Q1 = ⟨T3⟩.

Check remaining and empty:

isEmpty(Q1) = false, dequeue(Q1) = (Q0, T3), isEmpty(Q0) = true.

Summary of FIFO behavior:

• Tickets T1, T2, T3 were enqueued in that order.

• They were dequeued in the same order T1 → T2 → T3.

• At each step, isEmpty correctly reports whether any tickets remain.

This concrete example demonstrates the FIFO discipline and all queue operations in detail.

The definition of a hyperqueue is given below.

Definition 10 (Hyperqueue). Let E be a nonempty set and let Q = E be the classical queue over E. Define the hyperqueue on E as the hyperstructure HQ = P(Q), ⊙enq, ⊙deq , where P(Q) is the powerset of Q, and the operations ⊙enq : P(Q) × E −→ P(Q), ⊙deq : P(Q) −→ P(Q × E) are given by ⊙enq e = enqueue(q,e) | q ∈ A , A ⊙deq = (q , e) | q ∈ A, (q , e) = dequeue(q) .

Remark 2. By construction, ⊙enq and ⊙deq are hyperoperations in the sense that their codomains are powersets of the classical domains. Hence HQ is a hyperstructure on the base set Q.

Theorem 1. The classical queue (Q, enqueue, dequeue) is recovered from HQ by restricting to singleton subsets. Concretely, for every q ∈ Q and e ∈ E, {q} ⊙enq e = enqueue(q, e) , {q}⊙deq = dequeue(q) .

Proof : Recall that the hyperqueue HQ = (P(Q), ⊙enq, ⊙deq) is defined by A ⊙enq e = enqueue(q, e) | q ∈ A , A ⊙deq = (q , e) | q ∈ A, (q , e) = dequeue(q) .

We must show that when A is the singleton {q}, these hyperoperations collapse exactly to the classical operations on q.

(i) Enqueue. Let q ∈ Q and e ∈ E. Then by definition {q} ⊙enq e = enqueue(x, e) | x ∈ {q} .

Since the only element of {q} is q itself, the comprehension reduces to {q} ⊙enq e = {enqueue(q, e)}.

Hence every element of the left-hand side is of the form enqueue(q, e), and clearly enqueue(q, e) lies in the set on the right. This establishes the equality {q} ⊙enq e = {enqueue(q, e)}.

(ii) Dequeue. Similarly, by definition, {q} ⊙deq = (x , f) | x ∈ {q}, (x , f) = dequeue(x) .

Again the only choice for x is q. Therefore {q} ⊙deq = { dequeue(q) }, where dequeue(q) denotes the unique pair (q , f) returned by the classical dequeue on q. Thus {q}⊙deq = {dequeue(q)}.

Since both hyperoperations agree on singletons with the classical enqueue and dequeue, we conclude that the entire classical queue (Q, enqueue, dequeue) is recovered by restricting HQ to singleton subsets.

Example 3 (Customer Ticket Hyperqueue). Let the set of ticket identifiers be E = {T1, T2}, Q = E = {⟨ ⟩, ⟨T1⟩, ⟨T2⟩, ⟨T1, T2⟩, . . . }.

Consider the subset of queues A = ⟨T1⟩, ⟨T2, T1⟩ ∈ P(Q).

Hyperenqueue Enqueueing ticket T2 on every queue in A gives A ⊙enq T2 = enqueue(⟨T1⟩, T2), enqueue(⟨T2, T1⟩, T2) = ⟨T1, T2⟩, ⟨T2, T1, T2⟩ .

Hyperdequeue Removing the head ticket from each queue in A yields pairs of (remaining-queue, dequeued-ticket):

A ⊙deq = (q , e) q ∈ A, (q , e) = dequeue(q) = (⟨ ⟩, T1), (⟨T1⟩, T2) .

Iterated Hyperoperations One can combine hyperenqueue and hyperdequeue on sets of queues. For example, first enqueue T2 then dequeue:

A ⊙enq T2 ⊙deq = (⟨T2⟩, T1), (⟨T1, T2⟩, T2) .

This demonstrates how HQ = (P(Q), ⊙enq, ⊙deq) systematically generalizes classical queue operations to sets of queues.

The definition of a n-SuperHyperqueue is given below.

Definition 11 (n-SuperHyperqueue). Let E be a nonempty set, and let Q = E be the classical queue over E. For each integer n ≥ 0, define the n-SuperHyperqueue

(n) (n)

SNHQn = Pn(Q), ⊙enq, ⊙ ,

deq

where the two superhyperoperations are given by:

⊙enq : Pn(Q) × E −→ Pn(Q), ⊙ : Pn(Q) −→ Pn Q × E ,

deq (n) (n−1)

A ⊙enq e = X ⊙enq e X ∈ A ,

(n) (n−1)

A ⊙ = X ⊙ X ∈ A ,

deq deq

with the base case operations ⊙enq = enqueue and ⊙ = dequeue.

deq

Remark 3. By construction, each ⊙ takes its inputs in Pn(Q) (and E for enqueue) and returns values in Pn(Q) or Pn(Q × E). Hence SNHQn is an (m, n)-Superhyperstructure with m = 2.

Proposition 2.9. SNHQn indeed carries the structure of an n-Superhyperstructure on the base set Q.

(n) (n)

Proof : By Definition 2.7, both operations ⊙enq and ⊙ map into the n-th powerset Pn( · ). This exactly matches

deq

the requirement for an (m, n)-superhyperoperation (here m = 2), so SNHQn is an n-superhyperstructure. Theorem 2. For n = 0 and n = 1, the n-SuperHyperqueue specializes to the classical queue and the Hyperqueue, respectively:

SNHQ0 = Q, enqueue, dequeue , SNHQ1 = P(Q), ⊙enq, ⊙deq = HQ.

Proof : Case n = 0: By definition P0(Q) = Q, and ⊙enq = enqueue, ⊙ = dequeue. Thus SNHQ0 is exactly

deq

the classical queue.

Case n = 1: We have P1(Q) = P(Q). By the recursion A ⊙enq e = { X ⊙enq e | X ∈ A} = {enqueue(q, e) | q ∈ A}, and similarly for ⊙ . These coincide with the hyperqueue operations ⊙enq, ⊙deq. Therefore SNHQ1 = HQ.

deq

Theorem 3 (Closure under Enqueue and Dequeue). For every integer n ≥ 0, every e ∈ E, and every A ∈ Pn(Q):

(n) (n)

A ⊙enq e ∈ Pn(Q), A ⊙ ∈ Pn(Q × E).

deq

Proof : We prove both statements by induction on n.

Base case (n = 0). Here P0(Q) = Q, ⊙enq = enqueue, and ⊙ = dequeue. Clearly enqueue(q, e) ∈ Q and

deq

dequeue(q) ∈ Q × E for every q ∈ Q, so the closure holds.

Inductive step. Assume the claim holds for n − 1. Let A ∈ Pn(Q) = P(Pn−1(Q)). Then by definition

(n) (n−1)

A ⊙enq e = { X ⊙enq e | X ∈ A}.

(n−1)

Since each X ∈ Pn−1(Q), the inductive hypothesis ensures X ⊙enq e ∈ Pn−1(Q). Hence the entire set lies in P Pn−1(Q) = Pn(Q). The argument for ⊙ is identical, replacing enq by deq and Q by Q × E.

deq

Theorem 4 (Restriction to Singletons). For any n ≥ 1, any X ∈ Pn−1(Q), and any e ∈ E,

(n) (n−1) (n) (n−1)

{X} ⊙enq e = X ⊙enq e , {X} ⊙ = X ⊙ .

deq deq

Consequently, the level-n superhyperqueue restricted to singleton subsets recovers the level-(n−1) superhyperqueue.

Proof : Since {X} ⊆ Pn−1(Q), by definition

(n) (n−1) (n−1)

{X} ⊙enq e = { Y ⊙enq e | Y ∈ {X}} = { X ⊙enq e}.

The same reasoning applies to ⊙ . Hence every operation at level n on a singleton set exactly reproduces the

deq

corresponding level (n − 1) operation on its unique element.

Example 4 (A 2-SuperHyperqueue). Let E = {T1, T2}, Q = E = {⟨ ⟩, ⟨T1⟩, ⟨T2⟩, ⟨T1, T2⟩, ⟨T2, T1⟩, . . . }.

Then P1(Q) = P(Q) and P2(Q) = P(P(Q)). Choose the single second-level element A = { ⟨T1⟩, ⟨T2, T1⟩} ∈ P2(Q), where X = { ⟨T1⟩, ⟨T2, T1⟩} ∈ P1(Q).

Second-Level Hyperenqueue. For e = T2, A ⊙enq T2 = X ⊙enq T2 = { enqueue(q, T2) | q ∈ X} = { ⟨T1, T2⟩, ⟨T2, T1, T2⟩} .

Second-Level Hyperdequeue.

(2) (1) ′ ′

A ⊙ = X⊙ = { ( q , e) | q ∈ X, (q , e) = dequeue(q)}

deq deq

= { (⟨ ⟩, T1), (⟨T1⟩, T2)} .

Thus SNHQ2 encodes two nested levels of queue-like operations:

• Level 1 (⊙ ) is the hyperqueue on sets of queues, • Level 2 (⊙ ) lifts that to sets of sets of queues.

Each superhyperoperation systematically generalizes the classical queue operations across two hierarchical layers.

Example 5 (A real-world 3-superhyperqueue: Multi-Tier task pipelines). A task pipeline is a sequence of processing stages where tasks flow step-by-step, each stage performing a specific operation. Suppose a company processes support tickets through three hierarchical tiers: Agent Teams → Departments → Divisions. Let the basic ticket types be E = {T1, T2}, Q = E , and form the iterated powersets P1(Q) = P(Q), P2(Q) = P(P(Q)), P3(Q) = P(P2(Q)).

Level-1 (Agent Team) Queues:

X1 = {⟨T1⟩}, X2 = {⟨T2, T1⟩} ∈ P1(Q).

Level-2 (Department) Collections:

U = { X1, X2} ∈ P2(Q).

Level-3 (Division) Supercollection:

A = { U} ∈ P3(Q).

Third-Level Hyperenqueue. To enqueue a new ticket T2 at the division level:

A ⊙enq T2 = { U ⊙enq T2} = { X ⊙enq T2 | X ∈ U} = { ⟨T1, T2⟩, ⟨T2, T1, T2⟩} .

Third-Level Hyperdequeue. To dequeue at the division level:

A ⊙ = { U⊙ }

deq deq

= { (q , e) | q ∈ U, (q , e) = dequeue(q)} = { (⟨ ⟩, T1), (⟨T1⟩, T2)} .

Interpretation:

• Level 1 (P1) are the queues handled by individual agent teams.

• Level 2 groups those team-queues into department-wide collections.

• Level 3 collects departments into a division-wide superqueue.

• A single division-level hyperenqueue/dequeue simultaneously enqueues/dequeues on every department’s queues.

New applications of hyperstructures and superhyperstructures ...

This illustrates how a 3-SuperHyperqueue models three hierarchical tiers of queue processing in a real-world organization.

For reference, Table 2 presents an overview of queues and their hyper and superhyper extensions.

Notion Carrier (Universe) Enqueue Output Dequeue Output Queue Q := E (finite sequences over E) enqueue : Q × E → Q (single dequeue : Q → Q × E (single next queue) pair, undefined on ⟨ ⟩) Hyperqueue P(Q) (sets of queues) ⊙enq : P(Q) × E → P(Q), ⊙deq : P(Q) → P(Q × E), A ⊙enq e = {enqueue(q, e) | A⊙deq = {(q , e) | q ∈ q ∈ A} A, (q , e) = dequeue(q)}

(n) (n)

n-SuperHyperqueue Pn(Q) (iterated powerset, n ≥ 0) ⊙enq : Pn(Q) × E → Pn(Q), ⊙ : Pn(Q) → Pn(Q × E),

deq (n) (n−1) (n) (n−1)

A ⊙enq e = {X ⊙enq e | X ∈ A⊙ = {X⊙ | X ∈ A}

deq deq

a nonempty set, Q = E , and P0(Q) = Q, Pn+1(Q) = P(Pn(Q)). Special cases: SNHQ0 = (Q, enqueue, dequeue) and SNHQ1 = (P(Q), ⊙enq, ⊙deq).

A Markov chain is a stochastic process in which the next state depends only on the current state and not on the prior history [12, 13, 14]. In this subsection, we extend this concept to Markov HyperChains and Markov SuperHyperChains and investigate their properties.

Definition 12 (Markov Chain). [12, 13] Let (Ω, F,P ) be a probability space and let S be a countable (or finite) set called the state space. A discrete-time stochastic process {Xn}n=0, where each Xn : Ω −→ S, is called a (discrete-time) Markov chain if for every n ≥ 0 and every sequence of states i0, i1, . . . , in+1 ∈ S,

Definition 13 (Time-Homogeneity and Transition Matrix). A Markov chain {Xn} is time-homogeneous if there exists a function P : S × S −→ [0, 1], P(i, j) =(P Xn+1 = j | Xn = i), independent of n. The array P(i, j) is called the transition matrix.

i,j∈S

Definition 14 (Initial Distribution and n-Step Probabilities). The initial distribution π is given by πi =(P X0 = i), i ∈ S.

The n-step transition probability is P (i, j) = X P n = j | X0 = i , i, j ∈ S, n ≥ 0.

Theorem 15 (Chapman–Kolmogorov Equations). [15] For all m, n ≥ 0 and i, j ∈ S,

(m+n) (m) (n)

P (i, j) = P (i, k) P (k, j) .

In matrix form, if P is the one-step transition matrix, then

(n) n

P = P .

The property on the right is called the Markov property.

Remark 4. A state j is absorbing if P (j, j) = 1. A chain is irreducible if every state is reachable from every other. Additional classifications (recurrence/transience, periodicity, stationary distributions) can be developed from these definitions.

Example 6 (Weather Prediction Model). Weather prediction estimates future atmospheric conditions like temperature, rainfall, and wind using current data and numerical simulation models (cf.[16, 17, 18, 19]). A classical real-world application of a Markov chain is weather modeling, with a finite state space S = {Sunny, Cloudy, Rainy}.

We assume a discrete daily process {Xn}n=0, where Xn is the weather on day n.

Initial Distribution Suppose on day 0 the probabilities of each weather are

(0) (0) (0) (0)

π = π , π , π = (0.50, 0.30, 0.20).

Sunny Cloudy Rainy

Transition Matrix We posit a time-homogeneous transition matrix P capturing the probabilities of tomorrow’s weather given today’s:

0.80 0.15 0.05 P = P(i, j) = 0.20 0.60 0.20 , i,j∈S   0.20 0.30 0.50 where, for example, P(Sunny, Cloudy) = 0.15 and P(Rainy, Rainy) = 0.50.

One-Step Evolution The distribution on day 1 is 0.80 0.15 0.05 (1) (0)   π = π P = (0.50, 0.30, 0.20) 0.20 0.60 0.20 = (0.59, 0.345, 0.065).

0.20 0.30 0.50 Thus there is a 59% chance of Sunshine on day 1, 34.5% Cloudy, and 6.5% Rainy.

Two-Step Probabilities By the Chapman–Kolmogorov equation,

(2) 2

P = P , so, for instance, P (Sunny, Rainy) = P(Sunny, k) P(k, Rainy) = 0.8 · 0.05 + 0.15 · 0.20 + 0.05 · 0.50 = 0.095.

Hence if today is Sunny, the chance of Rain tomorrow (two days ahead) is 9.5%.

Interpretation This simple weather-prediction Markov chain illustrates how one can:

• Encode the “memoryless” Markov property: tomorrow’s weather depends only on today’s.

• Use an initial distribution and transition matrix to compute future distributions.

• Apply Chapman–Kolmogorov to obtain multi-step forecasts.

The definition of a Markov HyperChain is given below.

Definition 16 (Markov HyperChain). Let S be a nonempty countable (or finite) set, called the state space. A Markov HyperChain on S is a pair MHC = S, H , where H : S × S −→ P([0, 1]), (i, j) 7→ Hij, is called the hypertransition matrix, subject to:

(i) Nonemptiness:

Hij ̸= ∅, ∀ i, j ∈ S.

New applications of hyperstructures and superhyperstructures ...

(ii) Hypernormalization: For each source state i ∈ S, there exists at least one selection function pi : S → [0, 1] such that pi(j) ∈ Hij for every j ∈ S, and pi(j) = 1.

Remark 5. Because Hij ⊆ [0, 1], the map H takes values in the powerset P([0, 1]). Thus H is a hyperoperation in the sense of hyperstructure theory, and MHC forms a hyperstructure on S.

Theorem 5 (Recovery of Classical Markov Chain). If for all i, j ∈ S the hypertransition sets are singletons, Hij = P(i, j) , then MHC = (S, H) is equivalent to the usual time-homogeneous Markov chain with transition matrix P(i, j) .

Proof : Define pi(j) = P(i, j). Since Hij = {P(i, j)}, we have pi(j) ∈ Hij for each i, j. By the usual stochastic assumption, P(i, j) = 1. Hence all axioms (i)–(ii) of the Markov HyperChain are satisfied, and no additional choices arise because each Hij contains exactly one element. Thus the hyperstructure MHC reduces precisely to the classical Markov chain.

Theorem 6 (Hyperstructure Property). The mapping H : S × S −→ P([0, 1]) is a binary hyperoperation; therefore MHC = (S, H) is a hyperstructure in the sense that its operation takes values in a powerset.

Proof : By definition, for any (i, j) ∈ S × S, one has Hij ⊆ [0, 1]. Hence H(i, j) ∈ P([0, 1]), which is exactly the requirement for a hyperoperation on S. No further conditions are needed beyond (i)–(ii), so MHC is a legitimate hyperstructure.

Example 7 (Three-State Markov HyperChain: Machine Condition Model). Let S = { G, D, F} be the three-state space representing the health of a machine:

G = Good, D = Degraded, F = Failed.

We specify a hypertransition matrix H : S × S → P([0, 1]) by giving, for each i ∈ S, nonempty sets Hij ⊆ [0, 1], j ∈ S, as follows:

j = G j = D j = F i = G {0.7, 0.8} {0.2, 0.25} {0.0, 0.05} i = D {0.1, 0.15} {0.6, 0.7} {0.2, 0.3} i = F {0.05, 0.1} {0.1, 0.2} {0.7, 0.85}

Hypernormalization. For each source state i one must pick probabilities pi(j) ∈ Hij so that pi(j) = 1. Two concrete selections are:

(1) (1) (1)

(Selection 1) p = (0.8, 0.2, 0.0), p = (0.15, 0.6, 0.25), p = (0.10, 0.20, 0.70),

G D F (2) (2) (2)

(Selection 2) p = (0.7, 0.25, 0.05), p = (0.10, 0.70, 0.20), p = (0.05, 0.10, 0.85).

G D F Fujita |Opt. 3(2) (2026) 77-110

Induced Classical Chains. Each selection yields a classical transition matrix. For Selection 1:

0.80 0.20 0.00 P = p (j) = 0.15 0.60 0.25 ,

i i,j∈S

0.10 0.20 0.70 and for Selection 2:

0.70 0.25 0.05 (2)   P = 0.10 0.70 0.20 .

0.05 0.10 0.85

Interpretation.

• The sets Hij encode uncertainty or variability in transition probabilities.

• Choosing different pi corresponds to different plausible “worlds” or expert opinions.

• Each P is a valid stochastic matrix (rows sum to 1), recovering a classical Markov chain.

• This demonstrates the hyper nature: one hyperchain encodes multiple classical chains.

The definition of a Markov n-SuperHyperChain is given below.

Definition 17 (Markov n-SuperHyperChain). Let S be a nonempty countable (or finite) set (the state space) and fix an integer n ≥ 1. A Markov n-SuperHyperChain is a pair

(n) (n)

MHC = S, H , where

(n) (n)

H : S × S −→ Pn([0, 1]), (i, j) 7→ Hij , is called the n-superhypertransition mapping, satisfying:

(i) Nonemptiness:

Hij ̸= ∅ ∀ i, j ∈ S.

(ii) Nested Selection and Normalization: For each i ∈ S, there exist nested selection functions σi for k = n, n − 1, . . . , 1 such that:

(n) (n) (n)

σi : S → Pn−1([0, 1]), σi (j) ∈ Hij ,

(k) (k) (k+1)

σi : S → Pk−1([0, 1]), σi (j) ∈ σi (j) (1 < k < n),

(1) (1) (2)

σi : S → [0, 1], σi (j) ∈ σi (j), and finally the normalization σi (j) = 1. (1)

Remark 6. Since H takes values in the n-th powerset Pn([0, 1]), it is by definition an (m, n)-superhyperoperation (with m = 2). Hence MHC is an n-Superhyperstructure on S.

Theorem 7 (Recovery of Lower Levels). 1. When n = 1, MHC coincides with the Markov HyperChain MHC = (S, H) via Hij = Hij ⊆ [0, 1].

2. When n = 0, interpreting P0([0, 1]) = [0, 1], MHC is exactly a classical time-homogeneous Markov chain with numerical transition matrix P(i, j) = Hij .

Proof : Case n = 1. By definition P1([0, 1]) = P([0, 1]). The single nested selector σi : S → [0, 1] with (1) (1) P (1) σi (j) ∈ Hij and σi (j) = 1 is exactly the condition for a Markov HyperChain.

(0) (1) P (0) Case n = 0. Set P0([0, 1]) = [0, 1]. Then Hij ∈ [0, 1], and the single selector σi = id yields Hij = 1, recovering the usual transition probabilities P(i, j).

(n) (n)

Theorem 8 (n-SuperHyperstructure Property). MHC = (S, H ) is an (m, n)-Superhyperstructure with m = 2.

(n) (n)

Proof : By construction H : S × S → Pn([0, 1]). Thus H is a binary operation whose codomain is the n-th powerset of [0, 1], exactly the definition of a binary (m, n)-superhyperoperation. The nested selectors guarantee that one can extract a genuine probability distribution at the bottom level, satisfying all axioms of a classical Markov chain.

Example 8 (Machine Reliability as a 2-SuperHyperChain). Machine reliability measures the probability that a machine performs its intended function without failure over a specific time period (cf.[20, 21, 22]). Consider a machine whose daily condition is modeled by the state space S = { W, F}, W = Working, F = Failed.

We define a 2-superhypertransition H : S × S −→ P2([0, 1]) by the following table of subsets of subsets of [0, 1]:

j = W j = F i = W {0.85, 0.90}, {0.80, 0.95} {0.10, 0.15}, {0.05, 0.20} i = F {0.25, 0.35}, {0.30, 0.40} {0.65, 0.80}, {0.60, 0.70}

Nested Selection (k = 2, 1). We choose for each source state i ∈ {W, F} nested selectors σi ∈ P1([0, 1]) and σi ∈ [0, 1] as follows:

σi (W) σi (F) i = W {0.85, 0.90} ∈ H , {0.10, 0.15} ∈ H ,

W,W W,F

i = F {0.25, 0.35} ∈ H , {0.65, 0.80} ∈ H .

F,W F,F

Then pick σ (W) = 0.90 ∈ {0.85, 0.90}, σ (F) = 0.10 ∈ {0.10, 0.15}, and σ (W) = 0.35 ∈ {0.25, 0.35}, σ (F) = 0.65 ∈ {0.65, 0.80}.

(1) (1) (1) (1)

One checks σ (W) +σ (F) = 0.90 + 0.10 = 1 and σ (W) +σ (F) = 0.35 + 0.65 = 1, so nested normalization

W W F F

(1) holds.

Induced Classical Chain. The bottom-level selector σ yields the classical transition matrix P(i, j) = σi (j), i, j ∈ {W, F}, namely 0.90 0.10 P = .

0.35 0.65 Thus the 2-SuperHyperChain MHC encodes two levels of uncertainty—choices of probability-sets—and recovers a genuine Markov chain upon nested selection.

Example 9 (Weather Forecasting Ensemble as a 3-SuperHyperChain). We model daily weather with state space S = {S, R}, S = Sunny, R = Rainy.

Fix three levels of uncertainty: Model Family (level 3) → Parameter Set (level 2) → Probability Choice (level 1).

Level 3: Superhypertransition sets Define Hij ⊆ P2([0, 1]) by two “model families” for each i → j:

j = S j = R n o n o

(3) (3) (3) (3)

i = S F1 , F2 G1 , G2 n o n o

(3) (3) (3) (3)

i = R H1 , H2 K1 , K2 where, for instance, F1 = {0.7, 0.8}, {0.75, 0.85} , F2 = {0.65, 0.9}, {0.78, 0.82} ,

(3) (3) (3)

and similarly G , H , K each contain two subsets of [0, 1].

k k k

Level 2: Nested parameter-set selection Choose one parameter-set from each model family:

for i = S:

(3) (3) (3) (3)

σ (S) = F1 , σ (R) = G2 ; then pick

(2) (3) (2) (3)

σ (S) = {0.7, 0.8} ∈ F1 , σ (R) = {0.2, 0.25} ∈ G2 .

for i = R:

(3) (3) (3) (3)

σ (S) = H2 , σ (R) = K1 ; then pick

(2) (3) (2) (3)

σ (S) = {0.4, 0.5} ∈ H2 , σ (R) = {0.3, 0.7} ∈ K1 .

Level 1: Final probability choice and normalization Select a single probability from each parameter set so that they sum to 1:

σ (S) = 0.8 ∈ {0.7, 0.8}, σ (R) = 0.2 ∈ {0.2, 0.25}, 0.8 + 0.2 = 1.

σ (S) = 0.5 ∈ {0.4, 0.5}, σ (R) = 0.5 ∈ {0.3, 0.7}, 0.5 + 0.5 = 1.

An interval is a nonempty subset of a totally ordered set where, for any two elements, all elements between them are also included—representing continuous ranges (cf.[23]). In this subsection, we extend this concept to HyperIntervals and SuperHyperIntervals and investigate their properties.

Definition 18 (Totally Ordered Set). [24] A totally ordered set is a pair (X, ≤) where X is a set and ≤ is a binary relation on X satisfying, for all x, y,z ∈ X:

(i) Reflexivity: x ≤ x.

(ii) Antisymmetry: If x ≤ y and y ≤ x, then x = y.

(iii) Transitivity: If x ≤ y and y ≤ z, then x ≤ z.

(iv) Totality: Either x ≤ y or y ≤ x.

Definition 19 (Convex Subset). Let (X, ≤) be a totally ordered set. A subset I ⊆ X is convex if for every x, y ∈ I with x ≤ y, any z ∈ X satisfying x ≤ z ≤ y also lies in I:

x, y ∈ I, x ≤ z ≤ y =⇒ z ∈ I.

Definition 20 (Interval). An interval in a totally ordered set (X,≤) is any nonempty convex subset I ⊆ X. Equivalently, I is an interval if there exist (possibly equal or infinite) endpoints a, b ∈ X ∪ {−∞, +∞} with a ≤ b such that I = { x ∈ X | a ♢ x ♢ b}, where each of ♢, ♢ is chosen from {<, ≤}. In particular, in the real line (R, ≤) one has:

[a, b] = {x ∈ R | a ≤ x ≤ b}, (a, b) = {x ∈ R | a < x < b}, [a, b) = {x ∈ R | a ≤ x < b}, (a, b] = {x ∈ R | a < x ≤ b}, with the conventions (−∞, b] = {x | x ≤ b}, (a, ∞) = {x | x > a}, etc.

Remark 7. • A degenerate interval has a = b, e.g. [a, a] = {a} (cf.[25]).

• The whole set X is an interval with (a, b) = (−∞, +∞).

• Intervals are exactly the connected subsets of R under the usual topology.

Example 10. (a) [0, 1] ⊂ R is a closed bounded interval.

(b) (−∞, 0) ⊂ R is an unbounded open interval.

(c) {5} = [5, 5] is a degenerate (singleton) interval.

(d) In Z with the usual order, {2, 3, 4} = [2, 4] ∩ Z is an interval.

The definition of a Hyperinterval is given below.

Definition 21 (Hyperinterval Hyperoperation). Define a binary hyperoperation ⊠ : I(X) × I(X) −→ P I(X) by [a1, b1] ⊠ [a2, b2] = min(x, y), max(x, y) x ∈ [a1, b1], y ∈ [a2, b2] .

The pair I(X), ⊠ is called the hyperinterval hyperstructure on X.

Remark 8. For any x ∈ [a1, b1] and y ∈ [a2, b2] we have min(x, y) ≤ max(x, y), so each [min(x, y), max(x, y)] is indeed a closed interval in X. Hence ⊠ maps into P(I(X)) and is a legitimate hyperoperation.

Theorem 9 (Recovery of Classical Interval Span). For any two intervals I = [a1, b1] and J = [a2, b2], the smallest closed interval containing I ∪ J, hull(I, J) = min(a1, a2), max(b1, b2) , is recovered by choosing x = min(a1, a2) ∈ I ∪ J and y = max(b1, b2) ∈ I ∪ J, so that min(x, y), max(x, y) = min(a1, a2), max(b1, b2) ∈ I ⊠ J.

Thus ⊠ genuinely generalizes the classical “interval span” operation.

Proof : Since min(a1, a2) ≤ max(b1, b2), the chosen pair (x, y) satisfies x ∈ [a1, b1]∪[a2, b2] and y ∈ [a1, b1]∪[a2, b2]. By the definition of ⊠, [min(x, y), max(x, y)] = min(a1, a2), max(b1, b2) belongs to I ⊠ J. This interval is exactly the smallest closed interval containing both I and J.

Theorem 10 (Hyperstructure Property). I(X), ⊠ is a hyperstructure because its binary operation ⊠ takes each pair of intervals to a subset of I(X):

⊠ : I(X) × I(X) −→ P I(X) .

Proof : By construction, for any I, J ∈ I(X) the set I ⊠ J is a (possibly empty) subset of I(X). No further closure or algebraic axioms are required for a hyperstructure beyond this set-valued operation.

Example 11 (Hyperinterval for Scheduling Windows). Scheduling windows are specific time intervals during which tasks or operations are allowed, planned, or expected to be executed. Let X = R represent time in hours and consider two scheduling windows:

I = [10, 15], J = [12, 20].

Here I might be a maintenance window from 10:00 to 15:00, and J a delivery window from 12:00 to 20:00. Hyperinterval Computation By definition, I ⊠ J = min(x, y), max(x, y) | x ∈ [10, 15], y ∈ [12, 20] .

Concretely:

• If x = 10 and y = 12, then [min, max] = [10, 12].

• If x = 10 and y = 20, then [min, max] = [10, 20].

• If x = 15 and y = 12, then [min, max] = [12, 15].

• If x = 15 and y = 20, then [min, max] = [15, 20].

• If x = 13.5 and y = 14, then [min, max] = [13.5, 14].

Description of the Result One checks that I ⊠ J = [u, v] u ∈ [10, 20], v ∈ [12, 20], u ≤ v .

Thus:

• The earliest possible start is min(10, 12) = 10.

• The latest possible end is max(15, 20) = 20.

• Every interval [u, v] with 10 ≤ u ≤ v ≤ 20 and v ≥ 12 appears.

Interpretation In scheduling terms, the hyperinterval captures all possible combined windows: you might start as early as 10:00 (if maintenance begins before delivery) or as late as 12:00 (if delivery precedes maintenance), and finish as early as 12:00 or as late as 20:00. Every such candidate window [u, v] is a valid member of the hyperinterval I ⊠ J.

The definition of a n-SuperHyperinterval is given below.

Definition 22 ( n-th Powerset of Intervals ). Let (X,≤) be a totally ordered set and let I(X) = { [a, b] | a, b ∈ X ∪ {−∞, +∞}, a ≤ b} be the set of all closed intervals in X. Define recursively I0(X) = I(X), Ik+1(X) = P Ik(X) , k ≥ 0.

Definition 23 ( n-SuperHyperinterval Hyperoperation ). For each integer n ≥ 1, define a binary hyperoperation ⊠ : In(X) × In(X) −→ P In(X) by recursion on n. Given A, B ∈ In(X),

(n) (n−1)

A ⊠ B = U ⊠ V | U ∈ A, V ∈ B , where the base case n = 0 is the classical interval span:

I ⊠ J = [min(a1, a2), max(b1, b2)], I = [a1, b1], J = [a2, b2] ∈ I(X).

Definition 24 ( n-SuperHyperinterval Hyperstructure ). The pair In(X), ⊠ is called the n-SuperHyperinterval Hyperstructure on X.

Remark 9. Since ⊠ takes its values in P(In(X)), it is a binary (2, n)-superhyperoperation. Hence In(X), ⊠ is an n-Superhyperstructure.

Theorem 11 (Recovery of Lower Levels). (i) For n = 0, (I0(X), ⊠ ) is the classical interval span.

(ii) For n = 1, (I1(X), ⊠ ) coincides with the hyperinterval hyperstructure (I(X), ⊠).

Proof : (i) By definition I0(X) = I(X) and ⊠ is exactly I ⊠ J = [min(a1, a2), max(b1, b2)].

(ii) Since I1(X) = P(I(X)), the recursion gives A ⊠ B = { I ⊠ J | I ∈ A, J ∈ B}, which is precisely the hyperinterval operation ⊠.

Theorem 12 ( n-Superhyperstructure Property ). In(X), ⊠ is an n-Superhyperstructure, i.e. its operation has codomain P(In(X)).

Proof : By construction In(X) = P(In−1(X)), so for any A, B ∈ In(X), the set A ⊠ B belongs to P(In(X)), verifying the superhyperstructure condition.

Example 12 (2-SuperHyperinterval: Scheduling and Delivery Windows). Let X = R represent time in hours. Consider three sets of intervals:

S1 = {[9, 11], [14, 16]}, S2 = {[10, 12], [15, 18]}, S3 = {[11, 13]}.

Each Sk ⊆ I(X), so

A = { S1, S2} ∈ I2(X), B = { S3} ∈ I2(X).

By definition, A ⊠ B = { U ⊠ V | U ∈ A, V ∈ B}.

We compute each:

S1 ⊠ S3 = { I ⊠ J | I ∈ S1, J ∈ S3} = {[9, 11] ⊠ [11, 13], [14, 16] ⊠ [11, 13]} = {[ min(9, 11), max(11, 13) ], [ min(14, 11), max(16, 13) ]} = {[9, 13], [11, 16]}.

S2 ⊠ S3 = { I ⊠ J | I ∈ S2, J ∈ S3} = {[10, 12] ⊠ [11, 13], [15, 18] ⊠ [11, 13]} = {[10, 13], [11, 18]}.

Therefore the full 2-superhyperinterval is

A ⊠ B = {[9, 13], [11, 16]}, {[10, 13], [11, 18]} ⊆ I2(X).

Interpretation:

• S1 might represent two possible maintenance windows (9–11 h, 14–16 h).

• S2 two delivery windows (10–12 h, 15–18 h), and S3 a fixed inspection window (11–13 h).

• The first-level operation ⊠ combines each maintenance or delivery interval with the inspection window, yielding sets of possible combined windows.

• The second-level operation ⊠ collects these first-level results into a set of two candidate sets, reflecting two different scheduling scenarios.

Example 13 (A Real-World 3-SuperHyperinterval: Multi-Tier Scheduling). Multi-Tier Scheduling coordinates tasks across hierarchical levels—individual, team, and organizational—ensuring alignment, resource availability, and timing at each layer (cf.[26, 27, 27]). Let X = R be time in hours. We illustrate three hierarchical scheduling levels:

Level 0 (Basic Intervals):

S1 = [9, 11], S2 = [14, 16], S3 = [10, 12], S4 = [15, 18], S5 = [11, 13].

These lie in I0(X) = I(X).

Level 1 (Team-Level Sets): Form two teams’ windows:

U = {S1, S2}, V = {S3, S4}, W = {S5} ∈ I1(X) = P(I(X)).

Level 2 (Department-Level Collections): Group these team-sets into departments:

D1 = {U, V }, D2 = {W} ∈ I2(X) = P(I1(X)).

Level 3 (Division-Level Supercollections): Finally, form two divisions’ supercollections:

A = {D1}, B = {D2} ∈ I3(X) = P(I2(X)).

Third-Level Hyperoperation ⊠ :

A ⊠ B = { D1 ⊠ D2}.

Compute the inner 2-superhyperinterval:

(2) (1) (1)

D1 ⊠ D2 = U ⊠ W, V ⊠ W , where each ⊠ is the hyperinterval on sets of intervals:

U ⊠ W = { I ⊠ J | I ∈ U, J ∈ W} = { S1 ⊠ S5, S2 ⊠ S5} = {[9, 13], [11, 16]},

(1) (0) (0)

V ⊠ W = { S3 ⊠ S5, S4 ⊠ S5} = {[10, 13], [11, 18]}.

Thus D1 ⊠ D2 = {[9, 13], [11, 16]}, {[10, 13], [11, 18]} , and finally A ⊠ B = D1 ⊠ D2 = {{[9, 13], [11, 16]}, {[10, 13], [11, 18]}} .

Interpretation:

• Level 1 intervals Si are individual team work or delivery windows.

• Level 1 sets U, V, W group those into team schedules.

• Level 2 collections D1, D2 are department-wide scheduling options.

• Level 3 supercollections A, B represent division-level bundles.

• The 3-superhyperinterval A ⊠ B encodes all possible combined division schedules across three hierar-chical tiers.

For reference, Table 4 presents an overview of intervals and their hyper/superhyper extensions.

Concept Object and notation Extension mechanism (informal) Interval Nonempty convex I ⊆ (X, ≤) (e.g., in R: I = [a, b]). Order-convexity: x, y ∈ I, x ≤ z ≤ y ⇒ z ∈ I.

HyperInterval Universe I(X). Hyperoperation I ⊠ J := Replace the single span by a set of spans obtained from {[min(x, y), max(x, y)] | x ∈ I, y ∈ J} ⊆ I(X). internal choices x ∈ I, y ∈ J.

n-SuperHyperInterval Iteration: I0(X) = I(X), Ik+1(X) = P(Ik(X)). For Lift the operation level-by-level: objects are sets of lower- (n) (n−1) level objects; combine by taking all memberwise combi-A, B ∈ In(X), A ⊠ B := { U ⊠ V | U ∈ A, V ∈ (0) nations recursively.

B }. (Base n = 0: I ⊠ J = [min(a1, a2), max(b1, b2)].)

Propositional logic analyzes truth values of propositions using logical connectives such as AND, OR, and NOT, without considering internal structure (cf.[28, 29]). Abstract propositional logic studies logical consequence relations over arbitrary sets of formulas, independently of any specific syntax or semantics. In this paper, we extend this framework to Abstract Propositional HyperLogic and Superhyperlogic by employing HyperStructures and SuperHyperStructures.

Definition 25 (Language of Propositional Logic). Let Var be a nonempty set of propositional variables. The language Lprop is the smallest set of strings (called formulas) such that:

φ ::= p (p ∈ Var) (¬φ) (φ ∧ ψ) (φ ∨ ψ) (φ → ψ), where φ, ψ ∈ Lprop. We write Lprop(Var) when we wish to emphasize the dependence on Var.

Definition 26 (Abstract Logic). An abstract propositional logic is a pair L = Lprop, ⊢L , where • Lprop is the set of well-formed formulas (as above).

• ⊢L ⊆ P(Lprop) × Lprop is a consequence relation satisfying, for all Γ, ∆ ⊆ Lprop and φ, ψ ∈ Lprop:

(1) Reflexivity: If φ ∈ Γ, then Γ ⊢L φ.

(2) Monotonicity: If Γ ⊢L φ and Γ ⊆ ∆, then ∆ ⊢L φ.

(3) Cut (Transitivity): If Γ ⊢L ψ for every ψ ∈ ∆ and ∆ ⊢L φ, then Γ ⊢L φ.

(4) Substitution-Invariance: If Γ ⊢L φ then σ(Γ) ⊢L σ(φ) for every uniform substitution σ on propositional variables.

Definition 27 (Semantic Consequence). Let L = (Lprop, ⊢L) be an abstract logic. A valuation is a function v : Var → {0, 1}, extended to all of Lprop by:

v(¬φ) = 1 − v(φ), v(φ ∧ ψ) = min{v(φ), v(ψ)}, and similarly for ∨ and →. We write Γ |= φ ⇐⇒ for every valuation v, v(ψ) = 1 ∀ψ ∈ Γ implies v(φ) = 1.

Definition 28 (Soundness and Completeness). An abstract logic L = (Lprop, ⊢L) is called • sound if Γ ⊢L φ implies Γ |= φ.

• complete if Γ |= φ implies Γ ⊢L φ.

The definition of an Abstract Hyperlogic is given below.

Definition 29 (Abstract Hyperlogic). Let Var be a nonempty set of propositional variables, and let L = Lprop(Var) be the set of all well-formed propositional formulas over Var. Set H = P(L), the collection of all sets of formulas. An abstract hyperlogic is a pair H, ⊣H , where ⊣H : H −→ P(H) is a hyperconsequence operator (a unary hyperoperation) satisfying, for all Γ, ∆, Θ ∈ H:

(a) Hyperreflexivity: Γ ∈ ⊣H (Γ).

(b) Hypermonotonicity: If ∆ ∈⊣H (Γ) and ∆ ⊆ Θ, then Θ ∈⊣H (Γ).

(c) Hypercut: If ∆ ∈⊣H (Γ) and Θ ∈⊣H (∆), then Θ ∈⊣H (Γ).

Remark 10 (Hyperstructure Property). Since ⊣H: H → P(H), the pair (H, ⊣H) is by definition a hyper-structure with a unary hyperoperation.

Theorem 13 (Recovery of Abstract Logic). If there exists a classical consequence relation ⊢⊆ H × L whose associated closure operator Cl : H −→ H, Cl(Γ) = {φ ∈ L | Γ ⊢ φ}, then setting ⊣H (Γ) = Cl(Γ) (a singleton set) yields an abstract hyperlogic (H, ⊣H) which exactly generalizes the original logic.

Proof : Define ⊣H (Γ) = {Cl(Γ)}. Then for each Γ:

• Hyperreflexivity: Γ ⊆ Cl(Γ), so Γ ∈⊣H (Γ).

• Hypermonotonicity: If Cl(Γ) ⊆ Θ, then by monotonicity of Cl, Θ = Cl(Γ), so Θ ∈⊣H (Γ).

• Hypercut: Since ⊣H (Γ) and ⊣H (∆) are singletons, the transitivity of Cl ensures closure under two-step application.

Thus ⊣H satisfies (a)–(c) and recovers the original ⊢.

Example 14 (Concrete Nontrivial Hyperlogic). Let Var = {p, q} and consider the finite fragment of the propositional language L = p, q, p → q, q → p .

Fix the premise set Γ = { p → q, p}.

′ 2 ′

We define a hyperconsequence operator ⊣H: P(L ) → P (L ) by ⊣H (Γ) = Γ, Γ ∪ {q}, L .

| {z } |{z}

=C =T

Contents of each hyperconsequence set:

Γ = {p → q, p}, C = {p → q, p, q}, T = {p, q, p → q, q → p}.

Verification of hyperlogic axioms:

• Hyperreflexivity: Γ ∈⊣H (Γ) by construction.

• Hypermonotonicity: Any superset of Γ in L is either C or T, and both lie in ⊣H (Γ). Any superset of C is T, also in ⊣H (Γ).

• Hypercut: We need that if ∆ ∈⊣H (Γ) and Θ ∈⊣H (∆), then Θ ∈⊣H (Γ).

– If ∆ = Γ, then ⊣H (∆) = {Γ, C, T}, so any Θ ∈⊣H (Γ).

– If ∆ = C, then ⊣H (C) = {C, T} (by the same rule), and both lie in ⊣H (Γ).

– If ∆ = T, then ⊣H (T) = {T}, and T ∈⊣H (Γ).

Interpretation:

• Γ itself as a minimal consequence set (no inference).

• C is the classical propositional closure of Γ, since from p → q and p we infer q.

• T is the “explosive” or trivial logic closure, in which every formula follows.

Thus ⊣H (Γ) contains three nested consequence-sets, demonstrating genuine hyperbehavior beyond any single abstract logic.

The definition of an Abstract n-SuperHyperlogic is given below.

Definition 30 (Abstract n-SuperHyperlogic). Fix n ≥ 0. Define δ : D0 −→ D0, δ (Γ) = Cl(Γ), and for k = 0, 1, . . . , n − 1,

(k+1) (k+1) (k)

δ : D0 −→ Dk+1, δ (Γ) = δ (Γ) .

Then the pair D0, δ is called the abstract n-superhyperlogic.

Remark 11 (Superhyperstructure Property). Since δ maps D0 = P(L) into the n-fold powerset Dn =

n+1 (n)

P (L), it is a unary (1, n)-superhyperoperation, and thus D0, δ is a (1, n)-superhyperstructure.

Theorem 14 (Recovery of Logic and Hyperlogic). (i) If n = 0, then δ = Cl and D0, δ is exactly the original abstract logic.

(ii) If n = 1, then δ (Γ) = {Cl(Γ)}, so D0, δ coincides with the abstract hyperlogic.

Proof : • For n = 0, by definition δ (Γ) = Cl(Γ).

• For n = 1, δ (Γ) = {δ (Γ)} = {Cl(Γ)}, recovering the singleton-valued hyperconsequence.

Higher levels follow by the inductive construction of δ .

Theorem 15 (Nested Expansion). For every Γ ⊆ L and every n ≥ 0, (n) .

δ (Γ) = {{· · · {Cl(Γ)} · · · }} | {z }

n neste

Proof : We proceed by induction on n.

Base case (n = 0). By definition δ (Γ) = Cl(Γ), which is just one application of the closure.

Inductive step. Suppose the formula holds for some n = k, i.e.

δ (Γ) = {{· · · {Cl(Γ)} · · · }} .

| {z }

k braces (k+1)

Then by definition of δ ,

(k+1) (k)

δ (Γ) = {δ (Γ)} = {{· · · {Cl(Γ)} · · · }} , | {z }

k braces

which is exactly k + 1 nested braces around Cl(Γ). This completes the induction.

Theorem 15 (Idempotence). For every Γ ⊆ L and every n ≥ 0,

(n) (n) (n)

δ δ (Γ) = δ (Γ).

Proof : Again by induction on n.

Base case (n = 0).

(0) (0) (0)

δ δ (Γ) = Cl Cl(Γ) = Cl(Γ) = δ (Γ), using idempotence of the Tarskian closure Cl.

Inductive step. Assume δ is idempotent. Then n o n o

(k+1) (k+1) (k) (k+1) (k) (k+1)

δ δ (Γ) = δ δ (Γ) = δ (Γ) = δ (Γ),

(k+1) (k) (k) (k)

where the middle equality uses the fact that δ (Γ) = {δ (Γ)} and the inductive hypothesis δ (δ (Γ)) = δ (Γ).

Theorem 16 (Substitution Invariance). Let σ be any uniform substitution on propositional variables, extended pointwise to sets and iterated powersets. Then for every Γ ⊆ L and every n ≥ 0,

(n) (n)

δ σ(Γ) = σ δ (Γ) , where on the right σ acts level-wise on each nested set.

Proof : We prove by induction on n.

Base case (n = 0).

δ σ(Γ) = Cl σ(Γ) = σ Cl(Γ) = σ δ (Γ) , using substitution-invariance of the closure operator Cl.

Inductive step. Suppose the claim holds for n = k. Then n o n o

(k+1) (k) (k) (k) (k+1)

δ σ(Γ) = δ σ(Γ) = σ δ (Γ) = σ {δ (Γ)} = σ δ (Γ) , where we used the inductive hypothesis in the second equality and the natural action of σ on singleton-wrapped sets in the third.

Example 15 (Nested Consequences: Multi-Step Inference). Let Var = {p, q,r} and let L be the set of all propositional formulas built with ∧, ∨, →, ¬. Let Cl be the classical propositional closure operator (closing under modus ponens, conjunction introduction, etc.). Consider the premise set Γ = { p → q, q → r, p}.

• δ (Γ) = Cl(Γ) is the set of all formulas derivable from Γ. In particular:

Cl(Γ) = { p → q, q → r, p, q, r}.

Indeed, from p → q and p we infer q, and from q → r and q we infer r.

• δ (Γ) lifts this to a singleton set:

δ (Γ) = Cl(Γ) = {p → q, q → r, p, q, r} .

• δ (Γ) lifts one more level:

δ (Γ) = δ (Γ) = {{p → q, q → r, p, q, r}} .

Thus each application of δ moves the consequence set up one level in the powerset hierarchy, giving a fully detailed illustration of the 2-superhyperlogic for a nontrivial inference scenario.

Example 2.65 (Medical Diagnosis Workflow as a 3-SuperHyperlogic). Medical diagnosis identifies a patient’s disease or condition by analyzing symptoms, history, tests, and observations using clinical and scientific methods. Let Var = {Fever, Cough}, L = Lprop(Var), and introduce two diagnostic conclusions:

D1 ≡ “Diagnosis: Flu”, D2 ≡ “Diagnosis: Common Cold”.

Define the Tarskian closure Cl by medical inference rules so that Cl {Fever, Cough} = {D1, D2}.

Set Γ = {Fever, Cough}. Then the 3-superhyperlogic layers are:

δ (Γ) = Cl(Γ) = {D1, D2},

δ (Γ) = δ (Γ) = {D1, D2} ,

δ (Γ) = δ (Γ) = {{D1, D2}} ,

δ (Γ) = δ (Γ) = {{{D1, D2}}} .

Interpretation • δ yields the direct diagnostic conclusions from the symptoms.

• δ wraps those conclusions to represent a primary review (e.g., by a general practitioner).

• δ nests one more level to model a secondary review (e.g., by a specialist), encapsulating the primary’s output.

• δ adds a tertiary expert panel validation layer, nesting all prior reviews.

Thus the medical diagnosis process—symptoms → primary diagnosis → specialist review → panel validation—is captured as a 3-SuperHyperlogic with three tiers of nested inference.

For reference, Table 5 presents an overview of abstract propositional logic and its hyper/superhyper extensions.

A discrete-time system evolves in fixed time steps, updating states based on inputs or previous states at each time index (cf.[30, 31, 32]). In this subsection, we define Discrete-Time HyperSystems and n-SuperHyperSystems.

Definition 31 (Discrete-Time System). A (discrete-time) system is a tuple S = E, I, X, T, t0, x, F , where:

• E is a nonempty set of components.

• I ⊆ E × E is the influence relation: (j, i) ∈ I means component j influences component i.

• X is the state space.

• T is a nonempty totally ordered set (time index).

• t0 = min T is the initial time.

Concept Abstract Propositional Abstract Propositional Hy- Abstract Propositional Su-Logic perLogic perHyperLogic Carrier (universe) L = Lprop(Var) (formu- H = P(L) D0 = P(L), and Dk+1 = las) and H = P(L) (theo- P(Dk) = P (L) ries/premise sets) Core inference object Consequence relation ⊢L⊆ Hyperconsequence operator Iterated consequence lift-P(L) × L (unary hyperoperation) ⊣H : ing δ : D0 → Dn = H → P(H) P (L) Output type (from premises Γ) Set of consequences Family of consequence sets Nested family Cl(Γ) = {φ ∈ L | Γ ⊢L ⊣H (Γ) ⊆ H (depth n) δ (Γ) = φ} ∈ H {{· · · {Cl(Γ)} · · · }} ∈ Dn

| {z }

n nested braces

Axioms / properties (typical) Tarskian consequence: re- Hyperreflexivity: Recursively inherited prop-flexivity, monotonicity, cut, Γ ∈⊣H (Γ); hypermono- erties from Cl: idempotence

(n) (n) (n)

substitution invariance tonicity: ∆ ∈⊣H (Γ), ∆ ⊆ δ (δ (Γ)) = δ (Γ); Θ ⇒ Θ ∈⊣H (Γ); hypercut: substitution invariance

(n) (n)

∆ ∈⊣H (Γ), Θ ∈⊣H (∆) ⇒ δ (σ(Γ)) = σ(δ (Γ)) Θ ∈⊣H (Γ) Recovery of classical level Base notion If ⊣H (Γ) = {Cl(Γ)} n = 0 gives δ = Cl (singleton-valued), then hy- (abstract logic); n = 1 perlogic reduces to the orig- gives δ (Γ) = {Cl(Γ)} inal abstract logic (singleton-valued hyper-logic) Interpretation (one sentence) Single, deterministic clo- Multiple admissible clo- Multi-tiered (hierarchical) sure of a premise set sures/expansions of the wrapping of consequence in-same premise set (non- formation into n nested lev-deterministic consequence) els (meta-inference layers)

• x : E × T → X is the state mapping, where x(i, t) is the state of component i at time t.

• F = Fi is a family of local update functions, where for each i ∈ E:

Ni

Fi : X × T −→ X, Ni = { j ∈ E | (j, i) ∈ I} and Fi (x(j, s))j∈N , s<t, t determines the new state of i at time t.

These data satisfy, for all i ∈ E and all t ∈ T with t > t0, x(i, t) = Fi (x(j, s))j∈N , s<t, t .

Definition 32 (Subsystem). Let S = (E, I, X, T, t0, x, F) be a system. A subsystem of S is a tuple

′ ′ ′ ′ ′

S = E , I , X, T, t0, x , F where

′ ′ ′

• E ⊆ E satisfies Ni ⊆ E for every i ∈ E .

′ ′ ′

• I = I ∩ (E × E ).

• x = x|E′×T .

• F = (Fi)i∈E′ .

Theorem 17 (Subsystem Closure). Every subsystem S of a system S is itself a system in the sense of Definition 2.1.

We call S a discrete-time hypersystem if for every i ∈ E and t > t0, x(i, t) ∈ FH,i { x(j, s) | j ∈ Ni, s < t}, t .

Ni

Remark 12 (Hyperstructure Property). Each F : P(X ) × T → P(X) is a binary hyperoperation (its

Ni

codomain is a powerset), so P(X ) × T, FH,i is a hyperstructure. Hence the entire hypersystem S carries a hyperstructure of local updates.

Theorem 18 (Recovery of Classical System). If each hyperupdate always returns a singleton,

Ni

FH,i A, t = { Fi(A, t) } for all A ⊆ X , t ∈ T, then S reduces exactly to the usual discrete-time system E, I, X, T, t0, x, (Fi)i∈E .

Proof : Under the singleton assumption, the inclusion x(i, t) ∈ FH,i {x(j, s)}, t becomes the equality x(i, t) = Fi {x(j, s)}, t , so each FH,i behaves like a classical update function Fi. All other data remain the same, recovering the standard system.

Example 17 (Two-Room Thermal HyperSystem with Uncertain Conductance). A thermal system involves components that transfer, store, or convert heat energy, typically governed by thermodynamic principles and temperature variations (cf.[33, 34, 35]). Let E = {A, B}, I = {(A, A), (A, B), (B,A), (B,B)}, X = R, T = {0, 1, 2, . . . }, t0 = 0, where x(i, t) is the temperature (in °C) of room i at day t. Suppose the outside temperature at day t is

2πt

Text(t) = 10 + 5 sin .

We model uncertain heat-transfer coefficients by letting each local hyperupdate return two possible values. Concretely, for NA = NB = {A, B}, define FH,A {TA, TB}, t = 0.6 TA + 0.3 TB + 0.1 Text(t), 0.7 TA + 0.2 TB + 0.1 Text(t) , FH,B {TA, TB}, t = 0.2 TA + 0.7 TB + 0.1 Text(t), 0.3 TA + 0.6 TB + 0.1 Text(t) .

Thus S = (E, I, X, T, t0, x, FH) is a discrete-time hypersystem.

Initial condition and first update. Set x(A, 0) = 20.0, x(B, 0) = 18.0.

At t = 1, Text(1) = 10 + 5 sin ≈ 11.29.

Hence FH,A({20.0, 18.0}, 1) = { 0.6 · 20 + 0.3 · 18 + 0.1 · 11.29, 0.7 · 20 + 0.2 · 18 + 0.1 · 11.29} = {18.5, 18.7}, FH,B({20.0, 18.0}, 1) = { 0.2 · 20 + 0.7 · 18 + 0.1 · 11.29, 0.3 · 20 + 0.6 · 18 + 0.1 · 11.29} = {17.7, 17.9}. By the hypersystem law, x(A, 1) ∈ {18.5, 18.7}, x(B, 1) ∈ {17.7, 17.9}.

Thus the set of possible states at t = 1 is {(18.5, 17.7), (18.5, 17.9), (18.7, 17.7), (18.7, 17.9)} ⊆ X .

Interpretation I. The two values in each hyperupdate reflect uncertainty in the rooms’ thermal conductance.

II. The resulting “hyperstate” at each step is a set of classical temperature vectors.

III. This hyperstructure captures nondeterministic or interval-valued evolution in a single formalism.

Definition 34 (n-SuperHyperSystem). Fix an integer n ≥ 1. Define the iterated powersets P0(X) = X, Pk+1(X) = P Pk(X) , k ≥ 0.

An n-superhypersystem is a tuple

(n) (n)

S = E, I, X, T, t0, x, F , where

(n) (n) (n) N

F = Fi , Fi : Pn X × T −→ Pn(X), and the evolution law is, for t > t0, x(i, t) ∈ Fi { x(j, s)}, t , where {·}(n) indicates selection down through the n–fold powerset.

(n) N

Theorem 19 (n-SuperHyperstructure Property). Each Fi : Pn(X ) × T → Pn(X) is a binary

N (n)

(2, n)-superhyperoperation. Therefore Pn(X ) × T, Fi is a (2, n)-superhyperstructure of local updates.

Proof : By construction, the codomain of each Fi is Pn(X), the n–th iterated powerset of X. This exactly matches the definition of an (m, n)-superhyperoperation (here m = 2 arguments), so the collection F endows S with a superhyperstructure.

Example 18 (Two-Room 2-SuperHyperSystem: Calibration and Model Uncertainty). Consider again two adjacent rooms A, B with state-space X = R (temperature in °C), time-index T = {0, 1, 2, . . . }, initial time t0 = 0, and influence graph I = {(A, A), (A, B), (B, A), (B, B)}. We now model two levels of uncertainty: Level 1 (Hyperupdate): For each room i ∈ {A, B}, the local hyperupdate returns a set of two candidate temperatures reflecting measurement noise:

FH,i {TA, TB}, t = fi (TA, TB, t), fi (TA, TB, t) , where—for example—at t = 1 with TA = 20.0, TB = 18.0 and outside Text(1) ≈ 11.29:

f = 0.6 TA + 0.3 TB + 0.1 Text = 18.5, f = 0.7 TA + 0.2 TB + 0.1 Text = 18.7, FH,A({20, 18}, 1) = {18.5, 18.7}, FH,B({20, 18}, 1) = {17.7, 17.9}.

Level 2 (SuperHyperupdate): We further admit two distinct calibration models, each yielding its own Level 1 hyperupdate. Thus

(2) (cal1) (cal2)

Fi {TA, TB}, t = F ({TA, TB}, t), F ({TA, TB}, t) ,

H,i H,i (cal k)

where each F is itself a Level 1 hyperupdate set. Concretely at t = 1:

F ({20, 18}, 1) = {18.5, 18.7}, {19.0, 18.8} , F ({20, 18}, 1) = {17.7, 17.9}, {18.0, 17.8} .

Resulting 2-SuperHyperstate at t = 1. The collection of all possible Level 1 hyperstates under both calibration models is HA × HB HA ∈ F ({20, 18}, 1), HB ∈ F ({20, 18}, 1) , i.e. the four sets {18.5, 18.7} × {17.7, 17.9}, {18.5, 18.7} × {18.0, 17.8}, {19.0, 18.8} × {17.7, 17.9}, {19.0, 18.8} × {18.0, 17.8}.

Each of these is itself a set of two-room temperature pairs, giving a second-level hyperstructure.

Interpretation.

I. Level 1 captures sensor/measurement noise (two possible readings per room).

II. Level 2 captures calibration model uncertainty (two distinct hyperupdates).

III. The 2-superhyperstate at each time step is a set of hyperstates, each hyperstate being a set of possible temperature pairs.

IV. A nested selection (choose one calibration model, then one reading per room) recovers a classical temperature trajectory.

Example 19 (Discrete-Time Epidemiological SIR Model). The SIR model is an epidemiological model dividing a population into susceptible, infected, and recovered compartments to study disease spread (cf.[36, 37, 38]). We model the spread of an infectious disease in a fixed population of size N. Let E = {S, I,R}, X = [0, N] ⊂ R, T = {0, 1, 2, . . . }, t0 = 0.

Here x(S, t), x(I,t), x(R, t) denote the numbers of Susceptible, Infectious, and Recovered individuals at day t. Define the influence relation I = {(S, S), (S, I), (I,S), (I,I), (I,R), (R, R)}, so that NS = {S, I}, NI = {S, I}, NR = {I,R}.

Choose infection rate β > 0 and recovery rate γ > 0. The local update functions F = (FS, FI, FR) are St It FS (St, It), t = St − β , St It FI (St, It), t = It + β − γ It, FR (Rt, It), t = Rt + γ It.

Thus for each i ∈ {S, I, R} and t > 0, x(i, t) = Fi (x(j, t − 1))j∈N , t − 1 .

Parameters and Initial State Let N = 1000, β = 0.3/N, γ = 0.1, and x(S, 0) = 999, x(I, 0) = 1, x(R, 0) = 0.

First Day (t = 1) Compute new infections and recoveries:

999 · 1 β ≈ 0.2997, γ · 1 = 0.1.

Hence x(S, 1) = 999 − 0.2997 ≈ 998.7003, x(I, 1) = 1 + 0.2997 − 0.1 ≈ 1.1997, x(R, 1) = 0 + 0.1 = 0.1.

Second Day (t = 2) With S1 ≈ 998.7003, I1 ≈ 1.1997:

998.7003 · 1.1997 β ≈ 0.3595, γ · 1.1997 ≈ 0.11997.

Thus x(S, 2) ≈ 998.7003 − 0.3595 = 998.3408, x(I, 2) ≈ 1.1997 + 0.3595 − 0.11997 = 1.4392, x(R, 2) ≈ 0.1 + 0.11997 = 0.21997.

Interpretation I. St decreases by the number of new infections β StIt/N .

II. It increases by new infections and decreases by recoveries γIt.

III. Rt increases by γIt.

IV. This discrete-time system captures “memoryless” transitions in an epidemic, suitable for daily-step simulations.

For reference, an overview of discrete-time systems and their hyper/superhyper extensions is provided in Table 6.

Aspect Discrete-Time System Discrete-Time HyperSys- Discrete-Time n-tem SuperHyperSystem Base object A tuple S = A tuple SH = A tuple S = (E, I, X, T, t0, x, (Fi)i∈E) (E, I, X, T, t0, x, (FH,i)i∈E) (E, I, X, T, t0, x, (F )i∈E) with deterministic local with set-valued local updates. with n-fold set-valued local updates. updates.

Time index Discrete T = {0, 1, 2, . . . } (or Same T as classical. Same T as classical.

any totally ordered set with minimum t0).

Influence structure I ⊆ E × E, with neighbor- Same I and Ni. Same I and Ni.

hood Ni = {j ∈ E | (j, i) ∈ State space X (e.g., R, [0, N], finite alpha- X (underlying crisp state X (underlying crisp state bets). space remains unchanged). space remains unchanged).

N N (n) N

i i i

Local update type Fi : X × T → X. FH,i : P(X ) × T → P(X). F : Pn(X ) × T → Pn(X), where P0(Y ) = Y and Pk+1(Y ) = P(Pk(Y )). Evolution law (one step) For t > t0, x(i, t) = For t > t0, x(i, t) ∈ For t > t0, x(i, t) is obtained Fi (x(j, s))j∈N , s<t, t (or by nested selection from the i FH,i {x(j, s) | j ∈ Ni, s < in Markov form x(i, t) = n-level output F (·, t) ∈ Fi((x(j, t − 1))j∈N , t)). t}, t . Pn(X) (choose down through n powerset levels).

Uncertainty meaning No intrinsic uncertainty: one Nondeterministic / multi- Hierarchical uncertainty: sets next state per component at valued update: several admis- of sets of · · · of states (up each time. sible next states at each time. to n levels), e.g., uncertainty about models, then about pa-rameters, then about mea-surements.

Recovery of classical case — (baseline). If FH,i(A, t) = {Fi(A, t)} If each F is built as n (singleton output) for all i, nested singleton-lifts of a de-then the hypersystem col- terministic Fi, then S col-lapses to the classical system. lapses to the classical system after unwrapping.

Typical examples Discrete SIR, discrete ther- Thermal exchange with uncer- Two-level calibration mal exchange, finite-state au- tain conductance (two candi- + sensor-noise (a 2- tomata. date updates), nondetermin- superhypersystem), multi-tier istic control policies. model ensembles with nested scenario sets.