EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6834
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Hierarchical Uncertainty Modeling via
(m,n)-Superhyperuncertain

and (h, k)-ary (m,n)-Superhyperuncertain Sets:
Unified Extensions of Fuzzy, Neutrosophic, Soft, Rough,

and Plithogenic Set Theories

Takaaki Fujita1,∗, Florentin Smarandache2

1 Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan.
2 University of New Mexico, Gallup Campus, NM 87301, USA.

Abstract. Various set-theoretic frameworks have been widely recognized for their effectiveness in
handling uncertainty, including Fuzzy Sets, Neutrosophic Sets, Plithogenic Sets, Rough Sets, and
Soft Sets. These foundational models have been further extended through the use of hyperstruc-
tures—based on the powerset construction—and superhyperstructures—based on the n-th-order
powerset, obtained by iteratively applying the powerset operation [1, 2]. These extended con-
structs are collectively referred to as HyperUncertain Sets and SuperHyperUncertain Sets.
Research on SuperHyperUncertain Sets is still in its early stages, and investigations into their
properties, extended forms, and potential applications are expected to become increasingly sig-
nificant in the future. In this paper, we propose two new, more general frameworks: the (m,n)-
SuperHyperUncertain Set and the (h, k)-ary (m,n)-SuperHyperUncertain Set. These new struc-
tures represent a concrete and refined reconsideration of the foundational concepts introduced
in [1, 2]. It is anticipated that the concepts developed in this work can be effectively applied to
the modeling of more hierarchical forms of uncertainty, as well as to scenarios requiring complex
membership functions. Since this paper conducts only theoretical analysis, we also hope that
quantitative analysis using computational methods will be carried out in the future.

2020 Mathematics Subject Classifications: 03E72

Key Words and Phrases: HyperFuzzy Set, HyperNeutrosophic Set, HyperPlithogenic Set, Hy-
perRough Set, HyperSoft Set

1. Introduction

Set theory is one of the most fundamental areas of mathematics[3]. However, classical
sets cannot adequately represent the many forms of uncertainty present in the real world.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6834

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), 6834 2 of 69

Reasoning with incomplete, vague, or heterogeneous information has therefore motivated
the development of a rich family of uncertain set models [4]. Some of the most widely
studied examples include Fuzzy Sets [5], Intuitionistic Fuzzy Sets [6], Vague Sets [7],
Soft Sets [8], Rough Sets [9], Neutrosophic Sets [10], QuadriPartitioned Neutrosophic
Sets[11, 12], HeptaPartitioned Neutrosophic Sets[13, 14], and Plithogenic Sets [15]. For
the reader’s reference, a concise overview of common uncertain–set families is provided in
Table 1.

Table 1: Concise overview of common uncertain–set families.

Family Core semantics Key ref. Representative
extensions

Fuzzy Set Membership µ(x)∈ [0, 1]. [5] Hesitant [16]; Spherical
[17].

Intuitionistic / Vague Set Two degrees (µ, ν) for member-
ship / non–membership.

[7, 18] —

Soft Set Parameterized family E →
P(U).

[8] —

Rough Set Lower / upper approximations
via indiscernibility.

[9] Granular [19];
Probabilistic [20].

Neutrosophic Set Triple (T, I, F ) with 0 ≤
T+I+F ≤3.

[10] Bipolar [21];
Interval–Valued [22].

Plithogenic Set Multi–attribute appurtenance
with contradiction degree.

[15] —

These Uncertain Set frameworks have been successfully adapted to diverse mathemat-
ical and applied domains—including graph theory, topology, and knowledge representa-
tion—offering flexible abstractions for modeling uncertainty at multiple levels of granular-
ity. These Uncertain Sets are studied not only in mathematics but also in a wide range of
applications across decision science, engineering, and computer science (e.g.[23–25]).

Next, we explain the notions of hyperstructures, superhyperstructures, hyperuncertain
sets, and superhyperuncertain sets. Many classical mathematical structures admit two
natural and systematic forms of enrichment. The first replaces a base set S with its
powerset P(S) and endows it with hyperoperations, producing what are called hyperstruc-
tures [26, 27]. The second applies the powerset construction iteratively n times, yielding
superhyperstructures on P n(S), which are capable of encoding nested or hierarchical inter-
actions [28, 29]. This perspective is closely related to the theory of hypergraphs [30, 31] and
their higher-order analogues, superhypergraphs [32, 33], which generalize pairwise connec-
tions to multiway and multi-level relationships. Further examples of superhyperstructures
include SuperHyperAlgebra[34, 35] and Chemical SuperHyperStructures[36, 37]. Because
of their ability to represent hierarchical, multi-scale relationships, superhyperstructures
are expected to find applications in modeling a wide range of complex real-world systems.
For reference, a Concise overview of Classical, Hyper, and SuperHyper Structures is pro-
vided in Table 2. Unless otherwise specified, values such as n and m in this paper are
nonnegative integers. Moreover, throughout we work only with finite sets; issues related
to infinity are not considered.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 3 of 69

Table 2: Concise overview of Classical, Hyper, and SuperHyper Structures.

Notion Carrier Essence Typical instances / refs.

Classical
Structure

S (base set) Operations are defined directly on
elements of S (e.g., groups, rings,
topologies). Classical structures
capture pairwise or elementwise
interactions without higher-level
nesting.

Groups, Rings, Topological
Spaces, Graphs. See [30].

HyperStructure P(S) (powerset of
S)

Replace S with P(S) and
introduce hyperoperations acting
on subsets rather than single
elements. This enables
multi-valued or set-valued outputs,
capturing richer interactions.

Hypergroups, Hyperrings,
Hypergraphs. See [26, 27,
31].

SuperHyper
Structure

P n(S) (n≥1
iterated powerset)

Extend hyperstructures by
iterating the powerset n times,
thereby modeling nested and
hierarchical uncertainty and
multi-level interactions. Admits
multi-ary inputs and multi-layered
outputs.

SuperHypergraphs [32, 33];
SuperHyperAlgebra
[34, 35]; Chemical Super-
HyperStructures [36, 37].

In parallel with these structural enrichments, the notion of uncertain sets has itself
been ‘hyperized” and ‘superhyperized.” Specifically, HyperFuzzy [38, 39], HyperSoft [40–
42], HyperRough [43, 44], HyperNeutrosophic [45], and HyperPlithogenic Sets [46] arise
by imposing the corresponding semantics on the powerset P(S) [1, 2]. Their superhyper
counterparts—such as SuperHyperFuzzy [47], SuperHyperNeutrosophic[2], and related vari-
ants—are defined over the n-th powerset P n(S) and are designed to capture uncertainty
at multiple hierarchical levels [1]. For reference, a concise overview of Uncertain, Hype-
rUncertain, and SuperHyperUncertain Sets is provided in Table 3. Needless to say, these
frameworks are expected to find applications in various fields, such as decision–making and
beyond. Note that type-n uncertain sets refine uncertainty inside membership values—e.g.,
type-n fuzzy [48] treats degrees-of-membership as higher-order quantities. SuperHyper-
Uncertain sets move uncertainty to the elements themselves via iterated powersets, cap-
turing multi-level groupings and interactions; this enables explicit hierarchical modeling,
compositional operations, and scalable reasoning across nested attributes and real-world
scenarios.

From the above discussion, it is clear that research on HyperUncertain Sets and Super-
HyperUncertain Sets is of great importance, both mathematically and in applied math-
ematics. Nevertheless, the systematic study of HyperUncertain Sets—including Hyper-
Fuzzy, HyperVague, HyperSoft, HyperRough, HyperNeutrosophic, and HyperPlithogenic
models—remains comparatively young despite their clear potential for modeling complex
data. Existing frameworks such as HyperFuzzy or n–SuperHyperFuzzy cannot adequately
address real–world scenarios like multi–modal medical diagnostics, hierarchical IoT sensor
fusion, or cross–domain financial risk analysis involving layered uncertainties.

To bridge this gap, in this work we introduce and formalize the (m,n)-SuperHyperUncertain



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 4 of 69

Table 3: Concise overview of Uncertain, HyperUncertain, and SuperHyperUncertain Sets.

Notion Carrier Essence Typical instances /
refs.

Uncertain Set S (or
model-dependent
P(S))

Attach uncertainty semantics to
elements or subsets, such as graded
membership, parameterization,
lower/upper approximations,
neutrosophic triples, or
contradiction degree.

Fuzzy [5]; Soft [8];
Rough [9]; Neu-
trosophic [10];
Plithogenic [15].

HyperUncertain
Set

P(S) Impose these semantics on the
powerset; membership values
become set–valued, allowing
hesitation and multi–valued
appurtenance for richer modeling of
uncertainty.

HyperFuzzy [38, 39]; Hy-
perSoft [40, 42]; Hyper-
Rough [43, 44]; Hyper-
Neutrosophic [45]; Hy-
perPlithogenic [46].

SuperHyper-
Uncertain Set

P n(S) (n≥1) Lift semantics to the n-th iterated
powerset to capture hierarchical and
multi–level uncertainty, with the
ability to admit multi–ary inputs
and outputs simultaneously.

SuperHyperFuzzy, Su-
perHyperVague, and
other variants [1].

Set, which assigns uncertain semantics on the n-fold iterated powerset P n(·) and ad-
mits m-ary superhyperoperations. We further develop the more general (h, k)-ary (m,n)-
SuperHyperUncertain Set, enabling multiple input and output arities at each hierarchi-
cal level. Taken together, these constructions unify and extend existing uncertain–set
paradigms within the framework of superhyperstructures, yielding a scalable toolkit for
hierarchical and multi–attribute uncertainty while clarifying how classical models arise via
restriction, projection, and composition across levels.

For the reader’s reference, Table 4 provides a compact overview of n–, (m,n)–, and
(h, k)–ary (m,n)–SuperHyperUncertain Sets.

Table 4: Compact overview of n–, (m,n)–, and (h, k)–ary (m,n)–Superhyperuncertain Sets.

Notion Carrier / Signature Essence (very brief)

n–Superhyperuncertain Set Carrier: P n(S)
Sig.: F : P n(S) → D(n)

Uncertainty on the n-th powerset (hierar-
chical levels). Recovers classical for n=0,
hyper for n=1.

(m,n)–Superhyperuncertain
Set

Carrier: P n(S)
Sig.: F : (P n(S))m → D(n)

Adds m-ary interaction at level n; m=1
gives the n–case.

(h, k)–ary
(m,n)–Superhyperuncertain
Set

Carrier: inputs from P n(S)
Sig.:
F : (P n(S))h → (D(n))k

Most general: h inputs, k outputs at level
n. Reduces to (m,n)–case when h=1, k=1.

Note. D(n) denotes an n–level degree family (e.g., P̃n([0, 1]) for fuzzy, P̃n([0, 1]
3) for neutrosophic).

Together, these frameworks capture hierarchical uncertainty by assigning set-valued
memberships across iterated powersets and permitting variable input–output arities. They
unify fuzzy, neutrosophic, soft, rough, and plithogenic semantics, support compositional
reasoning, and scale to complex, nested attributes through well-defined operations and



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 5 of 69

cut-based analyses. Since this paper conducts only theoretical analysis, we also hope that
quantitative analysis using computational methods will be carried out in the future.

A concise, section-level outline is provided in Table 5.

Table 5: Section-level contents (Introduction omitted).

Section Summary

2. Preliminaries Notation and core constructions: powerset P(S) and iterates Pn(S); hy-
peroperations and hyperstructures; n-superhyperstructures on Pn(S);
(h, k)-ary SuperHyperstructure; brief, concrete examples.

3. Main Results Definition of (m,n)–Superhyperuncertain Sets. Subsections:
(i) SuperhyperFuzzy—restriction/projection in m,n, closure
(union/intersection), nested α-cuts, functoriality; (ii) Superhyper-
Neutrosophic—recovery of m=1 case, truth-projection to fuzzy, λ-cuts,
closure, functoriality; (iii) SuperhyperPlithogenic—HDAF/DCF ax-
ioms and decision-making examples.

4. Additional Result (h, k)-ary (m,n)–SuperhyperSoft and SuperhyperRough frameworks:
formal definitions over P̃m(S) and P̃n(U)/P̃n(X); unary reduction
lemmas; fixing/projection of inputs/outputs; closure under pointwise
union/intersection; functorial pushforwards (surjections/quotients); il-
lustrative cases (personalized recommendations; medical diagnosis).

5. Conclusion and Future Work Summary of the unified superhyper framework; prospective applications
to hierarchical uncertainty and complex memberships; outlook on algo-
rithms and domain-specific case studies.

2. Preliminaries

In this section, we summarize the basic definitions and notational conventions used in
the paper. Throughout, we work only with finite sets; issues related to infinity are not
considered.

2.1. SuperHyperstructure

In this context, a Classical Structure refers to any mathematical or real–world struc-
ture or concept. To capture such structures in a hierarchical manner, HyperStructures
are introduced. A Hyperstructure is a mathematical framework in which operations on
elements return sets rather than single outcomes, thereby enabling multivalued algebraic
relations on a base set[28, 29]. A SuperHyperstructure generalizes hyperstructures by
permitting operations on higher–order collections such as iterated powersets, supporting
multiple input and output arities, and modeling layered, hierarchical uncertainty [28, 29].

First, we record formal definitions of the powerset, the n-th powerset, and their non-
empty variants, followed by an illustrative example. Note that the n-th powerset of a set
S is obtained by iterating the powerset operator n times, producing Pn(S) = P(Pn−1(S)),
thereby encoding higher-order collections of subsets and supporting layered, hierarchical
structures [28, 29].



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 6 of 69

Definition 1 (Base Set). A base set S is the underlying set from which constructions
such as powersets, hyperstructures, and superhyperstructures are built. Formally,

S = {x | x is an element in the specified domain}.

All elements of objects like P(S) or Pn(S) are ultimately drawn from S.

Definition 2 (Powerset). The powerset of a set S, denoted P(S), is the set of all subsets
of S (including ∅ and S itself):

P(S) = {A | A ⊆ S}.

We also write the non-empty powerset as

P∗(S) = P(S) \ {∅}.

Definition 3 (n-th powerset and n-th non-empty powerset). (cf.,[28, 29]) For a set H
and an integer n ≥ 1, define recursively

P1(H) = P(H), Pn+1(H) = P
(
Pn(H)

)
.

Similarly, set

P∗
1 (H) = P∗(H), P∗

n+1(H) = P∗(P∗
n(H)

)
.

Thus Pn(H) (resp. P∗
n(H)) is obtained by iterating the (non-empty) powerset operator

n times.

Example 1 (n-th powerset: a concrete instance). Let H = {a, b}. Then

P1(H) = P(H) =
{
∅, {a}, {b}, {a, b}

}
.

The second powerset P2(H) = P
(
P1(H)

)
is the set of all subsets of P1(H). Since

P1(H) has four elements, P2(H) has 24 = 16 elements, which we list by cardinality:

size 0 :
{
∅
}
,

size 1 :
{
{∅}

}
,
{
{{a}}

}
,
{
{{b}}

}
,
{
{{a, b}}

}
,

size 2 :
{
{∅}, {{a}}

}
,
{
{∅}, {{b}}

}
,
{
{∅}, {{a, b}}

}
,{

{{a}}, {{b}}
}
,
{
{{a}}, {{a, b}}

}
,
{
{{b}}, {{a, b}}

}
,

size 3 :
{
{∅}, {{a}}, {{b}}

}
,
{
{∅}, {{a}}, {{a, b}}

}
,{

{∅}, {{b}}, {{a, b}}
}
,
{
{{a}}, {{b}}, {{a, b}}

}
,

size 4 :
{
{∅}, {{a}}, {{b}}, {{a, b}}

}
.

In particular, P2(H) is the Boolean algebra on the four generators ∅, {a}, {b}, {a, b}.
Higher iterations Pn(H) are obtained by repeating this construction.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 7 of 69

To establish a formal foundation for the concepts of Hyperstructures and Superhyper-
structures, we present the following definitions and propositions.

Definition 4 (Classical Structure). (cf.[28, 49]) A Classical Structure is a mathematical
framework defined on a non-empty set H, equipped with one or more Classical Operations
that satisfy specified Classical Axioms. Specifically:

A Classical Operation is a function of the form:

#0 : Hm → H,

where m ≥ 1 is a positive integer, and Hm denotes the m-fold Cartesian product of
H. Common examples include addition and multiplication in algebraic structures such
as groups, rings, and fields.

Definition 5 (Hyperoperation). (cf.[50, 51]) A hyperoperation is a generalization of a
binary operation where the result of combining two elements is a set, not a single element.
Formally, for a set S, a hyperoperation ◦ is defined as:

◦ : S × S → P(S),

where P(S) is the powerset of S.

Definition 6 (Hyperstructure). (cf.[26, 28, 52]) A Hyperstructure extends the notion of
a Classical Structure by operating on the powerset of a base set. Formally, it is defined as:

H = (P(S), ◦),

where S is the base set, P(S) is the powerset of S, and ◦ is an operation defined on subsets
of P(S). Hyperstructures allow for generalized operations that can apply to collections of
elements rather than single elements.

Example 2 (Grocery Substitution as a Hyperstructure). Let

S = {Milk, SoyMilk,Bread,GFBread,Eggs}.

Define a substitution map σ : S → P(S) by σ(Milk) = {Milk,SoyMilk}, σ(Bread) =
{Bread,GFBread}, and σ(x) = {x} for x ∈ {SoyMilk,GFBread,Eggs}. Set a hyperopera-
tion on items

⋆ : S × S → P(S), a ⋆ b = σ(a) ∪ σ(b),

and induce an operation on baskets (subsets) by

⊙ : P(S) × P(S) → P(S), A⊙B =
⋃

a∈A, b∈B
(a ⋆ b).

Then H = (P(S),⊙) is a Hyperstructure modelling shopping lists with permissible substi-
tutions. For instance,

{Milk,Bread} ⊙ {Eggs} = {Milk, SoyMilk,Bread,GFBread,Eggs}.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 8 of 69

Definition 7 (SuperHyperOperations). (cf.[28]) Let H be a non-empty set, and let P(H)
denote the powerset of H. The n-th powerset Pn(H) is defined recursively as follows:

P0(H) = H, Pk+1(H) = P(Pk(H)), for k ≥ 0.

A SuperHyperOperation of order (m,n) is an m-ary operation:

◦(m,n) : Hm → Pn
∗ (H),

where Pn
∗ (H) represents the n-th powerset of H, either excluding or including the empty

set, depending on the type of operation:

• If the codomain is Pn
∗ (H) excluding the empty set, it is called a classical-type (m,n)-

SuperHyperOperation.

• If the codomain is Pn(H) including the empty set, it is called a Neutrosophic (m,n)-
SuperHyperOperation.

These SuperHyperOperations are higher-order generalizations of hyperoperations, cap-
turing multi-level complexity through the construction of n-th powersets.

Definition 8 (n-Superhyperstructure). (cf.[28, 29, 49, 53]) An n-Superhyperstructure
further generalizes a Hyperstructure by incorporating the n-th powerset of a base set. It is
formally described as:

SHn = (Pn(S), ◦),

where S is the base set, Pn(S) is the n-th powerset of S, and ◦ represents an operation
defined on elements of Pn(S). This iterative framework allows for increasingly hierarchical
and complex representations of relationships within the base set.

Example 3 (Feature-Flag Cohorts as a 2-Superhyperstructure). In a modern microservice
architecture, a “feature flag” system often needs to manage nested groups of users for A/B
testing and gradual roll-outs. Let

S = {FeatureA, FeatureB, FeatureC}

be the base set of all flags. Then

P1(S) = P(S) and P2(S) = P
(
P(S)

)
are respectively all subsets of flags (cohorts) and all subsets of cohorts (collections of test
groups).

Define a binary operation

merge : P2(S) × P2(S) −→ P2(S)

by
merge(C1, C2) =

{
g1 ∪ g2 | g1 ∈ C1, g2 ∈ C2

}
.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 9 of 69

Then
SH2 =

(
P2(S), merge

)
is a concrete 2-Superhyperstructure: each element is a collection of cohorts, and merging
two collections produces every possible union of one cohort from each.

The definition of the (h, k)-ary SuperHyperstructure, which is known as a further
generalization of the above superhyper-framework, is provided below.

Definition 9 ((h, k)-ary SuperHyperstructure). Let S be a nonempty base set. For each
i ∈ {1, 2, . . . , h} choose a nonempty subset Ai ⊆ S and fix an integer mi ≥ 0. Denote by

Pmi(Ai)

the mi-th iterated powerset of Ai. Similarly, for each j ∈ {1, 2, . . . , k} choose a nonempty
subset Bj ⊆ S and fix an integer nj ≥ 0, and denote by

Pnj (Bj)

the nj-th iterated powerset of Bj.
Define the domain D and codomain C as

D = Pm1(A1) × Pm2(A2) × · · · × Pmh
(Ah),

C = Pn1(B1) × Pn2(B2) × · · · × Pnk
(Bk).

An (h, k)-ary SuperHyperstructure on S is an algebraic system

SH =
(
D, C, {◦α}α∈I

)
,

where {◦α}α∈I is an indexed family of SuperHyperoperations

◦α : D −→ C, α ∈ I.

For each (X1, . . . , Xh) ∈ D, one has

◦α(X1, . . . , Xh) = (Y1, . . . , Yk) ∈ C,

with Yj ∈ Pnj (Bj) for all j. The structural properties imposed on the maps ◦α—such
as associativity, commutativity, or distributivity—are specified according to the algebraic
framework adopted, thereby extending classical algebraic systems into a higher-order su-
perhyperstructural setting.

Example 4 (Collaborative-Filtering as a (2, 2)-ary SuperHyperstructure). In a recom-
mendation system, one often fuses information about user cohorts and item-sets to produce
both new item suggestions and peer cohorts. Let

S = U ∪ P

be the union of all users U = {Hiroko,Masahiro, Shinya, . . . } and all products P =
{iPhone,Galaxy,Pixel, . . . }.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 10 of 69

• Choose h = 2 inputs:

A1 = U, m1 = 1, A2 = P, m2 = 1.

Then Pm1(A1) = P(U) is the set of all user-cohorts, and Pm2(A2) = P(P ) is the
set of all item-sets.

• Choose k = 2 outputs:

B1 = P, n1 = 1, B2 = U, n2 = 1.

Then Pn1(B1) = P(P ) are the recommended items, and Pn2(B2) = P(U) are the
peer cohorts.

Thus the domain and codomain are

D = P(U) × P(P ), C = P(P ) × P(U).

Define a single SuperHyperOperation

◦ : D −→ C

by collaborative-filtering logic, for example

◦
(
X,Y

)
=

(
RecItems(X,Y ), SimUsers(X,Y )

)
,

where

RecItems(X,Y ) =
{
p ∈ P | ∃u ∈ X, u liked p′ ∈ Y, and other users in X also liked p},

SimUsers(X,Y ) =
{
u′ ∈ U | u′ /∈ X, ∃ p ∈ Y with u′ liked p

}
.

For instance,

◦
(
{Hiroko,Masahiro}, {iPhone,Galaxy}

)
=

(
{Pixel,OnePlus}, {Shinya,Dan}

)
.

Hence
SH =

(
D, C, {◦}

)
is a concrete (2, 2)-ary SuperHyperstructure modeling collaborative-filtering in practice.

Example 5 (Smart Home Automation as a (2, 2)-ary SuperHyperstructure). Consider a
smart home system where sensor events and notification preferences determine both device
actions and alert recipients. Let the base set be

S = {MotionSensor, DoorSensor, SmokeSensor,

LightSwitch, Alarm, OwnerApp, SecurityService}.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 11 of 69

Choose two input subsets:

A1 = {MotionSensor, DoorSensor, SmokeSensor}, m1 = 1,

A2 = {OwnerApp, SecurityService}, m2 = 1,

so that Pm1(A1) = P(A1) is the set of all active sensor-subsets, and Pm2(A2) = P(A2)
is the set of all notification-channel configurations. Similarly, choose two output subsets:

B1 = {LightSwitch, Alarm}, n1 = 1,

B2 = {OwnerApp, SecurityService}, n2 = 1,

so that Pn1(B1) = P(B1) are possible device-action sets, and Pn2(B2) = P(B2) are
possible recipient sets.

Thus the domain and codomain are

D = P(A1) × P(A2), C = P(B1) × P(B2).

Define a single SuperHyperOperation

◦ : D −→ C

by
◦(X,Y ) =

(
Actions(X), Recipients(X,Y )

)
,

where

Actions(X) =


{LightSwitch}, if MotionSensor ∈ X,

{Alarm}, if SmokeSensor ∈ X,

∅, otherwise,

Recipients(X,Y ) = { r ∈ Y | r is configured}.

For example,

◦
(
{MotionSensor,SmokeSensor}, {OwnerApp}

)
=

(
{LightSwitch,Alarm}, {OwnerApp}

)
.

Therefore,
SH =

(
D, C, {◦}

)
is a concrete (2, 2)-ary SuperHyperstructure modeling a real-world smart home automation
workflow.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 12 of 69

2.2. Fuzzy Set

The Fuzzy Set is a well-known concept used to address uncertainty in set theory[5,
54, 55]. These sets can be extended into Hyperfuzzy Sets[56, 57] and SuperHyperfuzzy
Sets[58] using hyperstructures and superhyperstructures. The definition is provided below
.

Definition 10 (Fuzzy Set). [5] A fuzzy set τ in a non-empty universe Y is a mapping
τ : Y → [0, 1]. A fuzzy relation on Y is a fuzzy subset δ in Y × Y . If τ is a fuzzy set in
Y and δ is a fuzzy relation on Y , then δ is called a fuzzy relation on τ if

δ(y, z) ≤ min{τ(y), τ(z)} for all y, z ∈ Y.

Definition 11 (HyperFuzzy Set). [2, 59–61] Let X be a non-empty set. A hyperfuzzy set
over X is defined as a mapping:

µ̃ : X → P([0, 1]) \ {∅},

where P([0, 1]) \ {∅} represents the family of all non-empty subsets of the interval [0, 1].
For each element x ∈ X, µ̃(x) assigns a non-empty subset of [0, 1], representing the

possible membership degrees of x in the hyperfuzzy set. This definition generalizes classical
fuzzy sets by allowing the membership degree of each element to be a range (set of values)
instead of a single scalar value.

Example 6 (Restaurant Customer Satisfaction). Let

X = {R1, R2, R3}

be a set of restaurants. We define a hyperfuzzy set µ̃ : X → P([0, 1]) \ {∅} modeling
customer satisfaction as follows:

µ̃(R1) = [0.7, 0.9], µ̃(R2) = {0.5, 0.6, 0.7}, µ̃(R3) = [0.2, 0.4] ∪ {0.6}.

Here:

• µ̃(R1) = [0.7, 0.9] reflects that customers’ satisfaction scores for R1 range continu-
ously from 0.7 to 0.9.

• µ̃(R2) = {0.5, 0.6, 0.7} indicates that satisfaction for R2 clusters at three discrete
levels.

• µ̃(R3) = [0.2, 0.4] ∪ {0.6} shows a mixed pattern: a continuous range for most cus-
tomers plus one high outlier at 0.6.

This hyperfuzzy set captures the variability and uncertainty in customer satisfaction across
different restaurants.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 13 of 69

Definition 12 (n-SuperHyperFuzzy Set). [1, 2] Let X be a non-empty set. The n-
SuperHyperFuzzy Set is a recursive generalization of fuzzy sets, hyperfuzzy sets, and su-
perhyperfuzzy sets. It is defined as:

µ̃n : P̃n(X) → P̃n([0, 1]),

where:

• P̃1(X) = P̃(X), and for k ≥ 2,

P̃k(X) = P̃(P̃k−1(X)),

represents the k-th nested family of non-empty subsets of X.

• P̃n([0, 1]) is similarly defined for the interval [0, 1].

• µ̃n assigns to each element A ∈ P̃n(X) a non-empty subset µ̃n(A) ⊆ [0, 1], repre-
senting the degrees of membership associated with A at the n-th level.

Example 7 (2-SuperHyperFuzzy Set for Equipment Reliability). Let

X = {Compressor, Pump}.

Then
P̃1(X) =

{
{Compressor}, {Pump}, {Compressor,Pump}

}
,

and
P̃2(X) = P̃

(
P̃1(X)

)
,

the family of all nonempty collections of nonempty subsets of X. We define a 2-SuperHyperFuzzy
Set µ̃2 : P̃2(X) → P̃2([0, 1]) by specifying its value on one representative element:

A =
{
{Compressor}, {Pump}

}
∈ P̃2(X),

µ̃2(A) =
{
{0.70, 0.75}, [0.60, 0.65], {0.50, 0.55}

}
.

Here each inner set (or interval) is a subset of [0, 1], representing possible reliability scores
from different assessment methods (e.g. expert judgment, sensor analytics, historical data).
Thus µ̃2(A) captures the multi-level uncertainty in the joint reliability evaluation of the
Compressor and Pump.

Example 8 (3-SuperHyperFuzzy Set for Predictive Maintenance). Let

X = {Sensor1, Sensor2}.

We construct the nested powersets:

P̃1(X) =
{
{Sensor1}, {Sensor2}, {Sensor1,Sensor2}

}
,



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 14 of 69

P̃2(X) = P̃
(
P̃1(X)

)
, P̃3(X) = P̃

(
P̃2(X)

)
.

Choose the element

A =
{
{{Sensor1}, {Sensor2}}, {{Sensor1, Sensor2}}

}
∈ P̃3(X).

A 3-SuperHyperFuzzy Set is a mapping

µ̃3 : P̃3(X) −→ P̃3([0, 1]).

We define its value on A by

µ̃3(A) =
{
B1, B2

}
⊆ P̃2([0, 1]),

where
B1 =

{
[0.80, 0.90], {0.85}

}
, B2 =

{
[0.60, 0.70], {0.65, 0.68}

}
.

Each Bi is a non-empty subset of P̃([0, 1]), representing multiple reliability scores from dif-
ferent diagnostic methods (e.g. expert judgment vs. sensor analytics). Thus µ̃3(A) captures
the third-level, nested uncertainty in the combined sensor evaluation.

2.3. Neutrosophic Set

A Neutrosophic Set models uncertainty using three membership functions: truth (T ),
indeterminacy (I), and falsity (F ), which satisfy:

0 ≤ T + I + F ≤ 3.

[62, 63]. These sets can be extended into HyperNeutrosophic Sets [64] and SuperHyper-
Neutrosophic Sets using hyperstructures and superhyperstructures.

Definition 13 (Neutrosophic Set). [10, 63] Let X be a non-empty set. A Neutrosophic
Set (NS) A on X is characterized by three membership functions:

TA : X → [0, 1], IA : X → [0, 1], FA : X → [0, 1],

where for each x ∈ X, the values TA(x), IA(x), and FA(x) represent the degrees of truth,
indeterminacy, and falsity, respectively. These values satisfy the following condition:

0 ≤ TA(x) + IA(x) + FA(x) ≤ 3.

Definition 14 (HyperNeutrosophic Set). [1, 2] Let X be a non-empty set. A mapping
µ̃ : X → P̃ ([0, 1]3) is called a HyperNeutrosophic Set over X, where P̃ ([0, 1]3) denotes
the family of all non-empty subsets of the unit cube [0, 1]3. For each x ∈ X, µ̃(x) ⊆
[0, 1]3 represents a set of neutrosophic membership degrees, each consisting of truth (T ),
indeterminacy (I), and falsity (F ) components, satisfying:

0 ≤ T + I + F ≤ 3.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 15 of 69

Example 9 (Medical Diagnosis Application). Let X = {Hiroko,Masahiro} be a set of
patients undergoing diagnosis for a particular disease. We model the uncertain evaluation
of each patient’s condition by a hyperneutrosophic set

µ̃ : X −→ P̃
(
[0, 1]3

)
.

Here each element of µ̃(x) ⊆ [0, 1]3 is a triple (T, I, F ) giving a possible assessment of
Truth (presence), Indeterminacy, and Falsity (absence) of the disease.

For example:

µ̃(Hiroko) =
{

(0.85, 0.10, 0.05), (0.80, 0.15, 0.05)
}
,

µ̃(Masahiro) =
{

(0.60, 0.20, 0.20), (0.65, 0.25, 0.10)
}
.

Each triple satisfies 0 ≤ T + I + F ≤ 3. The two points in µ̃(x) reflect, for instance, two
independent expert opinions or two different diagnostic tests, capturing both the variability
among assessments and the underlying neutrosophic uncertainty.

Definition 15 (n-SuperHyperNeutrosophic Set). [1, 2] Let X be a non-empty set. An
n-SuperHyperNeutrosophic Set is a recursive generalization of Neutrosophic Sets, Hyper-
Neutrosophic Sets, and SuperHyperNeutrosophic Sets. It is defined as:

Ãn : P̃n(X) → P̃n([0, 1]3),

where:

• P̃1(X) = P̃(X), and for k ≥ 2,

P̃k(X) = P̃(P̃k−1(X)),

represents the k-th nested family of non-empty subsets of X.

• P̃n([0, 1]3) is similarly defined for the unit cube [0, 1]3.

• The mapping Ãn assigns to each A ∈ P̃n(X) a subset Ãn(A) ⊆ [0, 1]3, representing
the degrees of truth (T ), indeterminacy (I), and falsity (F ) for the n-th level subsets
of X.

For each A ∈ P̃n(X) and (T, I, F ) ∈ Ãn(A), the following condition is satisfied:

0 ≤ T + I + F ≤ 3,

where T , I, and F represent the truth, indeterminacy, and falsity degrees, respectively.

Example 10 (Hierarchical Fault Diagnosis in an Industrial System). Let

X = {TempSensor , PressureSensor}



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 16 of 69

be the set of two critical sensors in an industrial process. We consider n = 2, so that

P̃1(X) =
{
{TempSensor}, {PressureSensor}, {TempSensor ,PressureSensor}

}
,

and
P̃2(X) = P̃

(
P̃1(X)

)
,

whose elements are nonempty collections of nonempty subsets of X. For example, choose

A =
{
{TempSensor}, {PressureSensor}

}
∈ P̃2(X).

A 2-SuperHyperNeutrosophic Set on X is a mapping

Ã2 : P̃2(X) −→ P̃2

(
[0, 1]3

)
.

We may define, for instance,

Ã2(A) =
{
{(0.85, 0.10, 0.05), (0.80, 0.15, 0.05)}, {(0.70, 0.20, 0.10)}

}
.

Here:

• The first inner set {(0.85, 0.10, 0.05), (0.80, 0.15, 0.05)} represents two expert judg-
ments about the joint status of the temperature and pressure sensors, each triple
(T, I, F ) satisfying 0 ≤ T + I + F ≤ 3.

• The second inner set {(0.70, 0.20, 0.10)} could correspond to an automated diagnostic
algorithm’s assessment.

Thus Ã2(A) captures the hierarchical uncertainty at the second level, aggregating multiple
neutrosophic evaluations for the group of sensors.

2.4. Plithogenic Set

A Plithogenic Set is a mathematical framework that incorporates multi-valued degrees
of appurtenance and contradictions, making it suitable for complex decision-making pro-
cesses. Various studies have been conducted on Plithogenic Sets [65, 66]. The definition
is presented below.

Definition 16 (Plithogenic Set). [15, 65] Let S be a universal set, and P ⊆ S. A
Plithogenic Set PS is defined as:

PS = (P, v, Pv, pdf, pCF )

where:

• v is an attribute.

• Pv is the range of possible values for the attribute v.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 17 of 69

• pdf : P × Pv → [0, 1]s is the Degree of Appurtenance Function (DAF).

• pCF : Pv × Pv → [0, 1]t is the Degree of Contradiction Function (DCF).

These functions satisfy the following axioms for all a, b ∈ Pv:

(i) Reflexivity of Contradiction Function:

pCF (a, a) = 0

(ii) Symmetry of Contradiction Function:

pCF (a, b) = pCF (b, a)

Example 11 (Job Candidate English Proficiency). Let

S = {Hiroko, Masahiro, Shinya}

be the universal set of job candidates, and set P = S. We evaluate each candidate’s English
proficiency using a plithogenic set.

PS =
(
P, v, Pv, pdf , pCF

)
,

where:

• v is the attribute “English proficiency level.”

• Pv = {Beginner, Intermediate, Advanced}.

• The Degree of Appurtenance Function pdf : P × Pv → [0, 1]3 assigns to each (x, ℓ)
a triple (g, w, f) of degrees for grammar (g), vocabulary (w), and fluency (f). For
example:

pdf (Hiroko,Beginner) = (0.10, 0.20, 0.10),

pdf (Hiroko, Intermediate) = (0.60, 0.70, 0.50),

pdf (Hiroko,Advanced) = (0.80, 0.90, 0.70).

• The Degree of Contradiction Function pCF : Pv × Pv → [0, 1] measures the contra-
diction between two proficiency levels. It satisfies:

pCF (a, a) = 0, pCF (Beginner,Advanced) = 1, pCF (Beginner, Intermediate) = 0.5,

and by symmetry pCF (b, a) = pCF (a, b).

This plithogenic set captures for each candidate not only a single “level” but a vector of
sub-scores (g, w, f), while the contradiction function quantifies how mutually incompatible
two levels are.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 18 of 69

These definitions establish the foundational framework necessary for exploring the
HyperPlithogenic Set and the SuperHyperPlithogenic Set. The definitions of the Hyper-
Plithogenic Set and the SuperHyperPlithogenic Set are presented below[46, 67].

Definition 17 (HyperPlithogenic Set). [1, 46] Let X be a non-empty set, and let A be
a set of attributes. For each attribute v ∈ A, let Pv be the set of possible values of v. A
HyperPlithogenic Set HPS over X is defined as:

HPS = (P, {vi}ni=1, {Pvi}ni=1, { ˜pdf i}ni=1, pCF )

where:

• P ⊆ X is a subset of the universe.

• For each attribute vi, Pvi is the set of possible values.

• For each attribute vi, ˜pdf i : P×Pvi → P̃ ([0, 1]s) is the Hyper Degree of Appurtenance
Function (HDAF), assigning to each element x ∈ P and attribute value ai ∈ Pvi a
set of membership degrees.

• pCF : (
⋃n

i=1 Pvi) × (
⋃n

i=1 Pvi) → [0, 1]t is the Degree of Contradiction Function
(DCF).

Example 12 (HyperPlithogenic Set in a Smartphone Purchase Decision). Let

X = {A, B, C}

be a set of smartphone models under consideration, and set P = X. We evaluate each
model with two attributes:

v1 = Battery Life, Pv1 = {Short, Moderate, Long},

v2 = Camera Quality, Pv2 = {Low, Medium, High}.

For simplicity, take s = 1 so that P̃([0, 1]s) = P̃([0, 1]).
The hyper degree of appurtenance functions are defined as follows:

˜pdf1(A,Short) = {0.2, 0.3}, ˜pdf1(A,Moderate) = {0.6, 0.7}, ˜pdf1(A,Long) = {0.4, 0.5},
˜pdf1(B,Short) = {0.3, 0.4}, ˜pdf1(B,Moderate) = {0.5, 0.6}, ˜pdf1(B,Long) = {0.7, 0.8},
˜pdf1(C,Short) = {0.1, 0.2}, ˜pdf1(C,Moderate) = {0.4, 0.5}, ˜pdf1(C,Long) = {0.6, 0.7},

˜pdf2(A,Low) = {0.1, 0.2}, ˜pdf2(A,Medium) = {0.5, 0.6}, ˜pdf2(A,High) = {0.7, 0.8},
˜pdf2(B,Low) = {0.2, 0.3}, ˜pdf2(B,Medium) = {0.6, 0.7}, ˜pdf2(B,High) = {0.8, 0.9},
˜pdf2(C,Low) = {0.3, 0.4}, ˜pdf2(C,Medium) = {0.4, 0.5}, ˜pdf2(C,High) = {0.6, 0.7}.

The degree of contradiction function pCF : (Pv1 ∪Pv2)× (Pv1 ∪Pv2) → [0, 1] is given by

pCF (Short,Long) = 0.9, pCF (Low,High) = 0.8,



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 19 of 69

with pCF (a, a) = 0 and symmetry pCF (a, b) = pCF (b, a).
Hence the HyperPlithogenic Set is

HPS =
(
P, {v1, v2}, {Pv1, Pv2}, { ˜pdf1,

˜pdf2}, pCF
)
,

which captures for each phone model both the hyper-valued membership degrees and the
contradictions among attribute levels.

Definition 18 (n-SuperHyperPlithogenic Set). [1, 68] Let X be a non-empty set, and let
V = {v1, v2, . . . , vn} be a set of attributes, each associated with a set of possible values Pvi.
An n-SuperHyperPlithogenic Set (SHPSn) is defined recursively as:

SHPSn = (Pn, V, {Pvi}ni=1, { ˜pdf
(n)
i }ni=1, pCF (n)),

where:

• P1 ⊆ X, and for k ≥ 2,
Pk = P̃(Pk−1),

represents the k-th nested family of non-empty subsets of P1.

• For each attribute vi ∈ V , Pvi is the set of possible values of the attribute vi.

• For each k-th level subset Pk, ˜pdf
(n)
i : Pn × Pvi → P̃([0, 1]s) is the Hyper Degree of

Appurtenance Function (HDAF), assigning to each element x ∈ Pn and attribute
value ai ∈ Pvi a subset of [0, 1]s.

• pCF (n) :
⋃n

i=1 Pvi ×
⋃n

i=1 Pvi → [0, 1]t is the Degree of Contradiction Function
(DCF), satisfying:

(i) Reflexivity: pCF (n)(a, a) = 0 for all a ∈
⋃n

i=1 Pvi,

(ii) Symmetry: pCF (n)(a, b) = pCF (n)(b, a) for all a, b ∈
⋃n

i=1 Pvi.

• s and t are positive integers representing the dimensions of the membership degrees
and contradiction degrees, respectively.

Example 13 (Nested Project Evaluation Under Uncertainty). Let

X = {Proj1, Proj2},

and take n = 2. Then

P1 = X, P2 = P̃(P1) =
{
{Proj1}, {Proj2}, {Proj1,Proj2}

}
.

Choose two attributes:

v1 = Risk Level, Pv1 = {Low, High},



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 20 of 69

v2 = Cost, Pv2 = {Cheap, Expensive}.

Let s = 1, so that P̃([0, 1]s) = P̃([0, 1]).

Define the hyper degree of appurtenance functions ˜pdf
(2)
i : P2 × Pvi → P̃([0, 1]) by:

˜pdf
(2)
1

(
{Proj1},Low

)
= {0.2, 0.3}, ˜pdf

(2)
1

(
{Proj1},High

)
= {0.7, 0.8},

˜pdf
(2)
1

(
{Proj2},Low

)
= {0.3, 0.4}, ˜pdf

(2)
1

(
{Proj2},High

)
= {0.6, 0.7},

˜pdf
(2)
1

(
{Proj1,Proj2},Low

)
= {0.4, 0.5}, ˜pdf

(2)
1

(
{Proj1,Proj2},High

)
= {0.8, 0.9},

˜pdf
(2)
2

(
{Proj1},Cheap

)
= {0.5, 0.6}, ˜pdf

(2)
2

(
{Proj1},Expensive

)
= {0.7, 0.8},

˜pdf
(2)
2

(
{Proj2},Cheap

)
= {0.4, 0.5}, ˜pdf

(2)
2

(
{Proj2},Expensive

)
= {0.6, 0.7},

˜pdf
(2)
2

(
{Proj1,Proj2},Cheap

)
= {0.6, 0.7}, ˜pdf

(2)
2

(
{Proj1,Proj2},Expensive

)
= {0.8, 0.9}.

Define the degree of contradiction pCF (2) : (Pv1 ∪ Pv2)2 → [0, 1] by:

pCF (2)(Low,High) = 0.9, pCF (2)(Cheap,Expensive) = 0.8,

with reflexivity pCF (2)(a, a) = 0 and symmetry pCF (2)(a, b) = pCF (2)(b, a).
Thus the 2-SuperHyperPlithogenic Set is

SHPS2 =
(
P2, {v1, v2}, {Pv1 , Pv2}, { ˜pdf

(2)
1 , ˜pdf

(2)
2 }, pCF (2)

)
,

which captures nested, hyper-valued appurtenance degrees and contradictions for multi-
criteria project evaluation.

3. Main Results of this paper

This section presents the main results of this paper.

3.1. (m,n)-Superhyperuncertain Set

We provide the definition of the (m,n)-Superhyperuncertain Set.

3.1.1. (m,n)-SuperhyperFuzzy Set

A (m,n)-Superhyperfuzzy Set maps m-level nested subsets of X to n-level nested fuzzy
membership-value subsets in [0, 1], capturing distinct hierarchical fuzzy uncertainty pat-
terns. The definition of the (m,n)-SuperhyperFuzzy Set is presented below.

Definition 19 ((m,n)-SuperhyperFuzzy Set). Let U be a universe of discourse and let
A ⊆ U be nonempty. Fix integers m,n ≥ 0. Define the m-th nested power-set of A by

P0(A) = A, Pk(A) = P
(
Pk−1(A)

)
(k ≥ 1),



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 21 of 69

and similarly define Pn([0, 1]) for the unit interval. An (m,n)-SuperhyperFuzzy Set on
A is a function

τ : Pm(A) −→ Pn([0, 1])

such that for each X ∈ Pm(A), the image τ(X) ⊆ [0, 1] is nonempty. In other words, the
“membership grade” of the (possibly nested) element X is not a single number but a set
of values—intervals and/or discrete points—drawn from [0, 1].

Example 14 (Student Group Assignment). Let U = {Hiroko,Masahiro, Shinya} be a
class roster, and set A = U . Choose m = 1 and n = 1. Then

P1(A) = P(A) =
{
∅, {Hiroko}, . . . , {Hiroko,Masahiro,Shinya}

}
, P1([0, 1]) = P([0, 1]).

Define

τ({Hiroko,Masahiro}) = [0.75, 0.85], τ({Shinya}) = {0.60, 0.65}, τ(∅) = {0},

and similarly for the remaining subsets. Here τ assigns each student group a range or
set of possible average grades (e.g. from different project evaluations), rather than a single
score.

Example 15 (Grocery–Delivery Freshness (simple (1, 1)-SuperhyperFuzzy Set)). Let the
universe be the set of items in a small grocery order,

U = {Milk, Lettuce, Fish },

and take A = U . Fix m = n = 1. We interpret

τ : P1(A) = P(A) −→ P1([0, 1]) = P([0, 1])

so that, for each subset X ⊆ A, the value τ(X) ⊆ [0, 1] is a set of plausible freshness
degrees for the delivery (aggregating uncertainty from traffic, temperature, and handling).
Concretely, define

τ(∅) = { 0 },
τ({Milk}) = [0.60, 0.80], τ({Lettuce}) = [0.50, 0.70], τ({Fish}) = [0.40, 0.60],

τ({Milk,Lettuce}) = [0.55, 0.75], τ({Milk,Fish}) = [0.45, 0.65],

τ({Lettuce,Fish}) = [0.40, 0.60], τ({Milk,Lettuce,Fish}) = [0.35, 0.55].

Each image is a nonempty subset of [0, 1], so τ is an (1, 1)-SuperhyperFuzzy Set. Intervals
narrow or widen to encode uncertainty (e.g., fish is more temperature–sensitive, hence a
lower range; mixing sensitive items widens risk and lowers the aggregated plausibility of
“fresh on arrival”).

Notation 1 (Iterated embeddings and flattenings). For a set S and integers r ≤ s with
r ≥ 1, define the canonical embedding

ιr→s : P r(S) −→ P s(S)



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 22 of 69

recursively by ιr→r = id and

ιr→(t+1)(X) = { ιr→t(X) } (t ≥ r).

Thus ιr→s nests X inside s− r successive singletons.
For a set S and integers s > r ≥ 0, define the level-flattening (union) map

Us→r : P s(S) −→ P r(S)

by Us→(s−1)(Y ) =
⋃
Y and, for r < s − 1, by Us→r = U(r+1)→r ◦ · · · ◦ Us→(s−1). When

S = [0, 1], nonemptiness is preserved because the families we consider consist of nonempty
members at every level (see the definitions of superhyperfuzzy codomains used in this pa-
per).

Theorem 1. If µ̃n : P n(X) → P n([0, 1]) is an n-SuperhyperFuzzy Set, then

τ := µ̃n ◦ ι1→n : P 1(X) −→ P n([0, 1])

is an (1, n)-SuperhyperFuzzy Set. Moreover, for every Z in the image of ι1→n, writing
Z = ι1→n(A) with A ∈ P(X), one has

τ(A) = µ̃n(Z),

i.e. τ agrees with µ̃n on all level-n inputs obtained by the canonical embedding from level
1.

Proof. By hypothesis µ̃n has codomain P n([0, 1]). The canonical embedding ι1→n :
P(X) → P n(X) is well-defined by the recursion in the notation: ι1→1 = id and ι1→(t+1)(A) =
{ι1→t(A)} for t ≥ 1. Therefore the composition

τ = µ̃n ◦ ι1→n

is a map P(X) → P n([0, 1]), which is precisely an (1, n)-SuperhyperFuzzy Set by defini-
tion.

For the agreement statement, take any Z in the image of ι1→n, so Z = ι1→n(A) for
some A ∈ P(X). Then by construction

τ(A) = µ̃n

(
ι1→n(A)

)
= µ̃n(Z),

which establishes that τ reproduces µ̃n on those level-n elements canonically obtained
from level 1.

Theorem 2 (Restriction to Lower m). Let τ : P m(A) → P n([0, 1]) be an (m,n)-
SuperhyperFuzzy Set and let m′ satisfy 0 ≤ m′ < m. Define the inclusion

ιm′→m : P m′
(A) ↪→ P m(A), ιm′→m = ι(m−1)→m ◦ · · · ◦ ιm′→(m′+1),

where each step ιt→(t+1)(X) = {X}. Then

τ ′ = τ ◦ ιm′→m : P m′
(A) −→ P n([0, 1])

is an (m′, n)-SuperhyperFuzzy Set.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 23 of 69

Proof. Fix X ∈ P m′
(A). By construction ιm′→m(X) ∈ P m(A), so τ

(
ιm′→m(X)

)
∈

P n([0, 1]). Hence τ ′ is well-defined with the required codomain. No additional axioms are
needed, so τ ′ is an (m′, n)-SuperhyperFuzzy Set.

Theorem 3 (Projection to Lower n). Let τ : P m(A) → P n([0, 1]) be an (m,n)-SuperhyperFuzzy
Set and let n′ satisfy 0 ≤ n′ < n. Define the level-flattening map

πn→n′ := Un→n′ : P n([0, 1]) −→ P n′
([0, 1]),

i.e. iterated unions taken (n− n′) times. Then

τ ′′ = πn→n′ ◦ τ : P m(A) −→ P n′
([0, 1])

is an (m,n′)-SuperhyperFuzzy Set.

Proof. Fix X ∈ P m(A). Since τ(X) ∈ P n([0, 1]), applying Un→(n−1) produces⋃
τ(X) ∈ P n−1([0, 1]). By the convention of this paper, members at each level are

nonempty, so the union is nonempty. Iterating this (n− n′) times yields

πn→n′
(
τ(X)

)
∈ P n′

([0, 1]).

Therefore τ ′′ is well-defined with codomain P n′
([0, 1]), and hence is an (m,n′)-SuperhyperFuzzy

Set.

Theorem 4 (Closure under Pointwise Union). If τ1, τ2 : P m(A) → P n([0, 1]) are (m,n)-
SuperhyperFuzzy Sets, then the map

(τ1 ∪ τ2)(X) := τ1(X) ∪ τ2(X) (X ∈ P m(A))

is also an (m,n)-SuperhyperFuzzy Set.

Proof. For each X, both τ1(X) and τ2(X) are elements of P n([0, 1]), i.e. subsets of
P n−1([0, 1]). Their union is again a subset of P n−1([0, 1]) and is nonempty because each
term is nonempty. Thus (τ1 ∪ τ2)(X) ∈ P n([0, 1]) for all X, which proves the claim.

Theorem 5 (Closure under Pointwise Intersection). Let τ1, τ2 : P m(A) → P n([0, 1]) be
(m,n)-SuperhyperFuzzy Sets. Define

(τ1 ∩ τ2)(X) := τ1(X) ∩ τ2(X) (X ∈ P m(A)).

Then (τ1 ∩ τ2) is an (m,n)-SuperhyperFuzzy Set provided each intersection is nonempty.
If some intersections are empty, one may either restrict the effective domain to those X
with nonempty intersection, or define the modified map

(τ1 ⊓ τ2)(X) :=

{
τ1(X) ∩ τ2(X), if τ1(X) ∩ τ2(X) ̸= ∅,

{ 0 }, otherwise,

which is again (m,n)-SuperhyperFuzzy.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 24 of 69

Proof. Fix X. Since τi(X) ∈ P n([0, 1]) (i = 1, 2), the intersection τ1(X) ∩ τ2(X) is
a subset of P n−1([0, 1]). If it is nonempty, then it lies in P n([0, 1]) and we are done. If
it happens to be empty for some X, the two remedies stated in the theorem ensure the
resulting value remains a (nonempty) element of P n([0, 1]).

Theorem 6 (Nested α-Cuts). Let τ : P m(A) → P n([0, 1]) be an (m,n)-SuperhyperFuzzy
Set with n ≥ 1. For α ∈ [0, 1], define

Cα =
{
X ∈ P m(A)

∣∣ ∃T ∈ Un→1

(
τ(X)

)
with T ≥ α

}
.

Then for α1 < α2 one has Cα2 ⊆ Cα1.

Proof. Suppose X ∈ Cα2 . By definition there exists T ∈ Un→1(τ(X)) ⊆ [0, 1] with
T ≥ α2. Since α2 > α1, the same T satisfies T ≥ α1, hence X ∈ Cα1 . Therefore
Cα2 ⊆ Cα1 .

Theorem 7 (Functoriality under Surjections). Let f : U → V be a surjection and let
τ : P m(U) → P n([0, 1]) be an (m,n)-SuperhyperFuzzy Set. Define the lifted preimage (by
recursion on m)

f−1
(1) : P(V ) → P(U), f−1

(1) (Y ) = {u ∈ U | f(u) ∈ Y },

and for t ≥ 1,

f−1
(t+1) : P t+1(V ) → P t+1(U), f−1

(t+1)(Y) = { f−1
(t) (Y ) | Y ∈ Y }.

Then the pushforward

f∗(τ) : P m(V ) −→ P n([0, 1]), f∗(τ)(Y) = τ
(
f−1
(m)(Y)

)
is an (m,n)-SuperhyperFuzzy Set on V .

Proof. Let Y ∈ P m(V ). By the recursive definition of f−1
(m), we have f−1

(m)(Y) ∈
P m(U). Therefore τ

(
f−1
(m)(Y)

)
∈ P n([0, 1]), showing f∗(τ) is well-defined with the re-

quired codomain. No additional properties are needed to conclude that f∗(τ) is an (m,n)-
SuperhyperFuzzy Set.

3.1.2. (m,n)-SuperhyperNeutrosophic Set

An (m,n)-SuperhyperNeutrosophic Set maps m-level nested subsets to nonempty families
of n-level neutrosophic triples (T, I, F ) ∈ [0, 1]3, thereby modeling hierarchical uncertainty
across levels.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 25 of 69

Definition 20 ((m,n)-SuperhyperNeutrosophic Set). Let U be a universe of discourse
and let A ⊆ U be nonempty. Fix m,n ∈ N ∪ {0}. Define recursively

P 0(A) = A, P k(A) = P
(
P k−1(A)

)
(k ≥ 1),

and analogously for P n([0, 1]3). An (m,n)-SuperhyperNeutrosophic Set on A is a mapping

µ : P m(A) −→ P n
(
[0, 1]3

)
such that for each X ∈ P m(A) the value µ(X) is nonempty and every triple (T, I, F )
occurring at level 1 inside µ(X) satisfies

0 ≤ T + I + F ≤ 3.

Equivalently, Un→1

(
µ(X)

)
⊆ [0, 1]3 is nonempty and consists of triples obeying T +I+F ≤

3, where Un→1 denotes iterated union (flattening) from level n to level 1.

Notation 2 (Canonical embedding and flattening). For r ≤ s with r ≥ 1, the canonical
embedding

ιr→s : P r(S) → P s(S)

is defined by ιr→r = id and ιr→(t+1)(X) = {ιr→t(X)} for t ≥ r; hence it nests X by s− r
singletons. For s > r ≥ 0, the level-flattening map

Us→r : P s(S) → P r(S)

is given by Us→(s−1)(Y ) =
⋃

Y and Us→r = U(r+1)→r ◦ · · · ◦ Us→(s−1).

Example 16 (Multi-Expert Risk Assessment). Let A = {a1, a2} and m = n = 1, so
P 1(A) = P(A) and P 1([0, 1]3) = P([0, 1]3). Suppose two experts assess {a1, a2}. Define

µ({a1, a2}) = {(0.80, 0.15, 0.05), (0.75, 0.20, 0.05)}, µ({a1}) = {(0.60, 0.25, 0.15)},

µ({a2}) = {(0.50, 0.30, 0.20)}, µ(∅) = {(0, 0, 0)}.

Each triple satisfies 0 ≤ T + I + F ≤ 3, so µ is (1, 1)-SuperhyperNeutrosophic.

Example 17 (Hierarchical Fault Diagnosis in an Industrial IoT Network). Let A =
{TempSensor,VibSensor}, m = 1, n = 2. Define

µ : P(A) −→ P
(
P([0, 1]3)

)
by, for X = {TempSensor,VibSensor},

µ(X) =
{
{(0.85, 0.10, 0.05), (0.80, 0.15, 0.05)}, {(0.60, 0.25, 0.15)}

}
.

Flattening gives U2→1(µ(X)) = {(0.85, 0.10, 0.05), (0.80, 0.15, 0.05), (0.60, 0.25, 0.15)}, all
obeying T + I + F ≤ 3.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 26 of 69

Theorem 8. Every n-SuperhyperNeutrosophic Set Ãn : P n(X) → P n([0, 1]3) is obtained
as the special case m = 1 of an (m,n)-SuperhyperNeutrosophic Set on X.

Proof. Define µ := Ãn ◦ ι1→n : P 1(X) → P n([0, 1]3). This composition is well-defined
since ι1→n maps P(X) into P n(X). For any A ∈ P(X) we have

µ(A) = Ãn

(
ι1→n(A)

)
,

hence µ agrees with Ãn on the level-n images of level-1 inputs. Because Ãn(A′) is nonempty
for each A′ ∈ P n(X) and its level-1 elements satisfy T + I + F ≤ 3, the same properties
hold for µ(A). Thus µ is (1, n)-SuperhyperNeutrosophic, realizing Ãn as the m = 1 case.

Theorem 9 (Restriction to Lower m). Let

µ : P m(A) → P n([0, 1]3)

be (m,n)-SuperhyperNeutrosophic and let 0 ≤ m′ < m. Let ιm′→m be the canonical
embedding. Then

µ′ = µ ◦ ιm′→m : P m′
(A) −→ P n([0, 1]3)

is (m′, n)-SuperhyperNeutrosophic.

Proof. Fix X ∈ P m′
(A). Then ιm′→m(X) ∈ P m(A) and µ

(
ιm′→m(X)

)
∈ P n([0, 1]3)

is nonempty with all (T, I, F ) at level 1 obeying T + I + F ≤ 3. Hence µ′(X) has the
required codomain and constraints.

Theorem 10 (Projection onto Truth Component). Let µ : P m(A) → P n([0, 1]3) be
(m,n)-SuperhyperNeutrosophic. Define πT : [0, 1]3 → [0, 1], πT (T, I, F ) = T , and lift it
levelwise to

ΠT : P n([0, 1]3) −→ P n([0, 1]), ΠT = P n(πT ).

Then τ := ΠT ◦ µ : P m(A) → P n([0, 1]) is an (m,n)-SuperhyperFuzzy Set.

Proof. For any X ∈ P m(A), the inner-most (level 1) set Un→1(µ(X)) ⊆ [0, 1]3 is
nonempty. Applying πT pointwise yields the nonempty set πT

(
Un→1(µ(X))

)
⊆ [0, 1].

Lifting this operation consistently to higher levels defines ΠT , so τ(X) = ΠT (µ(X)) ∈
P n([0, 1]). Thus τ is (m,n)-SuperhyperFuzzy.

Theorem 11 (Pointwise Union). If

µ1, µ2 : P m(A) → P n([0, 1]3)

are (m,n)-SuperhyperNeutrosophic, then

(µ1 ∪ µ2)(X) := µ1(X) ∪ µ2(X)

defines another (m,n)-SuperhyperNeutrosophic Set.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 27 of 69

Proof. For each X, both µ1(X) and µ2(X) lie in P n([0, 1]3) and are nonempty. Their
union is again a nonempty element of P n([0, 1]3). Every triple in the union comes from
either µ1(X) or µ2(X) and thus satisfies T + I + F ≤ 3.

Theorem 12 (Pointwise Intersection). Under the same hypotheses as Theorem 11, let

(µ1 ∩ µ2)(X) := µ1(X) ∩ µ2(X).

If µ1(X) ∩ µ2(X) ̸= ∅ for all X, then (µ1 ∩ µ2) is (m,n)-SuperhyperNeutrosophic.

Proof. For each X, the intersection is a (by assumption nonempty) subset of P n−1([0, 1]3),
hence an element of P n([0, 1]3). Any triple in the intersection already satisfies T+I+F ≤ 3
because it belongs to both µ1(X) and µ2(X).

Theorem 13 (Nested λ-Cuts). Let µ : P m(A) → P n([0, 1]3) be (m,n)-SuperhyperNeutrosophic.
For λ = (α, β, γ) ∈ [0, 1]3 define

Cλ =
{
X ∈ P m(A)

∣∣∣ ∃(T, I, F ) ∈ Un→1(µ(X)) with T ≥ α, I ≤ β, F ≤ γ
}
.

If λ′ = (α′, β′, γ′) satisfies α′ ≥ α, β′ ≤ β, γ′ ≤ γ, then Cλ′ ⊆ Cλ.

Proof. Take X ∈ Cλ′ . Then some (T, I, F ) ∈ Un→1(µ(X)) obeys T ≥ α′, I ≤ β′,
F ≤ γ′. Since α′ ≥ α, β′ ≤ β, and γ′ ≤ γ, the same triple witnesses X ∈ Cλ. Hence
Cλ′ ⊆ Cλ.

Theorem 14 (Functoriality under Surjections). Let f : A → B be surjective and let

µ : P m(A) −→ P n
(
[0, 1]3

)
be an (m,n)-SuperhyperNeutrosophic map. Define the lifted preimage at level 1 by

f−1
(1) : P(B) −→ P(A), f−1

(1) (Y ) := { a ∈ A : f(a) ∈ Y }.

Inductively
f−1
(t+1)(Y) =

{
f−1
(t) (Y ) : Y ∈ Y

}
(t ≥ 1).

Then
f∗µ : P m(B) −→ P n([0, 1]3), f∗µ(Y) = µ

(
f−1
(m)(Y)

)
is (m,n)-SuperhyperNeutrosophic on B.

Proof. Let Y ∈ P m(B). By construction f−1
(m)(Y) ∈ P m(A); surjectivity of f ensures

this recursion does not collapse to the empty family when Y ̸= ∅. Hence µ
(
f−1
(m)(Y)

)
∈

P n([0, 1]3) is nonempty, and all its level-1 elements satisfy T + I + F ≤ 3. Therefore f∗µ
has the required codomain and constraints, completing the proof.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 28 of 69

3.1.3. (m,n)-SuperhyperPlithogenic Set

A (m,n)-SuperhyperPlithogenic Set assigns m-level parameter-subsets and their attribute
values to n-level fuzzy-contradiction degree-sets, capturing multi-faceted membership and
inter-attribute conflicts and uncertainty patterns. The definition of the (m,n)-SuperhyperPlithogenic
Set is presented below.

Definition 21 ((m,n)-SuperhyperPlithogenic Set). Let X be a nonempty set and let
V = {v1, . . . , vk} be a finite set of attributes. For each v ∈ V , let Pv be the set of its
possible values. Fix positive integers m,n and positive dimensions s, t. Define the m-th
nested powerset of X by

P0(X) = X, Pr(X) = P
(
Pr−1(X)

)
(r ≥ 1),

and similarly Pn
(
[0, 1]s

)
for the s-dimensional unit cube. An (m,n)-SuperhyperPlithogenic

Set over X is the quintuple

SHP (m,n) =
(
Pm(X), V, {Pv}v∈V , { ˜pdf

(m,n)
v }v∈V , pCF (m,n)

)
,

where

(i) Pm(X) is the domain of “super-elements” of level m.

(ii) For each v ∈ V , Pv is the finite set of its values.

(iii) The Hyper Degree of Appurtenance Function

˜pdf
(m,n)
v : Pm(X) × Pv −→ Pn

(
[0, 1]s

)
assigns to each (A, a) with A ∈ Pm(X) and a ∈ Pv a nonempty subset ˜pdf

(m,n)
v (A, a) ⊆

[0, 1]s representing all possible membership-degree vectors of dimension s.

(iv) The Degree of Contradiction Function

pCF (m,n) :
(⋃
v∈V

Pv

)
×
(⋃
v∈V

Pv

)
−→ [0, 1]t

satisfies for all a, b:

(a) pCF (m,n)(a, a) = 0 (reflexivity),

(b) pCF (m,n)(a, b) = pCF (m,n)(b, a) (symmetry).

Example 18 (Simple (1, 1)-SuperhyperPlithogenic Set). Let X = {a1, a2} and V = {v}
with Pv = {p1, p2}. Take m = n = 1 and s = t = 1. Then

P1(X) = P(X), P1([0, 1]) = P([0, 1]).



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 29 of 69

Define
˜pdf

(1,1)
v : P(X) × Pv → P([0, 1])

by
˜pdf

(1,1)
v ({a1, a2}, p1) = [0.7, 0.8], ˜pdf

(1,1)
v ({a1}, p1) = {0.5, 0.6},

˜pdf
(1,1)
v ({a2}, p1) = [0.3, 0.4], ˜pdf

(1,1)
v (∅, p1) = {0},

and similarly for p2. For the contradiction,

pCF (1,1) : Pv × Pv → [0, 1],

set
pCF (1,1)(p1, p1) = 0, pCF (1,1)(p1, p2) = 0.2, pCF (1,1)(p2, p2) = 0,

with symmetry pCF (p2, p1) = 0.2. Then

SHP (1,1) =
(
P(X), {v}, {Pv}, { ˜pdf

(1,1)
v }, pCF (1,1)

)
is a valid (1, 1)-SuperhyperPlithogenic Set.

Example 19 (Hackathon Team Evaluation). Let

X = {Hiroko, Masahiro, Shinya, Dave},

and take m = n = 1, so that P1(X) = P(X). We consider two evaluation attributes:

v1 = Innovation, Pv1 = {Low, Medium, High},

v2 = Feasibility, Pv2 = {Low, Medium, High}.

For richness, let the membership–degree vectors be two–dimensional (peer score, judge
score), so s = 2, and let the contradiction measure be scalar, so t = 1. Then

˜pdf
(1,1)
vi : P(X) × Pvi −→ P

(
[0, 1]2

)
assigns to each team A ⊆ X and level a ∈ Pvi a nonempty set of vectors in [0, 1]2.

For example, for the team A = {Hiroko,Masahiro}:

˜pdf
(1,1)
Innovation(A,High) =

{
(0.90, 0.85), (0.88, 0.82)

}
,

˜pdf
(1,1)
Innovation(A,Medium) =

{
(0.70, 0.75)

}
,

˜pdf
(1,1)
Feasibility(A,High) =

{
(0.80, 0.78), (0.82, 0.80)

}
,

˜pdf
(1,1)
Feasibility(A,Low) =

{
(0.50, 0.55)

}
.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 30 of 69

The degree of contradiction function

pCF (1,1) :
(
Pv1 ∪ Pv2

)
×
(
Pv1 ∪ Pv2

)
−→ [0, 1]

is defined by

pCF (1,1)(High,Low) = 1.0, pCF (1,1)(High,Medium) = 0.5, pCF (1,1)(a, a) = 0,

with symmetry pCF (b, a) = pCF (a, b).
Hence

SHP (1,1) =
(
P(X), {v1, v2}, {Pv1 , Pv2}, { ˜pdf

(1,1)
v1 , ˜pdf

(1,1)
v2 }, pCF (1,1)

)
is a concrete (1, 1)-SuperhyperPlithogenic Set modeling multi-dimensional team evaluations
in a hackathon.

Example 20 (Supply Chain Resilience Evaluation with (2, 2)-SuperhyperPlithogenic Set).
Let

X = {FactoryA, FactoryB},

and take m = n = 2. Then

P1(X) = P(X) = {∅, {FactoryA}, {FactoryB}, {FactoryA,FactoryB}},

P2(X) = P
(
P1(X)

)
,

the set of all subsets of P1(X). For concreteness, choose the level-2 element

A =
{
{FactoryA}, {FactoryA,FactoryB}

}
∈ P2(X).

We evaluate two attributes:

v1 = Cost Efficiency, Pv1 = {Low,Medium,High},

v2 = Delivery Reliability, Pv2 = {Slow,Moderate,Fast}.

Let the membership–degree vectors be one-dimensional (s = 1) and the contradiction mea-
sure scalar (t = 1).

Define the hyper–degree functions

˜pdf
(2,2)
vi : P2(X) × Pvi −→ P2([0, 1])

by giving their values on A:

˜pdf
(2,2)
v1 (A,Low) =

{
{0.40, 0.50}, {0.45, 0.55}

}
,

˜pdf
(2,2)
v1 (A,Medium) =

{
[0.55, 0.65]

}
,

˜pdf
(2,2)
v1 (A,High) =

{
[0.70, 0.80], {0.75}

}
,



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 31 of 69

˜pdf
(2,2)
v2 (A,Slow) =

{
{0.30, 0.40}

}
,

˜pdf
(2,2)
v2 (A,Moderate) =

{
{0.60}, [0.60, 0.70]

}
,

˜pdf
(2,2)
v2 (A,Fast) =

{
[0.80, 0.90], {0.85}

}
.

The degree of contradiction function

pCF (2,2) :
(
Pv1 ∪ Pv2

)
×
(
Pv1 ∪ Pv2

)
−→ [0, 1]

is specified by:

pCF (2,2)(Low,High) = 1.0, pCF (2,2)(Low,Medium) = 0.3, pCF (2,2)(Medium,High) = 0.5,

pCF (2,2)(Slow,Fast) = 0.8, pCF (2,2)(a, a) = 0, pCF (2,2)(a, b) = pCF (2,2)(b, a).

Thus the structure

SHP (2,2) =
(
P2(X), {v1, v2}, {Pv1 , Pv2}, { ˜pdf

(2,2)
v1 , ˜pdf

(2,2)
v2 }, pCF (2,2)

)
models, at two hierarchical levels, the cost-efficiency and delivery-reliability uncertainties
of the supply chain with hyper-valued degrees and quantified contradictions.

Theorem 15. Every n-SuperhyperPlithogenic Set
(
P1(X), V, {Pv}, { ˜pdf

(1,n)
v }, pCF (1,n)

)
arises as the special case m = 1 of an (m,n)-SuperhyperPlithogenic Set.

Proof. By definition, an n-SuperhyperPlithogenic Set has domain P1(X) = P(X). In
the general (m,n) framework, setting m = 1 replaces Pm(X) by P1(X) and leaves all
other components identical. Hence the two structures coincide exactly when m = 1.

Theorem 16 (Restriction to Lower m). Let

SHP (m,n) =
(
P m(X), V, {Pv}, { ˜pdf

(m,n)
v }, pCF (m,n)

)
be an (m,n)-SuperhyperPlithogenic Set and let m′ satisfy 0 ≤ m′ < m. Define, for each
v ∈ V and a ∈ Pv,

˜pdf
(m′,n)
v,↓m (A, a) := ˜pdf

(m,n)
v

(
ιm′→m(A) , a

)
for A ∈ P m′

(X),

and keep pCF (m,n) unchanged. Then

SHP
(m′,n)
↓ =

(
P m′

(X), V, {Pv}, { ˜pdf
(m′,n)
v,↓m }, pCF (m,n)

)
is an (m′, n)-SuperhyperPlithogenic Set.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 32 of 69

Proof. Fix A ∈ P m′
(X), v ∈ V , a ∈ Pv. Since ιm′→m(A) ∈ P m(X), the value

˜pdf
(m,n)
v

(
ιm′→m(A), a

)
∈ P n([0, 1]s) is well-defined and nonempty by the (m,n)-definition.

Hence ˜pdf
(m′,n)
v,↓m (A, a) has the required codomain. The contradiction map pCF (m,n) is

unchanged, so all axioms are preserved.

Theorem 17 (Projection to Lower n). Let SHP (m,n) be as above and let n′ satisfy 0 ≤
n′ < n. For each v ∈ V , a ∈ Pv, define

˜pdf
(m,n′)
v,↓n (A, a) := Un→n′

(
˜pdf

(m,n)
v (A, a)

)
∈ P n′

([0, 1]s),

and keep pCF (m,n) unchanged. Then

SHP
(m,n′)
↓ =

(
P m(X), V, {Pv}, { ˜pdf

(m,n′)
v,↓n }, pCF (m,n)

)
is an (m,n′)-SuperhyperPlithogenic Set.

Proof. Fix A ∈ P m(X), v, a. Since ˜pdf
(m,n)
v (A, a) ∈ P n([0, 1]s) is nonempty and

every level is nonempty by definition, applying Un→(n−1) gives a nonempty element of

P n−1([0, 1]s). Iterating (n − n′) times yields a nonempty element of P n′
([0, 1]s). The

contradiction map is unaffected, so the structure satisfies the definition at level n′.

Theorem 18 (Closure under Pointwise Union). Let SHP
(m,n)
1 and SHP

(m,n)
2 be (m,n)-

SuperhyperPlithogenic Sets on the same (X,V, {Pv}) and with the same pCF (m,n). Define,
for each v ∈ V , a ∈ Pv,(

˜pdf
(m,n)
v

)∪
(A, a) := ˜pdf

(m,n)
v,1 (A, a) ∪ ˜pdf

(m,n)
v,2 (A, a) for A ∈ P m(X).

Then
SHP

(m,n)
∪ =

(
P m(X), V, {Pv}, {

(
˜pdf

(m,n)
v

)∪}, pCF (m,n)
)

is an (m,n)-SuperhyperPlithogenic Set.

Proof. For each (A, a), both ˜pdf
(m,n)
v,1 (A, a) and ˜pdf

(m,n)
v,2 (A, a) lie in P n([0, 1]s) and

are nonempty. Their union is a nonempty subset of P n−1([0, 1]s), hence an element of
P n([0, 1]s). The shared pCF (m,n) remains valid.

Theorem 19 (Attribute–Value α-Cuts are Nested). Fix v ∈ V and a ∈ Pv. For a
threshold vector α = (α1, . . . , αs) ∈ [0, 1]s, define

Cv,a(α) =
{

A ∈ P m(X)
∣∣∣ ∃d ∈ Un→1

(
˜pdf

(m,n)
v (A, a)

)
with d ≥ α (componentwise)

}
.

If α′,α ∈ [0, 1]s satisfy α′ ≥ α componentwise, then

Cv,a(α′) ⊆ Cv,a(α).



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 33 of 69

Proof. Let A ∈ Cv,a(α′). Then there exists d = (d1, . . . , ds) ∈ Un→1( ˜pdf
(m,n)
v (A, a))

with di ≥ α′
i for all i. Since α′

i ≥ αi for each i, we also have di ≥ αi; hence the same d
witnesses A ∈ Cv,a(α).

Theorem 20 (Functoriality under Surjections on the Base). Let f : X → Y be surjective
and let

SHP
(m,n)
X =

(
P m(X), V, {Pv}, { ˜pdf

(m,n)
v }, pCF (m,n)

)
be an (m,n)-SuperhyperPlithogenic Set on X. Define lifted preimages recursively:

f−1
(1) : P(Y ) → P(X), f−1

(1) (B) = {x ∈ X : f(x) ∈ B},

f−1
(t+1)(B) = { f−1

(t) (B) | B ∈ B } (t ≥ 1).

Define, for A′ ∈ P m(Y ), v ∈ V , a ∈ Pv,

˜pdf
(m,n)
v,f (A′, a) := ˜pdf

(m,n)
v

(
f−1
(m)(A

′), a
)
,

and keep pCF (m,n) unchanged. Then

SHP
(m,n)
Y =

(
P m(Y ), V, {Pv}, { ˜pdf

(m,n)
v,f }, pCF (m,n)

)
is an (m,n)-SuperhyperPlithogenic Set on Y .

Proof. For each A′ ∈ P m(Y ), the recursive construction yields f−1
(m)(A

′) ∈ P m(X).

Hence ˜pdf
(m,n)
v

(
f−1
(m)(A

′), a
)
∈ P n([0, 1]s) is nonempty, so ˜pdf

(m,n)
v,f (A′, a) is well-defined

with the required codomain. The contradiction map is copied verbatim, preserving its
axioms.

Theorem 21 (Monotone Scalarization to SuperhyperFuzzy). Let ϕ : [0, 1]s → [0, 1] be
componentwise nondecreasing. Lift ϕ levelwise to

Φ : P n([0, 1]s) −→ P n([0, 1]), Φ = P n(ϕ).

For fixed v ∈ V , a ∈ Pv, define

τv,a(A) := Φ
(

˜pdf
(m,n)
v (A, a)

)
for A ∈ P m(X).

Then τv,a : P m(X) → P n([0, 1]) is an (m,n)-SuperhyperFuzzy Set (parametrized by
(v, a)).

Proof. Fix A. Since ˜pdf
(m,n)
v (A, a) ∈ P n([0, 1]s) is nonempty, applying ϕ pointwise at

level 1 yields a nonempty subset of [0, 1]. Lifting to higher levels preserves nonemptiness
and nesting, so τv,a(A) ∈ P n([0, 1]). No further axioms are needed for the fuzzy case;
hence τv,a is (m,n)-SuperhyperFuzzy.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 34 of 69

Theorem 22 (Contradiction-Bounded Selections are Stable). Fix v ∈ V and a bound
δ ∈ [0, 1]t. For A ∈ P m(X) define the δ-admissible value set

Av,δ(A) =
{
a ∈ Pv

∣∣∣ ∀b ∈ Pv, ∥pCF (m,n)(a, b)∥∞ ≤ ∥δ∥∞
}
.

Then for any δ′ ≤ δ (componentwise) one has

Av,δ′(A) ⊆ Av,δ(A).

Proof. If a ∈ Av,δ′(A), then ∥pCF (m,n)(a, b)∥∞ ≤ ∥δ′∥∞ for all b ∈ Pv. Since ∥δ′∥∞ ≤
∥δ∥∞, the same a satisfies the bound for δ, hence a ∈ Av,δ(A).

3.2. (h, k)-ary (m,n)-Superhyperuncertain Set

We provide an explanation of the (h, k)-ary (m,n)-Superhyperuncertain Set below.

3.2.1. (h, k)-ary (m,n)-SuperhyperFuzzy Set

A (h, k)-ary (m,n)-SuperhyperFuzzy Set generalizes the (m,n)-SuperhyperFuzzy notion
by accepting h inputs and producing k outputs, where the inputs are m-level nested subsets
of a base set and the outputs are n-level nested fuzzy–degree sets.

Definition 22 ((h, k)-ary (m,n)-SuperhyperFuzzy Set). Let X be a nonempty set and let
m,n, h, k ∈ N with m,n, h, k ≥ 1. Define

P0(X) = X, Pr(X) = P
(
Pr−1(X)

)
(r ≥ 1),

and analogously Pr([0, 1]) for the unit interval. An (h, k)-ary (m,n)-SuperhyperFuzzy Set
on X is a mapping

µ
(m,n)
(h,k) :

(
Pm(X)

)h −→
(
Pn([0, 1])

)k
,

which to each input tuple A = (A1, . . . , Ah) ∈ (Pm(X))h assigns an output tuple µ
(m,n)
(h,k) (A) =

(B1, . . . , Bk) with Bj ∈ Pn([0, 1]) for every 1 ≤ j ≤ k. We additionally require nonempti-
ness at the outer level: for each A and each j, Bj ̸= ∅. Equivalently, writing the coordinate
maps as µ

(m,n)
j : (Pm(X))h → Pn([0, 1]), we demand µ

(m,n)
j (A) ∈ Pn([0, 1]) \ {∅} for all

A and all j.

Example 21 (Smart building comfort control as a (2, 2)-ary (1, 1)-SuperhyperFuzzy Set).
Let X = {Room1,Room2,Room3} and fix m = n = 1, h = k = 2. Thus P1(X) = P(X)
and P1([0, 1]) = P([0, 1]), so the domain is D = P(X) × P(X) and the codomain is
C = P([0, 1]) × P([0, 1]). Interpret an input (A1, A2) ∈ D as: A1 is the set of occupied
rooms, and A2 the set of rooms with windows open. Define the following concrete mapping:

µ
(1,1)
(2,2)(A1, A2) =

(
Bcomfort(A1, A2), Benergy(A1, A2)

)
,



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 35 of 69

where, with the shorthand

θ(A1, A2) :=
|A1 ∩A2|

max{1, |A1|}
, ω(A2) :=

|A2|
|X|

,

we set

Bcomfort(A1, A2) =
[

0.60 + 0.30 θ(A1, A2), 0.70 + 0.25 θ(A1, A2)
]
⊂ [0, 1],

Benergy(A1, A2) =
[

0.20 + 0.40ω(A2), 0.30 + 0.45ω(A2)
]
⊂ [0, 1].

Thus, if A1 = {Room1,Room2} and A2 = {Room2}, then θ = 1/2 and ω = 1/3, hence

Bcomfort = [0.60 + 0.15, 0.70 + 0.125] = [0.75, 0.825],

Benergy = [0.20 + 0.1333, 0.30 + 0.15] = [0.3333, 0.45],

so one may choose, for instance, µ
(1,1)
(2,2)(A1, A2) =

(
[0.75, 0.825], [0.3333, 0.45]

)
∈ C. This

implements a concrete (2, 2)-ary (1, 1)-SuperhyperFuzzy Set with explicitly computed fuzzy
outputs.

Example 22 (Two–symptom, two–disease risk assessment). Let X = {Fever,Cough,Fatigue},
again with m = n = 1 and h = k = 2. Given two symptom–subsets (A1, A2) ∈
P(X) × P(X), define

µ
(1,1)
(2,2)(A1, A2) =

(
BDisA(A1), BDisB(A2)

)
,

where
BDisA(A1) =

[
0.30 + 0.20 |A1|

3 , 0.50 + 0.30 |A1|
3

]
,

BDisB(A2) =
{

0.25 + 0.15 |A2|
3 , 0.35 + 0.20 |A2|

3

}
.

For instance, µ
(1,1)
(2,2)

(
{Fever,Cough}, {Fatigue}

)
=

(
[0.30+0.20 · 23 , 0.50+0.30 · 23 ], {0.25+

0.15 · 1
3 , 0.35 + 0.20 · 1

3}
)

=
(
[0.4333, 0.70], {0.30, 0.4166}

)
.

Theorem 23. If h = k = 1, then any (h, k)-ary (m,n)-SuperhyperFuzzy Set reduces to
an ordinary (m,n)-SuperhyperFuzzy Set.

Proof. Let µ
(m,n)
(1,1) : (Pm(X))1 → (Pn([0, 1]))1 be given. Define the canonical bijections

ϕh : (Pm(X))1 −→ Pm(X), ϕh((A)) = A,

ϕk : (Pn([0, 1]))1 −→ Pn([0, 1]), ϕk((B)) = B.

Both ϕh and ϕk are bijective with inverses A 7→ (A) and B 7→ (B), respectively. Define

µ̃ := ϕk ◦ µ
(m,n)
(1,1) ◦ ϕ−1

h : Pm(X) −→ Pn([0, 1]).



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 36 of 69

For any A ∈ Pm(X) we have ϕ−1
h (A) = (A) and thus

µ̃(A) = ϕk

(
µ
(m,n)
(1,1) ((A))

)
= B,

where µ
(m,n)
(1,1) ((A)) = (B) with B ∈ Pn([0, 1]) nonempty by definition of an (h, k)-ary

(m,n)-SuperhyperFuzzy Set. Hence µ̃ is a well-defined map Pm(X) → Pn([0, 1]) with
nonempty outer values; that is, an ordinary (m,n)-SuperhyperFuzzy Set.

Theorem 24 (Fixing inputs). Fix indices 1 ≤ i1 < · · · < ir ≤ h and elements Aij ∈
Pm(X). Define µfix : (Pm(X))h−r → (Pn([0, 1]))k by inserting the fixed Aij in the

corresponding coordinates and applying µ
(m,n)
(h,k) . Then µfix is an (h − r, k)-ary (m,n)-

SuperhyperFuzzy Set.

Proof. Write Afree = (Aℓ)ℓ∈Ifree ∈ (Pm(X))h−r, where I free = {1, . . . , h} \ {i1, . . . , ir}
(ordered increasingly). Define the insertion operator

ι : (Pm(X))h−r −→ (Pm(X))h, ι(Afree) = B = (B1, . . . , Bh),

by

Bu =

{
Aij if u = ij for some j,

Aℓ if u = ℓ ∈ I free.

Then set µfix := µ
(m,n)
(h,k) ◦ ι. For any Afree, ι(Afree) ∈ (Pm(X))h; hence µ

(m,n)
(h,k) yields

a k-tuple
(
µ
(m,n)
1 (ι(Afree)), . . . , µ

(m,n)
k (ι(Afree))

)
with each coordinate in Pn([0, 1]) and

nonempty. Therefore µfix is a well-defined (h− r, k)-ary (m,n)-SuperhyperFuzzy Set.

Theorem 25 (Projection to a subset of outputs). Let 1 ≤ j1 < · · · < js ≤ k. The
coordinate projection

µproj : (Pm(X))h −→ (Pn([0, 1]))s, µproj(A) =
(
µ
(m,n)
j1

(A), . . . , µ
(m,n)
js

(A)
)
,

is an (h, s)-ary (m,n)-SuperhyperFuzzy Set.

Proof. Let prJ : (Pn([0, 1]))k → (Pn([0, 1]))s be the coordinate projection to J =

{j1, . . . , js}. Then µproj = prJ ◦ µ
(m,n)
(h,k) . Since each output coordinate µ

(m,n)
j (A) is a

nonempty element of Pn([0, 1]), their s-tuple lies in (Pn([0, 1]))s with nonempty coordi-
nates. Hence µproj has the required type.

Theorem 26 (Pointwise union). If µ, ν : (Pm(X))h → (Pn([0, 1]))k are (h, k)-ary (m,n)-
SuperhyperFuzzy Sets, then the map (µ ∪ ν)(A) =

(
µ1(A) ∪ ν1(A), . . . , µk(A) ∪ νk(A)

)
is

again an (h, k)-ary (m,n)-SuperhyperFuzzy Set.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 37 of 69

Proof. Fix A ∈ (Pm(X))h and a coordinate 1 ≤ j ≤ k. By hypothesis, µj(A), νj(A) ∈
Pn([0, 1]) and are nonempty. Recall that Pn([0, 1]) consists of sets of level n− 1 objects;
the union of two sets at the same level is again a set at that level. Hence µj(A)∪ νj(A) ∈
Pn([0, 1]), and it is nonempty because the union of two nonempty sets is nonempty. Do-
ing this for each j produces a k-tuple in (Pn([0, 1]))k with all coordinates nonempty, as
required.

Theorem 27 (Nested α-cuts). For α ∈ [0, 1], define

Cα =
{
A ∈ (Pm(X))h

∣∣∣ ∃j ∃t ∈ µ
(m,n)
j (A) with t ≥ α

}
.

If 0 ≤ α1 < α2 ≤ 1, then Cα2 ⊆ Cα1.

Proof. Let A ∈ Cα2 . By definition, there exists an index j and a value t ∈ µ
(m,n)
j (A)

such that t ≥ α2. Since α2 > α1, we have t ≥ α2 > α1, hence the same j and t witness
that A ∈ Cα1 . Therefore Cα2 ⊆ Cα1 .

Theorem 28 (Compatibility with surjections). Let f : X → Y be surjective. Define
f−1
(1) : P(Y ) → P(X) by f−1

(1) (B) = {x ∈ X : f(x) ∈ B} and recursively

f−1
(r+1)(B) =

{
f−1
(r) (B)

∣∣ B ∈ B
}

(r ≥ 1).

Given µ
(m,n)
(h,k) on X, the pushforward

(f∗µ)
(m,n)
(h,k) : (Pm(Y ))h −→ (Pn([0, 1]))k,

(f∗µ)
(m,n)
(h,k) (A′

1, . . . , A
′
h) = µ

(m,n)
(h,k)

(
f−1
(m)(A

′
1), . . . , f

−1
(m)(A

′
h)
)
,

is an (h, k)-ary (m,n)-SuperhyperFuzzy Set on Y .

Proof. We first show by induction on r ≥ 1 that f−1
(r) : Pr(Y ) → Pr(X) is well defined.

For r = 1, this is the usual preimage map on subsets: if B ⊆ Y , then f−1
(1) (B) ⊆ X, so

f−1
(1) (B) ∈ P(X) = P1(X). Assume f−1

(r) is defined on Pr(Y ) with values in Pr(X). If

B ∈ Pr+1(Y ), then B is a set of elements B ∈ Pr(Y ), and by induction f−1
(r) (B) ∈ Pr(X)

for each such B. Therefore the set { f−1
(r) (B) | B ∈ B } belongs to P(Pr(X)) = Pr+1(X),

which establishes the inductive step.
Now fix (A′

1, . . . , A
′
h) ∈ (Pm(Y ))h. By the previous paragraph, f−1

(m)(A
′
i) ∈ Pm(X) for

each i, so the h-tuple
(
f−1
(m)(A

′
1), . . . , f

−1
(m)(A

′
h)
)

lies in (Pm(X))h. Applying µ
(m,n)
(h,k) yields a

k-tuple in (Pn([0, 1]))k with nonempty coordinates (by the defining nonemptiness property

of µ
(m,n)
(h,k) ). Therefore (f∗µ)

(m,n)
(h,k) is a well-defined (h, k)-ary (m,n)-SuperhyperFuzzy Set

on Y .



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 38 of 69

3.2.2. (h, k)-ary (m,n)-SuperhyperNeutrosophic Set

An (h, k)-ary (m,n)-SuperhyperNeutrosophic Set accepts h inputs taken from the m-th
nonempty iterated powerset of a base set and returns k outputs that are elements of the
n-th nonempty iterated powerset of the unit cube [0, 1]3 of neutrosophic triples. In this
way, it models multi-ary interactions between hierarchical (nested) collections of objects
and produces hierarchical collections of neutrosophic assessments—each assessment being
a triple (T, I, F ) ∈ [0, 1]3 of truth, indeterminacy, and falsity degrees satisfying 0 ≤ T +
I + F ≤ 3.

Definition 23 ((h, k)-ary (m,n)-SuperhyperNeutrosophic Set). Let X be a nonempty set,
and let m,n, h, k ∈ N with m,n, h, k ≥ 1. An (h, k)-ary (m,n)-SuperhyperNeutrosophic
Set on X is a mapping

Ã
(m,n)
(h,k) :

(
P̃m(X)

)h −→
(
P̃n([0, 1]3)

)k
with the following property: for every input h-tuple (A1, . . . , Ah) ∈ (P̃m(X))h the image is
a k-tuple (C1, . . . , Ck) with each Cj ∈ P̃n([0, 1]3); in particular, every base-level element
of each Cj is a triple (T, I, F ) ∈ [0, 1]3 satisfying

0 ≤ T + I + F ≤ 3.

Nonemptiness is enforced at every nested level by the use of P̃•.

Example 23 (Unary case (h, k) = (1, 1)). Let X = {a, b} and take m = n = 1. Then
P̃1(X) = P̃(X) =

{
{a}, {b}, {a, b}

}
and P̃1([0, 1]3) = P̃([0, 1]3). A map

Ã
(1,1)
(1,1) : P̃(X) −→ P̃([0, 1]3)

is given, for instance, by

Ã
(1,1)
(1,1)({a}) = {(0.6, 0.2, 0.1)}, Ã

(1,1)
(1,1)({b}) = {(0.4, 0.3, 0.2)},

Ã
(1,1)
(1,1)({a, b}) = {(0.7, 0.1, 0.1), (0.8, 0.0, 0.1)}.

Each triple lies in [0, 1]3 and satisfies T + I + F ≤ 3, as required.

Example 24 (Credit & Liquidity Risk as a (2, 2)-ary (1, 1)-system). Consider banking
indicators

X = {DebtToIncome, CreditScore, MarketVolatility, InterestRate},

and set m = n = 1, h = k = 2. The domain and codomain are

D = P̃(X) × P̃(X),

C = P̃([0, 1]3) × P̃([0, 1]3).



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 39 of 69

Given (A1, A2) ∈ D—say

A1 = {DebtToIncome,CreditScore}

and
A2 = {MarketVolatility, InterestRate}

—define

Ã
(1,1)
(2,2)(A1, A2) =

(
Cdefault, Cliquidity

)
,

with
Cdefault = {(0.70, 0.20, 0.10), (0.65, 0.25, 0.10)},

Cliquidity = {(0.40, 0.40, 0.20)}.

All triples satisfy 0 ≤ T + I + F ≤ 3. Thus Ã
(1,1)
(2,2) is a concrete (2, 2)-ary (1, 1)-

SuperhyperNeutrosophic Set.

Theorem 29 (Relation to classical and fuzzy cases). Every (h, k)-ary (m,n)-SuperhyperNeutrosophic
Set generalizes both (i) the classical (m,n)-SuperhyperNeutrosophic Set (when h = k = 1)
and (ii) the (h, k)-ary (m,n)-SuperhyperFuzzy Set (by projecting triples to their truth com-
ponents).

Proof. Let à = Ã
(m,n)
(h,k) : (P̃m(X))h → (P̃n([0, 1]3))k.

(i) Classical case. Assume h = k = 1. Then the Cartesian powers collapse via the
canonical bijections

ιh : (P̃m(X))1
∼=−−→ P̃m(X),

ιk : (P̃n([0, 1]3))1
∼=−−→ P̃n([0, 1]3),

given by ιh(A) = A and ιk(C) = C. Under these identifications, Ã is precisely a map

à : P̃m(X) −→ P̃n([0, 1]3),

whose values are nonempty n-level neutrosophic sets. This is exactly the classical (m,n)-
SuperhyperNeutrosophic definition.

(ii) Fuzzy case via truth–projection. Define πT : [0, 1]3 → [0, 1] by πT (T, I, F ) = T .
We extend πT to all nesting levels by recursion on r ≥ 1:

Π
(1)
T : P̃1([0, 1]3) → P̃1([0, 1]), Π

(1)
T (S) := { πT (t) | t ∈ S },

and, for r ≥ 1,

Π
(r+1)
T : P̃r+1([0, 1]3) → P̃r+1([0, 1]),

Π
(r+1)
T (S) := { Π

(r)
T (S) | S ∈ S }.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 40 of 69

Nonemptiness is preserved at each step: if S ̸= ∅, then Π
(1)
T (S) ̸= ∅; if S ̸= ∅, then

Π
(r+1)
T (S) ̸= ∅ because it collects the nonempty images of members of S. Put ΠT := Π

(n)
T

and define

Π̂T : (P̃n([0, 1]3))k → (P̃n([0, 1]))k, Π̂T (C1, . . . , Ck) := (ΠT (C1), . . . ,ΠT (Ck)).

Then Π̂T ◦ Ã maps any A ∈ (P̃m(X))h to a k-tuple of nonempty n-level subsets of [0, 1],
i.e. an (h, k)-ary (m,n)-SuperhyperFuzzy Set.

Theorem 30 (Fixing some inputs). Fix indices 1 ≤ i1 < · · · < ir ≤ h and super-elements
Aij ∈ P̃m(X). The map

Ãfix :
(
P̃m(X)

)h−r −→
(
P̃n([0, 1]3)

)k
,

defined by inserting the fixed Aij into the corresponding coordinates of an (h − r)-tuple

and then applying Ã, is an (h− r, k)-ary (m,n)-SuperhyperNeutrosophic Set.

Proof. Let I = {1, . . . , h} and J = {i1, . . . , ir}. For a free-input tuple B = (Bℓ)ℓ∈I\J ∈
(P̃m(X))h−r, define the insertion map

ι(B) = A = (A1, . . . , Ah) ∈ (P̃m(X))h, Au =

{
Aij , u ∈ J with u = ij ,

Bu, u ∈ I \ J.

Then ι is well-defined and ι(B) has every coordinate in P̃m(X). Set Ãfix := Ã ◦ ι. For any
B, Ãfix(B) = Ã(ι(B)) is a k-tuple whose jth coordinate lies in P̃n([0, 1]3) by definition of
Ã. Thus Ãfix is an (h− r, k)-ary (m,n)-SuperhyperNeutrosophic Set.

Theorem 31 (Projection to selected outputs). Let 1 ≤ j1 < · · · < js ≤ k. The coordinate
projection

Ãproj(A) =
(
Cj1 , . . . , Cjs

)
, where Ã(A) = (C1, . . . , Ck),

defines an (h, s)-ary (m,n)-SuperhyperNeutrosophic Set Ãproj : (P̃m(X))h → (P̃n([0, 1]3))s.

Proof. Define PJ : (P̃n([0, 1]3))k → (P̃n([0, 1]3))s by PJ(C1, . . . , Ck) = (Cj1 , . . . , Cjs).
Each Cjr ∈ P̃n([0, 1]3) and is nonempty, so PJ is well-defined. Put Ãproj := PJ ◦ Ã to
obtain a map with domain (P̃m(X))h and codomain (P̃n([0, 1]3))s, as required.

Theorem 32 (Pointwise union). If Ã1, Ã2 : (P̃m(X))h → (P̃n([0, 1]3))k are (h, k)-ary
(m,n)-SuperhyperNeutrosophic Sets, then

(Ã1 ∪ Ã2)(A) :=
(
Ã1(A)j ∪ Ã2(A)j

)k
j=1

is again an (h, k)-ary (m,n)-SuperhyperNeutrosophic Set.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 41 of 69

Proof. We prove a general lemma by induction on r ≥ 1:
Lemma. If U, V ∈ P̃r(S), then U ∪ V ∈ P̃r(S).
r = 1: U, V are nonempty subsets of S, hence U ∪ V is a nonempty subset of S, i.e. in

P̃1(S).
r → r + 1: U, V are nonempty families of r-level elements. Thus U ∪ V is a nonempty

family whose members all lie in P̃r(S); hence U ∪ V ∈ P̃r+1(S).
Apply the lemma with S = [0, 1]3 and r = n. For each input A and coordinate j,

both Ã1(A)j and Ã2(A)j lie in P̃n([0, 1]3) and are nonempty, so their union does as well.
Moreover, every base-level element of the union is still a triple (T, I, F ) with 0 ≤ T+I+F ≤
3 (the constraint is preserved under set union). Therefore (Ã1 ∪ Ã2)(A) ∈ (P̃n([0, 1]3))k

for all A.

Theorem 33 (Pointwise intersection). With the same hypotheses, define

(Ã1 ∩ Ã2)(A) :=
(
Ã1(A)j ∩ Ã2(A)j

)k
j=1

.

If every coordinate-wise intersection is nonempty, then (Ã1 ∩ Ã2) is an (h, k)-ary (m,n)-
SuperhyperNeutrosophic Set.

Proof. We use the companion lemma (proved by induction on r ≥ 1):
Lemma. If U, V ∈ P̃r(S) and U ∩ V ̸= ∅, then U ∩ V ∈ P̃r(S).
r = 1: U, V are nonempty subsets of S. If U ∩V ̸= ∅, then U ∩V is a nonempty subset

of S, i.e. in P̃1(S).
r → r+ 1: U, V are nonempty families of r-level elements. If U ∩V ̸= ∅, every element

of U ∩ V is itself in P̃r(S), and U ∩ V is nonempty; hence U ∩ V ∈ P̃r+1(S).
Apply the lemma with S = [0, 1]3 and r = n. The nonemptiness hypothesis ensures

each coordinate-wise intersection lies in P̃n([0, 1]3). Neutrosophic constraints are point-
wise and are preserved by set-theoretic intersection. Therefore (Ã1 ∩ Ã2) has the desired
codomain.

Theorem 34 (Neutrosophic λ-cuts). For λ = (α, β, γ) ∈ [0, 1]3, define the cut

Cλ =
{
A ∈ (P̃m(X))h

∣∣∣ ∃j, ∃(T, I, F ) ∈ Ã(A)j with T ≥ α, I ≤ β, F ≤ γ
}
.

If λ′ = (α′, β′, γ′) satisfies α′ ≥ α, β′ ≤ β, and γ′ ≤ γ, then Cλ′ ⊆ Cλ.

Proof. Take any A ∈ Cλ′ . By definition, there exist an index j and a triple (T, I, F ) ∈
Ã(A)j with T ≥ α′, I ≤ β′, F ≤ γ′. Since α′ ≥ α, β′ ≤ β, and γ′ ≤ γ, the same triple
satisfies T ≥ α, I ≤ β, F ≤ γ, so A ∈ Cλ. Thus Cλ′ ⊆ Cλ.

Theorem 35 (Truth-projection yields fuzzy structure). Let πT : [0, 1]3 → [0, 1] be πT (T, I, F ) =
T and extend it levelwise to ΠT : (P̃n([0, 1]3))k → (P̃n([0, 1]))k. Then ΠT ◦ Ã is an (h, k)-
ary (m,n)-SuperhyperFuzzy Set.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 42 of 69

Proof. Define ΠT recursively as in the proof of the Relation Theorem: base level n = 1
by taking elementwise truth components, and for n > 1 by applying the previous level to
each member of a nonempty family. Nonemptiness and the nesting level are preserved at
each step. Hence for every input A, (ΠT ◦ Ã)(A) is a k-tuple of nonempty n-level subsets
of [0, 1], i.e. an element of (P̃n([0, 1]))k. Therefore ΠT ◦ Ã satisfies the definition of an
(h, k)-ary (m,n)-SuperhyperFuzzy Set.

Theorem 36 (Functoriality under surjections). Let f : X → Y be surjective. Define
recursively the nonempty preimage maps f−1

(r) : P̃r(Y ) → P̃r(X) by

f−1
(1) (B) := {x ∈ X : f(x) ∈ B} (B ∈ P̃(Y )), f−1

(r+1)(B) :=
{
f−1
(r) (B)

∣∣ B ∈ B
}
.

Then the pushforward

(f∗Ã)(B1, . . . , Bh) := Ã
(
f−1
(m)(B1), . . . , f

−1
(m)(Bh)

)
, (Bi ∈ P̃m(Y )),

defines an (h, k)-ary (m,n)-SuperhyperNeutrosophic Set on Y .

Proof. We show by induction on r ≥ 1 that f−1
(r) maps P̃r(Y ) into P̃r(X) and preserves

nonemptiness.
r = 1: Let B ∈ P̃(Y ) be nonempty. Pick y ∈ B; surjectivity gives x ∈ X with

f(x) = y, hence x ∈ f−1
(1) (B) and f−1

(1) (B) ∈ P̃(X).

r → r + 1: Let B ∈ P̃r+1(Y ) be nonempty. For each B ∈ B, the inductive hypothesis
yields f−1

(r) (B) ∈ P̃r(X). Since B ̸= ∅, the collection {f−1
(r) (B) | B ∈ B} is a nonempty

family in P̃r(X), i.e. an element of P̃r+1(X).
Now let (B1, . . . , Bh) ∈ (P̃m(Y ))h. Then f−1

(m)(Bi) ∈ P̃m(X) for each i, so à applies

to yield a k-tuple in (P̃n([0, 1]3))k with nonempty coordinates. Hence f∗Ã is well-defined
and has the required domain and codomain.

Theorem 37 (Reduction to the classical unary case). If h = k = 1, then an (h, k)-ary
(m,n)-SuperhyperNeutrosophic Set à reduces to the classical (m,n)-SuperhyperNeutrosophic
mapping à : P̃m(X) → P̃n([0, 1]3).

Proof. The canonical bijections (P̃m(X))1 ∼= P̃m(X) and (P̃n([0, 1]3))1 ∼= P̃n([0, 1]3)
identify the (h, k)-ary map with a unary map of the stated type. No additional properties
are needed, since nonemptiness and neutrosophic constraints are part of Ã’s codomain by
definition.

3.2.3. (h, k)-ary (m,n)-SuperhyperPlithogenic Set

An (h, k)-ary (m,n)-SuperhyperPlithogenic Set assigns to each h–tuple of m–level (nested)
parameter–subsets a k–tuple of n–level sets of attribute–based membership vectors in
[0, 1]s, together with a quantified, symmetric degree of contradiction on attribute values.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 43 of 69

For a nonempty set S, write

P0
+(S) := S, Pr+1

+ (S) := {T ⊆ Pr
+(S) | T ̸= ∅ } (r ≥ 0),

so P1
+(S) is the collection of all nonempty subsets of S, P2

+(S) is the collection of all
nonempty families of nonempty subsets of S, and so on. On [0, 1]s we use the componen-
twise order: for x, y ∈ [0, 1]s,

x ⪯ y ⇐⇒ xi ≤ yi for all i = 1, . . . , s.

Definition 24 ((h, k)-ary (m,n)-SuperhyperPlithogenic Set). Let P be a nonempty base
set of parameters, fix integers m,n, h, k ∈ N with m,n, h, k ≥ 1, and fix dimensions s, t ∈ N
with s, t ≥ 1. Let v be an attribute with value set Pv (nonempty). Define

D :=
(
Pm
+ (P )

)h
, C :=

(
Pn
+([0, 1]s)

)k
.

An (h, k)-ary (m,n)-SuperhyperPlithogenic Set (with respect to v) is a quintuple

SHP
(h,k)
pl (m,n) =

(
D, v, Pv, ˜pdf

(m,n)
v , pCF

)
,

where:

(i) ˜pdf
(m,n)
v : D × Pv → C (the Hyper Degree of Appurtenance) assigns, to each (A, a)

with A = (A1, . . . , Ah) ∈ D and a ∈ Pv, a k–tuple

˜pdf
(m,n)
v (A, a) = (B1, . . . , Bk) ∈ C,

where each Bj ∈ Pn
+([0, 1]s) is a nonempty, level–n collection of membership vectors

x ∈ [0, 1]s;

(ii) pCF : Pv × Pv → [0, 1]t (the Degree of Contradiction) is reflexive and symmetric:

pCF (a, a) = 0 ∈ [0, 1]t, pCF (a, b) = pCF (b, a) for all a, b ∈ Pv.

Example 25 (Smartphone selection: (2, 2)-ary (1, 1) case). Let P = {iPhone,Galaxy,Pixel}
and the attribute

v = BatteryLifeCategory, Pv = {Low,Medium,High}.

Fix m = n = 1, h = k = 2, s = 1, so

D = P1
+(P ) × P1

+(P ) = P(P ) × P(P )

and
C = P([0, 1]) × P([0, 1])



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 44 of 69

. Interpret an input A = (A1, A2) ∈ D as: A1 = phones within the preferred price range;
A2 = phones from preferred brands. Define, for the value a = High,

˜pdf
(1,1)
v

(
({iPhone,Galaxy}, {iPhone,Pixel}), a

)
=

(
[0.7, 0.9], {0.8}

)
,

and for a = Medium,

˜pdf
(1,1)
v

(
({iPhone,Galaxy}, {iPhone,Pixel}), a

)
=

(
{0.5, 0.6}, [0.4, 0.6]

)
.

Here the first coordinate encodes the fuzzy membership to “High battery life”, while the sec-
ond aggregates a secondary satisfaction score. Set pCF (Low,High) = 1, pCF (Medium,Low) =

0.5, pCF (a, a) = 0, and extend by symmetry. Then SHP
(2,2)
pl (1, 1) models two–stage, at-

tribute–based assessments with quantified contradictions.

Example 26 (Customer pricing preference: (1, 1)-ary (1, 1) case). Let P = {Laptop, Smartphone}
and v = PricingTier with Pv = {Budget,Premium}. Fix m = n = h = k = s = t = 1.
Then D = P(P ) and C = P([0, 1]). Define

˜pdf
(1,1)
v ({Laptop},Premium) = [0.70, 0.90],

˜pdf
(1,1)
v ({Laptop},Budget) = {0.40, 0.50},

˜pdf
(1,1)
v ({Smartphone},Premium) = {0.80},
˜pdf

(1,1)
v ({Smartphone},Budget) = [0.30, 0.45],

˜pdf
(1,1)
v ({Laptop, Smartphone},Premium) = [0.65, 0.85],

˜pdf
(1,1)
v ({Laptop, Smartphone},Budget) = {0.35, 0.55}.

With pCF (Budget,Premium) = pCF (Premium,Budget) = 0.8 and pCF (a, a) = 0, the
structure records fuzzy membership to pricing tiers and their quantified contradiction.

Example 27 (Filtered product satisfaction: (2, 2)-ary (1, 1) with vector degrees). Let
P = {Laptop,Tablet,Smartphone}, and fix m = n = 1, h = k = 2, s = 2, t = 1. Let v =
Satisfaction with Pv = {Low,Medium,High}. For the input

(
{Laptop,Tablet}, {Premium}

)
at a = High, set

˜pdf
(1,1)
v

((
{Laptop,Tablet}, {Premium}

)
,High

)
=

(
{(0.85, 0.80), (0.88, 0.82)}, {(0.90, 0.85)}

)
,

where each (x, y) ∈ [0, 1]2 denotes (reviewScore, returnRateScore).
Let pCF (High,Low) = 1, pCF (High,Medium) = 0.6, pCF (Medium,Low) = 0.4,

pCF (a, a) = 0, with symmetry.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 45 of 69

Example 28 (Hierarchical sensor–analytics confidence: (2, 2)-ary (3, 3)). Let P = {Temp,Pressure},
m = n = 3, h = k = 2, s = 2, t = 1, and v = ConfidenceType with Pv = {Low,Medium,High}.
Then D = P3

+(P ) × P3
+(P ) and C = (P3

+([0, 1]2))2. Pick

A =
{
{{Temp}, {Temp,Pressure}}, {{Pressure}}

}
,

B =
{
{{Pressure}, {Temp,Pressure}}

}
,

and define, at A = (A,B) and a = High,

˜pdf
(3,3)
v (A, a) = (E1, E2),

where
E1 =

{
{(0.90, 0.85), (0.92, 0.88)}, {(0.88, 0.80)}

}
,

E2 =
{
{(0.75, 0.70)}, {(0.78, 0.73), (0.80, 0.75)}

}
.

Set pCF (Low,High) = 0.9, pCF (Low,Medium) = 0.5, pCF (Medium,High) = 0.6, pCF (a, a) =
0, with symmetry.

Example 29 (Urban Route Planning as a (2, 2)-ary (2, 2)-SuperhyperPlithogenic Set).
Setting. Consider an urban mobility platform that recommends public–transport routes.
Let the base parameter set be

P = {CostCap, TimeWindow, MaxTransfers, EmissionCap}.

Fix (m,n, h, k) = (2, 2, 2, 2) and dimensions s = 2, t = 1. We interpret the two member-
ship coordinates x = (x1, x2) ∈ [0, 1]2 as

x1 = (commuter satisfaction), x2 = (operational reliability).

The two output coordinates (k = 2) will represent two stakeholder views:

output 1 = commuter view, output 2 = operator view.

Domain and codomain. The input space is

D =
(
P2
+(P )

)2
,

so each input is a pair A = (A1, A2) of level–2 hyper–parameters, with Ai ∈ P2
+(P ) =

{ Γ ⊆ P1
+(P ) | Γ ̸= ∅ }. The output space is

C =
(
P2
+([0, 1]2)

)2
,

so each output coordinate is a nonempty family (level–2) of nonempty subsets (level–1) of
[0, 1]2.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 46 of 69

Attribute and its values. Let the plithogenic attribute be

v = RouteQuality, Pv = {Eco, Fast, Cheap}.

We quantify incompatibility between attribute values via the degree–of–contradiction

pCF : Pv × Pv → [0, 1], pCF (a, a) = 0, pCF (a, b) = pCF (b, a),

specified by

pCF (Eco,Fast) = 0.6, pCF (Eco,Cheap) = 0.4, pCF (Fast,Cheap) = 0.5.

A concrete input. Interpret A1 as a policy/constraint cluster and A2 as a user–preference
cluster. Choose

A1 =
{

{CostCap,EmissionCap}, {TimeWindow}
}
,

A2 =
{

{MaxTransfers}, {TimeWindow,EmissionCap}
}
,

so A = (A1, A2) ∈ D.

Hyper Degree of Appurtenance (HDAF). For each attribute value a ∈ Pv, the
map

˜pdf
(2,2)
v : D × Pv −→ C

returns a pair
(
B

(a)
1 , B

(a)
2

)
with B

(a)
j ∈ P2

+([0, 1]2). At the input A above, define:

Eco:
B

(Eco)
1 =

{
{(0.88, 0.84), (0.86, 0.83)}, {(0.82, 0.80)}

}
,

B
(Eco)
2 =

{
{(0.78, 0.75)}, {(0.81, 0.79)}

}
.

Fast:
B

(Fast)
1 =

{
{(0.90, 0.72)}, {(0.85, 0.70), (0.83, 0.69)}

}
,

B
(Fast)
2 =

{
{(0.76, 0.68)}

}
.

Cheap:

B
(Cheap)
1 =

{
{(0.80, 0.77)}, {(0.74, 0.73)}

}
,

B
(Cheap)
2 =

{
{(0.88, 0.65)}

}
.

Each B
(a)
j is a level–2 object: a nonempty family of nonempty sets of two–dimensional

membership vectors (satisfaction, reliability) ∈ [0, 1]2. The first coordinate (j=1) aggre-
gates commuter–side appraisals; the second (j=2) aggregates operator–side appraisals.
The plithogenic nature appears in the simultaneous presence of multiple attribute values a
together with the quantified contradictions pCF between them.

The choice (A1, A2) prioritizes emission limits, moderate cost caps, and few transfers
within a usable time window. Consequently, Eco achieves high (x1, x2) on both outputs



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 47 of 69

(good satisfaction without sacrificing reliability), whereas Fast increases satisfaction but
lowers reliability for the operator; Cheap is relatively reliable for the operator yet yields only
moderate commuter satisfaction. The contradiction matrix pCF quantifies these trade–offs
(e.g., Eco vs. Fast at 0.6), enabling plithogenic decision aggregation across attribute values.
Thus, all components (

D, v, Pv, ˜pdf
(2,2)
v , pCF

)
instantiate a concrete (2, 2)-ary (2, 2)-SuperhyperPlithogenic Set for multi–criteria route
planning.

Theorem 38 (Unary reduction). Every (m,n)-SuperhyperPlithogenic Set is the special
case h = k = 1 of Definition 24.

Proof. By Definition 24, the multi–ary structure is

SHP
(h,k)
pl (m,n) =

(
D, v, Pv, ˜pdf

(m,n)
v , pCF

)
, D = (Pm

+ (P ))h, C = (Pn
+([0, 1]s))k.

Set h = k = 1. There are canonical bijections

Φ : (Pm
+ (P ))1 → Pm

+ (P ), Ψ :
(
Pn
+([0, 1]s)

)1 → Pn
+([0, 1]s),

given by Φ(A) = A and Ψ(B) = B. Under Φ and Ψ, the map

˜pdf
(m,n)
v : (Pm

+ (P ))1 × Pv −→
(
Pn
+([0, 1]s)

)1
is identified with

˜pdf
(m,n)
v : Pm

+ (P ) × Pv −→ Pn
+([0, 1]s),

which is precisely the classical (m,n)-SuperhyperPlithogenic mapping (with the same
attribute v, value set Pv, and contradiction function pCF ). Reflexivity pCF (a, a) = 0
and symmetry pCF (a, b) = pCF (b, a) are unaffected by the identifications. Hence the
unary case is exactly the classical one.

Theorem 39 (Fuzzy and neutrosophic specializations). An (h, k)-ary (m,n)-SuperhyperPlithogenic
Set specializes to

(i) an (h, k)-ary (m,n)-SuperhyperFuzzy Set when s = 1, and

(ii) an (h, k)-ary (m,n)-SuperhyperNeutrosophic Set when s = 3 (with triples (T, I, F )).

Proof. Let ˜pdf
(m,n)
v : D × Pv → C with D = (Pm

+ (P ))h and C = (Pn
+([0, 1]s))k.

(i) If s = 1, then [0, 1]s = [0, 1] and C = (Pn
+([0, 1]))k. Thus

˜pdf
(m,n)
v : D × Pv −→

(
Pn
+([0, 1])

)k
assigns to each input a k–tuple of nonempty, level–n fuzzy degree sets, which is exactly
the codomain required by the (h, k)-ary (m,n)-SuperhyperFuzzy structure. The domain
and pCF are unchanged, so all axioms remain valid.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 48 of 69

(ii) If s = 3, then [0, 1]s = [0, 1]3 and C = (Pn
+([0, 1]3))k. Hence

˜pdf
(m,n)
v : D × Pv −→

(
Pn
+([0, 1]3)

)k
returns k level–n families of neutrosophic triples (T, I, F ) ∈ [0, 1]3 (with any standard
neutrosophic constraint, e.g. 0 ≤ T + I + F ≤ 3, enforced elementwise when imposed).
This is precisely the codomain of the (h, k)-ary (m,n)-SuperhyperNeutrosophic structure.

Theorem 40 (Fixing some inputs). Fix indices 1 ≤ i1 < · · · < ir ≤ h and super–elements
Aij ∈ Pm

+ (P ). Define ι : (Pm
+ (P ))h−r ↪→ (Pm

+ (P ))h by inserting the fixed Aij in the
positions ij. Then

˜pdf
(m,n)
fix (B, a) := ˜pdf

(m,n)
v (ι(B), a)

together with pCF yields an (h− r, k)-ary (m,n)-SuperhyperPlithogenic Set.

Proof. Let B = (Bℓ)ℓ∈Ifree ∈ (Pm
+ (P ))h−r, where I free = {1, . . . , h}\{i1, . . . , ir}. Define

the h–tuple ι(B) = (C1, . . . , Ch) by

Cu =

{
Aij , u = ij for some j,

Bℓ, u = ℓ ∈ I free.

Thus ι(B) ∈ (Pm
+ (P ))h. For any a ∈ Pv,

˜pdf
(m,n)
fix (B, a) = ˜pdf

(m,n)
v (ι(B), a) ∈ (Pn

+([0, 1]s))k

has nonempty coordinates because ˜pdf
(m,n)
v does so for every h–tuple. Hence ˜pdf

(m,n)
fix is a

valid hyper degree function on the smaller domain (Pm
+ (P ))h−r × Pv. The contradiction

function pCF is defined only on Pv ×Pv and does not depend on the arity of the domain;
its axioms (reflexivity, symmetry) are preserved.

Theorem 41 (Projection to fewer outputs). Let 1 ≤ j1 < · · · < js ≤ k. Define πj1,...,js :
C →

(
Pn
+([0, 1]s)

)s
by

πj1,...,js(B1, . . . , Bk) := (Bj1 , . . . , Bjs).

Then (A, a) 7→ πj1,...,js
(

˜pdf
(m,n)
v (A, a)

)
(with D and the same pCF ) defines an (h, s)-ary

(m,n)-SuperhyperPlithogenic Set.

Proof. Fix (A, a) ∈ D × Pv. Since ˜pdf
(m,n)
v (A, a) = (B1, . . . , Bk) ∈ C, each Bjr is a

nonempty element of Pn
+([0, 1]s). Therefore

πj1,...,js
(

˜pdf
(m,n)
v (A, a)

)
= (Bj1 , . . . , Bjs) ∈

(
Pn
+([0, 1]s)

)s
.

Nonemptiness and level–n nesting are preserved coordinatewise. The domain D and the
contradiction function pCF remain unchanged, so the projected structure satisfies Defini-
tion 24 with output arity s.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 49 of 69

Theorem 42 (Closure under pointwise union). If ˜pdf, ˜pdf
′
: D×Pv → C are hyper degree

functions (with the same pCF ), then

( ˜pdf ∪ ˜pdf
′
)(A, a) := ˜pdf(A, a) ∪ ˜pdf

′
(A, a)

(where the union is taken coordinatewise in C) defines another hyper degree function,
hence an (h, k)-ary (m,n)-SuperhyperPlithogenic Set.

Proof. Fix (A, a). Write ˜pdf(A, a) = (B1, . . . , Bk) and ˜pdf
′
(A, a) = (B′

1, . . . , B
′
k), with

Bj , B
′
j ∈ Pn

+([0, 1]s). We show by induction on n ≥ 1 that Bj ∪B′
j ∈ Pn

+([0, 1]s).
Base n = 1: Bj and B′

j are nonempty subsets of [0, 1]s, so Bj ∪ B′
j is a nonempty

subset of [0, 1]s, i.e. an element of P1
+([0, 1]s).

Inductive step: suppose the claim holds for level n. Let n+1. Then Bj , B
′
j ⊆ Pn

+([0, 1]s)
are nonempty families of level–n elements. Their union Bj ∪ B′

j is nonempty (as a union
of two nonempty sets) and every element of it still lies in Pn

+([0, 1]s). Hence Bj ∪ B′
j ∈

Pn+1
+ ([0, 1]s).

Applying this coordinatewise yields ( ˜pdf ∪ ˜pdf
′
)(A, a) ∈ C with nonempty coordinates.

The domain D, attribute data (v, Pv), and pCF are unchanged, so ( ˜pdf ∪ ˜pdf
′
) is a valid

hyper degree function.

Theorem 43 (Closure under pointwise intersection). With the same hypotheses as The-
orem 42, define

( ˜pdf ∩ ˜pdf
′
)(A, a) := ˜pdf(A, a) ∩ ˜pdf

′
(A, a)

(coordinatewise). If every coordinatewise intersection is nonempty, this again defines a
valid hyper degree function.

Proof. Fix (A, a) and j. Assume Bj ∩ B′
j ̸= ∅, where Bj , B

′
j ∈ Pn

+([0, 1]s). We prove
by induction on n ≥ 1 that Bj ∩B′

j ∈ Pn
+([0, 1]s).

Base n = 1: Bj , B
′
j ⊆ [0, 1]s are nonempty and Bj ∩ B′

j ̸= ∅ by hypothesis, so

Bj ∩B′
j ∈ P1

+([0, 1]s).
Inductive step: suppose the claim holds for level n. For level n+1, Bj , B

′
j ⊆ Pn

+([0, 1]s)
are nonempty families. The intersection Bj ∩ B′

j is a family of level–n elements; by the

nonemptiness hypothesis it belongs to Pn+1
+ ([0, 1]s).

Thus each coordinate intersection is (by assumption) nonempty and of the correct

level. Collecting the k coordinates gives ( ˜pdf ∩ ˜pdf
′
)(A, a) ∈ C, so the result is a valid

hyper degree function.

Theorem 44 (Nested α-cuts). Fix a ∈ Pv and α ∈ [0, 1]s. Define

C a
α :=

{
A ∈ D

∣∣∣ ∃ j ∈ {1, . . . , k} ∃x ∈
[

˜pdf
(m,n)
v (A, a)

]
j

with x ⪰ α
}
,

where ⪰ is the componentwise order on [0, 1]s. If α′, α ∈ [0, 1]s with α′ ⪰ α, then C a
α′ ⊆

C a
α .



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 50 of 69

Proof. Let A ∈ C a
α′ . Then there exist a coordinate j and a vector x ∈ [ ˜pdf

(m,n)
v (A, a)]j

such that x ⪰ α′. Since α′ ⪰ α, transitivity of ⪰ implies x ⪰ α. Hence the same j and x
witness A ∈ C a

α . Therefore C a
α′ ⊆ C a

α , i.e. the family {C a
α }α is nested decreasing in the

threshold α.

Theorem 45 (Compatibility with surjective mappings). Let f : P → Q be surjective. For
r ≥ 1 define recursively

f−1
(1) (B) := {x ∈ P | f(x) ∈ B } (B ∈ P1

+(Q)), f−1
(r+1)(B) := { f−1

(r) (B) | B ∈ B }.

Given ˜pdf
(m,n)
v : DP × Pv → C on P , set

˜pdf
(m,n)
v;Q (B, a) := ˜pdf

(m,n)
v

(
f−1
(m)(B), a

)
, B ∈ DQ := (Pm

+ (Q))h.

Then
(
DQ, v, Pv, ˜pdf

(m,n)
v;Q , pCF

)
is an (h, k)-ary (m,n)-SuperhyperPlithogenic Set on Q.

Proof. Step 1 (well-definedness and nonemptiness of f−1
(r) ). We prove by induc-

tion on r ≥ 1 that f−1
(r) : Pr

+(Q) → Pr
+(P ) is well defined and preserves nonemptiness.

Base r = 1: If B ∈ P1
+(Q), then B ̸= ∅. Surjectivity of f ensures ∃x ∈ P with

f(x) ∈ B, hence f−1
(1) (B) ̸= ∅ and f−1

(1) (B) ⊆ P , so f−1
(1) (B) ∈ P1

+(P ).

Inductive step: suppose f−1
(r) maps Pr

+(Q) into Pr
+(P ) and preserves nonemptiness.

Let B ∈ Pr+1
+ (Q) be nonempty. For each B ∈ B, f−1

(r) (B) ∈ Pr
+(P ) and the family

{ f−1
(r) (B) | B ∈ B } is nonempty. Hence f−1

(r+1)(B) ∈ Pr+1
+ (P ).

Step 2 (type checking for the pushforward). Take any B = (B1, . . . , Bh) ∈ DQ =
(Pm

+ (Q))h. By Step 1, f−1
(m)(Bi) ∈ Pm

+ (P ) for every i, so f−1
(m)(B) := (f−1

(m)(B1), . . . , f
−1
(m)(Bh)) ∈

DP . Therefore

˜pdf
(m,n)
v;Q (B, a) = ˜pdf

(m,n)
v

(
f−1
(m)(B), a

)
∈ C = (Pn

+([0, 1]s))k

with nonempty coordinates. The attribute v, its value set Pv, and the contradiction
function pCF are unchanged. Hence all clauses of Definition 24 hold on Q.

Theorem 46 (Reduction to the classical case). If h = k = 1, then ˜pdf
(m,n)
v : Pm

+ (P ) ×
Pv → Pn

+([0, 1]s) is exactly the classical (m,n)-SuperhyperPlithogenic Set.

Proof. As in the proof of Unary reduction, apply the canonical identifications

(Pm
+ (P ))1 ∼= Pm

+ (P ),
(
Pn
+([0, 1]s)

)1 ∼= Pn
+([0, 1]s),

to rewrite the multi–ary map on D × Pv into the unary map

˜pdf
(m,n)
v : Pm

+ (P ) × Pv −→ Pn
+([0, 1]s).

All structural data (v, Pv, pCF ) are preserved verbatim, and nonemptiness/level con-
straints in the codomain are unchanged. This is precisely the classical definition.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 51 of 69

4. Additional Result: (h,k)-ary (m,n)-SuperhyperSoft Set and (h,k)-ary
(m,n)-SuperhyperRough Set

Although SuperHyperSoft Sets and SuperHyperRough Sets have already been studied
in existing literature, their formulations exhibit slight conceptual differences from the
framework of SuperHyperUncertain Sets presented above. Therefore, in this paper, we
deliberately reconsider and refine their definitions to better suit the context and objectives
of the present study. It is important to note that the existing definitions of SuperHyperSoft
Sets and SuperHyperRough Sets are, of course, mathematically correct.

Note that soft Sets assign parameters to subsets of a universe, while k-ary Soft Sets
extend this mapping to tuples. In contrast, (h,k)-ary (m,n)-SuperhyperSoft Sets embed
hierarchical multi-uncertainty using iterated powersets and superhyperoperations, cap-
turing complex interactions across multiple structural levels. Rough Sets approximate
subsets by lower and upper bounds under an indiscernibility relation. (h,k)-ary (m,n)-
SuperhyperRough Sets extend this by encoding hierarchical, multi-ary superhyperstruc-
tures, enabling refined layered approximations, multi-level granularity, and advanced at-
tribute interactions for complex uncertain domains.

4.1. (h,k)-ary (m,n)-SuperhyperSoft Set

For a nonempty set X, write

P+(X) := {A ⊆ X | A ̸= ∅ }

for the nonempty powerset. Define the rth nonempty iterated powerset recursively by

P̃0(X) := X, P̃r+1(X) := P+

(
P̃r(X)

)
(r ≥ 0).

We use the componentwise inclusion order on Cartesian powers: for A = (A1, . . . , Ah), A′ =
(A′

1, . . . , A
′
h) ∈ (P̃m(S))h, write A ⊆ A′ if Ai ⊆ A′

i for all i.

In classical soft set theory [8, 69], one fixes a parameter set S and uses a mapping
F : S → P(U). To capture hierarchical parameter groupings and multi-output responses,
we lift parameters to the mth iterated nonempty powerset and allow h inputs (interacting
parameter aggregates), while the outputs are lifted to the nth iterated nonempty powerset
and replicated in k coordinates.

Definition 25 ((h, k)-ary (m,n)-SuperhyperSoft Set). Let U be a nonempty universe
and S a nonempty parameter set. Fix integers m,n, h, k ≥ 1. An (h, k)-ary (m,n)-
SuperhyperSoft Set over U (with respect to S) is a mapping

F :
(
P̃m(S)

)h −→
(
P̃n(U)

)k
.

For every input A = (A1, . . . , Ah) ∈ (P̃m(S))h the value

F (A) =
(
B1, . . . , Bk

)



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 52 of 69

is a k-tuple with each Bj ∈ P̃n(U) (in particular, each Bj is nonempty and n-nested).
When m = n = h = k = 1 this reduces (up to the conventional treatment of the empty set)
to the classical soft set.

Example 30 (Unary case). Let U = {u1, u2, u3} and S = {a, b}. Then P̃1(S) =
{{a}, {b}, {a, b}} and P̃1(U) = {B ⊆ U | B ̸= ∅ }. For m = n = h = k = 1, an
(1, 1)-ary (1, 1)-SuperhyperSoft Set is a map F : P̃1(S) → P̃1(U), e.g.

F ({a}) = {u1, u2}, F ({b}) = {u2, u3}, F ({a, b}) = {u1, u2, u3}.

If one increases h and k, inputs become tuples of parameter-aggregates and outputs become
tuples of nonempty U -subsets (or, for n > 1, of nested nonempty families of such).

Example 31 (Personalized movie recommendations). Let U = {M1, . . . ,M6} be a catalog
and S = {Comedy,Action,Drama}. Choose m = 2, n = 1, h = 2, k = 2. Then

P̃1(S) = P+(S), P̃2(S) = P+

(
P+(S)

)
,

so each element of P̃2(S) is a nonempty collection of nonempty genre-subsets. Set

D =
(
P̃2(S)

)2
, C =

(
P̃1(U)

)2
.

Pick

γ(1) =
{
{Comedy}, {Action,Drama}

}
, γ(2) =

{
{Action}, {Drama,Comedy}

}
∈ P̃2(S).

A (2, 2)-ary (2, 1)-SuperhyperSoft map F : D → C may assign

F
(
γ(1), γ(2)

)
=

(
B1, B2

)
, B1 = {M1,M3,M5}, B2 = {M2,M4},

interpreted as primary and secondary recommendations driven by two hierarchical genre
profiles.

Example 32 (University Course Planning as a (2, 2)-ary (2, 1)-SuperhyperSoft Set). Let
the universe of objects be the catalog of next-term courses

U = {CS101,CS102,MATH201,STAT210,AI310,NLP320,SYS220,HCI230}.

Let the parameter set encode requirement tags and scheduling constraints:

S = {Req-Core,Req-Math,Req-AI,Req-Systems,

Time-Morning,Time-Afternoon,Mode-InPerson,No-Friday}.
We choose m = 2, n = 1, h = 2, k = 2. Thus the domain is D = (P̃2(S))2, where

P̃1(S) = {A ⊆ S | A ̸= ∅},

P̃2(S) = {Γ ⊆ P̃1(S) | Γ ̸= ∅ }.

The codomain is (P̃1(U))2, i.e., ordered pairs of nonempty subsets of U .
Interpret the two input coordinates as:



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 53 of 69

• coordinate 1: a cluster of degree requirements (hyper–parameter on S),

• coordinate 2: a cluster of personal constraints/preferences (hyper–parameter on S).

Consider the concrete hyper–parameters

Γreq =
{
{Req-Core}, {Req-Math,Req-AI}

}
∈ P̃2(S),

Γpref =
{
{Time-Afternoon}, {Mode-InPerson}, {No-Friday}

}
∈ P̃2(S).

An (h, k)-ary (m,n)-SuperhyperSoft Set is a mapping

F : D = (P̃2(S))2 −→ (P̃1(U))2,

(Γreq,Γpref) 7−→
(
B1, B2

)
,

where B1, B2 ⊆ U are nonempty. For the above input, define (one valid instantiation)

B1 = {CS101,MATH201,AI310}
(primary plan: meets core/math/AI while fitting afternoon, in–person, no–Friday),

B2 = {STAT210,NLP320,HCI230}
(contingency plan under the same preferences).

Hence
F
(
Γreq,Γpref

)
=

(
B1, B2

)
∈
(
P̃1(U)

)2
.

The nested, set–of–subsets form of Γreq and Γpref lets a program advisor encode hi-
erarchical requirement/preference groupings (e.g., “satisfy Req-Core and at least one of
{Req-Math,Req-AI}”) together with scheduling clusters (e.g., “Afternoon and In–Person
and No–Friday”). The two outputs B1, B2 provide a primary and a backup recommenda-
tion set, each a nonempty element of P̃1(U). This realizes, in a real advising scenario, a
(2, 2)-ary (2, 1)-SuperhyperSoft Set in the sense of the definition above.

Theorem 47 (Classical case as a special instance). Every classical soft set Fcl : S → P(U)
is naturally represented by an (h, k)-ary (m,n)-SuperhyperSoft Set with m = n = h = k =
1.

Proof. Set m = n = h = k = 1. The domain of a (1, 1)-ary (1, 1)-SuperhyperSoft Set is
P̃1(S) = P+(S) (nonempty subsets of S) and its codomain is P̃1(U) = P+(U) (nonempty
subsets of U).

Step 1 (embedding singletons). Define the canonical injection

ιS : S −→ P+(S), ιS(s) := {s}.

Although ιS is not surjective (multielement parameter blocks are not hit), it embeds the
classical parameter space into the level–1 parameter space on which SuperhyperSoft sets
act.

Step 2 (lifting the codomain to nonempty subsets). If Fcl(s) ̸= ∅ for all s ∈ S, define

LU : P(U) −→ P+(U), LU (B) := B (which lies in P+(U)).

If some Fcl(s) may be empty, one may either:



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 54 of 69

• (a) adjoin a dummy element ⋆ /∈ U and lift to U⋆ := U ⊔ {⋆} by LU⋆(B) := B if
B ̸= ∅, and LU⋆(∅) := {⋆}, or

• (b) relax the codomain to P(U) (dropping the “nonempty” requirement).

To stay within the present definition P̃1(U) = P+(U), we adopt (a).

Step 3 (a canonical extension to P+(S)). Define the union–lift F : P+(S) → P+(U⋆) by

F (A) := LU⋆

( ⋃
s∈A

Fcl(s)
)
.

Then F is well defined and nonempty-valued: if
⋃

s∈A Fcl(s) ̸= ∅, we keep that set;
otherwise LU⋆ returns {⋆}.

Step 4 (agreement with the classical soft set on singletons). For s ∈ S,

F ({s}) = LU⋆

(
Fcl(s)

)
=

{
Fcl(s), if Fcl(s) ̸= ∅,

{⋆}, if Fcl(s) = ∅.

Hence, modulo the harmless convention of representing the classical empty image by {⋆},
the mapping F reproduces Fcl on the embedded copy ιS(S) of S, and extends it canon-
ically to all nonempty parameter blocks via union. (One could just as well choose the
intersection–lift A 7→

⋂
s∈A Fcl(s) if conjunctive semantics are desired.)

Therefore F : (P̃1(S))1 → (P̃1(U
⋆))1 is a (1, 1)-ary (1, 1)-SuperhyperSoft Set that

represents the given classical soft set.

Theorem 48 (Fixing some parameters). Fix indices 1 ≤ i1 < · · · < ir ≤ h and elements
Aij ∈ P̃m(S). Define

Ffix :
(
P̃m(S)

)h−r −→
(
P̃n(U)

)k
by inserting the fixed Aij into the corresponding coordinates of each (h− r)-tuple and then
applying F . Then Ffix is an (h− r, k)-ary (m,n)-SuperhyperSoft Set.

Proof. Let I = {1, . . . , h} and I free := I \ {i1, . . . , ir}. Define the insertion map

ιfix :
(
P̃m(S)

)h−r −→
(
P̃m(S)

)h
as follows: for B = (Bℓ)ℓ∈Ifree , put ιfix(B) = A with

Au :=

{
Aij , u = ij for some j,

Bℓ, u = ℓ ∈ I free.

This is well defined and injective. Set Ffix := F ◦ ιfix. Since F takes values in (P̃n(U))k, so
does Ffix. Thus Ffix is a mapping with domain (P̃m(S))h−r and codomain (P̃n(U))k, i.e.,
an (h− r, k)-ary (m,n)-SuperhyperSoft Set.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 55 of 69

Theorem 49 (Projection to selected outputs). Let 1 ≤ j1 < · · · < js ≤ k. The coordinate
projection

πj1,...,js :
(
P̃n(U)

)k −→
(
P̃n(U)

)s
, (B1, . . . , Bk) 7→ (Bj1 , . . . , Bjs),

composed with F , yields an (h, s)-ary (m,n)-SuperhyperSoft Set.

Proof. By definition, each Bjr ∈ P̃n(U) is nonempty and level–n. Hence πj1,...,js maps

(P̃n(U))k into (P̃n(U))s, preserving nonemptiness coordinatewise. Therefore πj1,...,js ◦ F :

(P̃m(S))h → (P̃n(U))s is an (h, s)-ary (m,n)-SuperhyperSoft Set.

Theorem 50 (Pointwise union). If F1, F2 : (P̃m(S))h → (P̃n(U))k are (h, k)-ary (m,n)-
SuperhyperSoft Sets, then

(F1 ∪ F2)(A) :=
(
F1(A)j ∪ F2(A)j

)k
j=1

is again an (h, k)-ary (m,n)-SuperhyperSoft Set.

Proof. Fix A and a coordinate j. We prove by induction on n that F1(A)j ∪F2(A)j ∈
P̃n(U).

Base n = 1. Then Fi(A)j ∈ P+(U), so each is a nonempty subset of U . The union of two

nonempty subsets of U is nonempty, hence lies in P+(U) = P̃1(U).

Step n ⇒ n+1. Now Fi(A)j ∈ P̃n+1(U) = P+(P̃n(U)), i.e., each is a nonempty family
of level–n elements. Their union is a nonempty family of level–n elements and therefore
belongs to P+(P̃n(U)) = P̃n+1(U).

Doing this in each coordinate j = 1, . . . , k gives the claim.

Theorem 51 (Pointwise intersection). With the same hypotheses, define

(F1 ∩ F2)(A) :=
(
F1(A)j ∩ F2(A)j

)k
j=1

.

If each intersection is nonempty, then (F1∩F2) is an (h, k)-ary (m,n)-SuperhyperSoft Set.

Proof. Fix A and j, and assume F1(A)j ∩ F2(A)j ̸= ∅. Induct on n.

Base n = 1. Two nonempty subsets of U that meet have a nonempty intersection, which
lies in P+(U) = P̃1(U).

Step n ⇒ n+1. Here Fi(A)j are nonempty families of level–n elements. A nontrivial
intersection of such families is again a nonempty family of level–n elements; hence it lies
in P+(P̃n(U)) = P̃n+1(U).

Coordinatewise application yields the result in (P̃n(U))k.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 56 of 69

Theorem 52 (Functoriality under surjections). Let f : U → V be surjective. Define

recursively maps f
(1)
∗ : P̃1(U) → P̃1(V ) by f

(1)
∗ (B) := f [B] = { f(u) | u ∈ B } and, for

r ≥ 1,

f
(r+1)
∗ : P̃r+1(U) −→ P̃r+1(V ), f

(r+1)
∗ (B) :=

{
f
(r)
∗ (B)

∣∣ B ∈ B
}
.

Then the pushforward

f∗F : (P̃m(S))h −→ (P̃n(V ))k, f∗F (A) =
(
f
(n)
∗ (F (A)j)

)k
j=1

,

is an (h, k)-ary (m,n)-SuperhyperSoft Set over V .

Proof. We prove by induction on r ≥ 1 that f
(r)
∗ is well defined and preserves nonempti-

ness.

Base r = 1. If B ∈ P̃1(U) then B ̸= ∅; surjectivity of f implies f [B] ̸= ∅, so f
(1)
∗ (B) ∈

P̃1(V ).

Step r ⇒ r+1. Let B ∈ P̃r+1(U), so B ̸= ∅ and each B ∈ B lies in P̃r(U). By induction,

each f
(r)
∗ (B) ∈ P̃r(V ); since B ̸= ∅, the set {f (r)

∗ (B) | B ∈ B} is a nonempty subset of
P̃r(V ), hence belongs to P+(P̃r(V )) = P̃r+1(V ).

Now, for any input A ∈ (P̃m(S))h, F (A) ∈ (P̃n(U))k; applying f
(n)
∗ to each coordinate

yields an element of (P̃n(V ))k. Thus f∗F has the required domain and codomain and is
an (h, k)-ary (m,n)-SuperhyperSoft Set over V .

Theorem 53 (Reduction to the classical soft set). If m = n = h = k = 1, then F :
P̃1(S) → P̃1(U) corresponds naturally to a classical soft set Fcl : S → P(U) (up to the
treatment of the empty set).

Proof. Consider the restriction of F to singleton parameters via ιS : S → P+(S),
s 7→ {s}. Define Fcl(s) := F ({s}) if F ({s}) ̸= ∅ and Fcl(s) := ∅ otherwise (if one insists
on allowing empties in the classical codomain). This recovers a map Fcl : S → P(U).
Conversely, given Fcl, the union– or intersection–lift (as in the proof of the first theorem)
produces an F : P+(S) → P+(U) whose restriction to singletons coincides with Fcl (up to
the chosen convention for the empty set). Hence the two notions correspond naturally in
the unary, level–1 case.

Theorem 54 (Nested inclusion under antitone monotonicity). Suppose F is antitone in
each parameter coordinate, i.e., for every i ∈ {1, . . . , h} and tuples A,A′ that coincide in
all coordinates except possibly i,

Ai ⊆ A′
i =⇒ F (A)j ⊇ F (A′)j for all j = 1, . . . , k.

Then for any A,A′ ∈ (P̃m(S))h with A ⊆ A′ coordinatewise, one has

F (A)j ⊇ F (A′)j (j = 1, . . . , k).



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 57 of 69

Proof. Enumerate the coordinates so that we can move from A to A′ by changing one
coordinate at a time. Define a chain of tuples

A(0) := A, A(t) := (A′
1, . . . , A

′
t, At+1, . . . , Ah) (1 ≤ t ≤ h),

so A(h) = A′ and A(t−1) and A(t) differ only in coordinate t, with At ⊆ A′
t. By antitonicity

in coordinate t,
F
(
A(t−1)

)
j

⊇ F
(
A(t)

)
j

for all j.

Chaining these inclusions for t = 1, . . . , h yields F (A)j = F (A(0))j ⊇ F (A(h))j = F (A′)j
for all j, as required.

4.2. (h, k)-ary (m,n)-SuperhyperRough Set

An (h, k)-ary (m,n)-SuperhyperRough Set extends classical rough sets by allowing
multiblock, higher–order parameterization and nested outputs. Concretely, we collect k
input blocks, each block being an h-tuple of m-level (nonempty) parameter subsets, and
we output an n-level (nonempty) subset of the universe. Lower and upper approximations
are then taken on the ground subset of the universe obtained by flattening the nested
output.

Notation (nonempty nested powerset and flattening). For a nonempty set Y and
r ∈ N≥1, define recursively

P̃1(Y ) := {A ⊆ Y | A ̸= ∅ }, P̃r+1(Y ) := {A ⊆ P̃r(Y ) | A ̸= ∅ }.

The level–r flattening Flatr : P̃r(Y ) → P(Y ) is defined by

Flat1(W ) := W, Flatr+1(W) :=
⋃

W∈W
Flatr(W ).

Thus Flatr(W ) ⊆ Y for every W ∈ P̃r(Y ), and Flatr(W1 ∪W2) = Flatr(W1)∪Flatr(W2).
Moreover, Flatr(W1 ∩W2) ⊆ Flatr(W1) ∩ Flatr(W2).

Definition 26 ((h, k)-ary (m,n)-SuperhyperRough Set). Let X be a nonempty finite
universe and let R ⊆ X × X be an equivalence relation; write [x]R for the R-class of x.
Let J be a nonempty parameter space (e.g. a Cartesian product of attribute sets). Fix
integers m,n, h, k ≥ 1.

Input space. Put

D :=
(
P̃m(J)

)h
and D k := D × · · · ×D︸ ︷︷ ︸

k factors

,

so an input to our map is a k-tuple γ = (γ1, . . . , γk), each

γℓ = (Aℓ,1, . . . , Aℓ,h) ∈ D



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 58 of 69

with
Aℓ,i ∈ P̃m(J)

.

Output space. Put C := P̃n(X). Its elements are nonempty level–n families of subsets
of X.

An (h, k)-ary (m,n)-SuperhyperRough Set is a function

F : D k −→ C, γ 7−→ W := F (γ).

The rough lower and upper approximations of W (relative to R) are defined on the ground
subset Flatn(W ) ⊆ X by

W := {x ∈ X | [x]R ⊆ Flatn(W ) },

W := {x ∈ X | [x]R ∩ Flatn(W ) ̸= ∅ }.

When n = 1, Flat1(W ) = W and these are the classical rough approximations of W ⊆ X.

Example 33 (2-ary (1, 1) SuperhyperRough Set). Let

X = {u1, u2, u3, u4}

with R-classes
[u1]R = [u2]R = {u1, u2}

and
[u3]R = [u4]R = {u3, u4}

. Choose m = n = h = 1 and k = 2. Take any nonempty parameter set J (its internal
structure is irrelevant here), so D = P̃1(J) ∼= J , hence D2 ∼= J × J , and C = P̃1(X) =
P+(X). Define F : J × J → P+(X) by

F (a, b) := {u1, u2, u3} (for all (a, b)).

Then W = F (a, b) has approximations

W = {u1, u2}, W = {u1, u2, u3, u4}.

Thus F is a concrete 2-ary (1, 1)-SuperhyperRough Set.

Example 34 (Medical diagnosis as a 2-ary (2, 2)-SuperhyperRough Set). Let

X = {P1,P2,P3,P4}

and let R group patients by age: [P1]R = [P2]R, [P3]R = [P4]R. Let

J = {Fever,Cough} × {Pain,Nausea}



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 59 of 69

. Take m = 2, n = 1, h = 2, k = 2. Then

D =
(
P̃2(J)

)2
, C = P̃1(X) = P+(X).

Choose

γ1 =
{
{(Fever,Pain)}, {(Cough,Nausea)}

}
, γ2 =

{
{(Fever,Nausea)}

}
.

Define F (γ1, γ2) := {P1,P2,P3}. Then (with n = 1) the approximations are

W = {P1,P2}, W = {P1,P2,P3,P4}.

Hence F models a two–block, hierarchical symptom selection yielding a rough decision set.

Example 35 (Household Contact Tracing as a (2, 2)-ary (2, 1)-SuperhyperRough Set).
Universe and indiscernibility. Let the finite universe X be six residents labeled

X = {A,B,C,D,E, F}.

Let R ⊆ X ×X encode the household equivalence:

[A]R = [B]R = {A,B}, [C]R = [D]R = {C,D}, [E]R = [F ]R = {E,F}.

Parameter space and hyperparameters. Consider the attribute space

J = Ssymp×Sexpo, Ssymp = {Fever,Cough,Asympt.}, Sexpo = {Close,Casual,Travel}.

Fix (m,n, h, k) = (2, 1, 2, 2). Then

P̃1(J) = {U ⊆ J | U ̸= ∅ }, P̃2(J) = {Γ ⊆ P̃1(J) | Γ ̸= ∅ }.

The input space is D = (P̃2(J))2 (two coordinates), and an input to F is a pair γ =
(γ1, γ2) ∈ D2 with γℓ = (Aℓ,1, Aℓ,2), Aℓ,i ∈ P̃2(J). The output space is C = P̃1(X) =
{W ⊆ X | W ̸= ∅ }.

Interpret the two coordinates in each γℓ as:

• coordinate 1: a cluster of symptom patterns,

• coordinate 2: a cluster of exposure contexts.

Choose the concrete hyperparameters

A1,1 =
{
{(Fever,Close)}, {(Cough,Close)}

}
, A1,2 =

{
{(Asympt.,Close)}

}
,

A2,1 =
{
{(Fever,Travel)}

}
, A2,2 =

{
{(Cough,Casual)}, {(Asympt.,Travel)}

}
.

Thus γ1 = (A1,1, A1,2) captures high-risk local symptoms/close contacts, while γ2 =
(A2,1, A2,2) captures travel–related or casual exposures.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 60 of 69

SuperhyperRough mapping. Define

F : D2 −→ C = P̃1(X), (γ1, γ2) 7−→ W,

and, for the above input, set (one valid epidemiological decision rule)

W = {A,B,D}.

Intuitively, {A,B} are flagged due to close symptomatic contact (household cluster), and
D is flagged due to a travel–symptom combination matching A2,1.

Rough approximations (with n = 1). Since n = 1, Flat1(W ) = W . The lower
and upper rough approximations of W (relative to R) are

W = {x ∈ X | [x]R ⊆ W }, W = {x ∈ X | [x]R ∩W ̸= ∅ }.

Compute using the three households:

[A]R = [B]R = {A,B} ⊆ W =⇒ {A,B} ⊆ W,

[C]R = [D]R = {C,D} ̸⊆ W but {C,D} ∩W = {D} ̸= ∅,

[E]R = [F ]R = {E,F} ̸⊆ W and {E,F} ∩W = ∅.

Hence
W = {A,B}, W = {A,B,C,D}.

Interpretation. The lower approximation W contains residents whose entire house-
hold is flagged ({A,B}); they are quarantined with certainty under the household–indiscernibility
R. The upper approximation W additionally includes residents whose household has some
flagged member (C is included because D is in W ). This example realizes a real–world
(2, 2)-ary (2, 1)-SuperhyperRough Set: multi–level, clustered parameters (m=2, h=2), two
input tuples (k=2), and rough reasoning on the ground set X with n=1.

Theorem 55 (Classical rough sets as a special case). If m = h = k = 1 and n =
1, then any classical rough set mapping Fcl : J → P(X) is exactly a (1, 1)-ary (1, 1)-
SuperhyperRough Set. More generally, for arbitrary n ≥ 1, the composition Flatn ◦ F
recovers the classical setting on X.

Proof. When m = h = k = n = 1 we have D1 = P̃1(J) ∼= J and C = P̃1(X) = P+(X).
Thus every Fcl : J → P+(X) is a (1, 1)-ary (1, 1)-SuperhyperRough Set. If n > 1, then
F : D1 → P̃n(X) and Flatn ◦ F : D1 → P(X) produces ground subsets on which the
classical lower/upper approximations are taken. Hence the generalized model subsumes
the classical one.

Theorem 56 (Reduction of input arity). Fix indices 1 ≤ i1 < · · · < ir ≤ k and blocks
γ∗ij ∈ D. Define

ι : D k−r −→ D k, ι(δ1, . . . , δk−r) = γ

by inserting the fixed blocks at slots i1, . . . , ir and the free blocks elsewhere, and put Ffix :=
F ◦ ι. Then Ffix : D k−r → P̃n(X) is an (h, k − r)-ary (m,n)-SuperhyperRough Set.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 61 of 69

Proof. The insertion map ι is well defined; composing with F yields a map into C.
Nonemptiness and the definition of ·, · depend only on the image W = F (γ), which is
unaffected by viewing the fixed coordinates as parameters of the new map. Thus the
structure and approximations remain valid with arity k − r.

Theorem 57 (Union). If F1, F2 : D k → P̃n(X) are (h, k)-ary (m,n)-SuperhyperRough
Sets, then

(F1 ∪ F2)(γ) := F1(γ) ∪ F2(γ)

is again an (h, k)-ary (m,n)-SuperhyperRough Set. Moreover,

Flatn
(
(F1 ∪ F2)(γ)

)
= Flatn

(
F1(γ)

)
∪ Flatn

(
F2(γ)

)
,

hence F1 ∪ F2 = F1 ∪ F2 and F1 ∪ F2 = F1 ∪ F2.

Proof. By induction on n, the union of two nonempty level–n families is nonempty
level–n. Thus (F1∪F2)(γ) ∈ P̃n(X). The displayed equality follows from the definition of
Flatn and distributivity of ordinary unions. The identities for lower/upper approximations
are the classical rough equalities applied to the ground sets Flatn(·).

Theorem 58 (Intersection). With the same hypotheses, define

(F1 ∩ F2)(γ) := F1(γ) ∩ F2(γ).

If this intersection is nonempty for all γ, then F1∩F2 is an (h, k)-ary (m,n)-SuperhyperRough
Set. Furthermore,

Flatn
(
(F1 ∩ F2)(γ)

)
⊆ Flatn

(
F1(γ)

)
∩ Flatn

(
F2(γ)

)
,

whence F1 ∩ F2 ⊆ F1 ∩ F2 and F1 ∩ F2 ⊆ F1 ∩ F2. Equality holds in all three inclusions
when n = 1.

Proof. As for union, the intersection of two level–n families that meets nontrivially is
again a nonempty level–n family, by induction on n. The inclusion for Flatn is immediate
from the definition (the base elements surviving the intersection are a subfamily of those
present in each factor). The inclusions for rough approximations follow from the mono-
tonicity of ·, · with respect to set inclusion on ground subsets. When n = 1, Flat1 is the
identity and all inclusions become equalities.

Theorem 59 (Monotonicity of approximations). If γ,γ ′ ∈ D k satisfy F (γ ′) ⊆ F (γ) (in
P̃n(X)), then

F (γ ′) ⊆ F (γ), F (γ ′) ⊆ F (γ).

Proof. From F (γ ′) ⊆ F (γ) we get Flatn(F (γ ′)) ⊆ Flatn(F (γ)). If [x]R ⊆ Flatn(F (γ ′)),
then [x]R ⊆ Flatn(F (γ)), hence x ∈ F (γ). Similarly, if [x]R ∩ Flatn(F (γ ′)) ̸= ∅, then
[x]R ∩ Flatn(F (γ)) ̸= ∅, whence the upper inclusion.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 62 of 69

Theorem 60 (Functoriality under quotients). Let f : X → Y be a surjection that is
constant on R-classes (i.e., xRx′ ⇒ f(x) = f(x′)). Let S be the induced equivalence on
Y whose classes are the images f([x]R) (x ∈ X). Define the level–n direct image

f
(1)
∗ : P̃1(X) → P̃1(Y ), B 7→ f [B],

f
(r+1)
∗ (B) := { f (r)

∗ (B) | B ∈ B }.

Then
(f∗F )(γ) := f

(n)
∗

(
F (γ)

)
∈ P̃n(Y )

defines an (h, k)-ary (m,n)-SuperhyperRough Set on Y . Moreover,

Flatn
(
(f∗F )(γ)

)
= f(Flatn(F (γ))) ,

and for rough approximations with respect to S one has

f
(
F (γ)

)
⊆ (f∗F )(γ),

f
(
F (γ)

)
⊆ (f∗F )(γ),

with equality whenever f is the quotient map X → X/R followed by a bijection onto Y .

Proof. The construction of f
(n)
∗ preserves nonemptiness by induction on n (surjectivity

gives f [B] ̸= ∅ when B ̸= ∅; passing from n to n+1 takes nonempty families to nonempty

families). The identity Flatn(f
(n)
∗ (W )) = f(Flatn(W )) follows directly from the recursive

definitions.
Let WX := Flatn(F (γ)) ⊆ X and WY := f(WX) = Flatn((f∗F )(γ)). If x ∈ F (γ) then

[x]R ⊆ WX . By construction of S, f([x]R) = [f(x)]S ⊆ f(WX) = WY , so f(x) ∈ (f∗F )(γ),

proving the first inclusion. If x ∈ F (γ) then [x]R∩WX ̸= ∅, hence [f(x)]S = f([x]R) meets
WY = f(WX), and f(x) ∈ (f∗F )(γ). If, moreover, f is the canonical quotient collapsing
R-classes (up to bijection), these inclusions are equalities by standard properties of rough
sets on quotient spaces.

5. Conclusion

In this paper, we introduced two new and more general frameworks:
the (m,n)–SuperHyperUncertain Set and the (h, k)–ary (m,n)–SuperHyperUncertain Set.
In particular, we formally defined the concepts summarized in Table 6. These frameworks
are expected to provide a refined means of representing real–world notions involving hier-
archical uncertainty. Modern datasets often exhibit hierarchical, multi–source uncertainty
with interacting attributes. The (m,n) and (h, k)–ary SuperHyperUncertain Sets unify
classical models, support multi–ary composition, and enable provably consistent and scal-
able reasoning across multiple levels and modalities.

Future work on the (m,n)-SuperHyperUncertain Set and
the (h, k)-ary (m,n)-SuperHyperUncertain Set includes:



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 63 of 69

Table 6: Concise overview of (h, k)-ary (m,n)-SuperHyperSet families

Family Essence

(h, k)-ary (m,n)-SuperHyperFuzzy Set Maps hyper-parameters to fuzzy degrees [0, 1].
(h, k)-ary (m,n)-SuperHyperNeutrosophic Set Encodes triples (T, I, F ) for truth, indeterminacy, falsity.
(h, k)-ary (m,n)-SuperHyperPlithogenic Set Adds contradiction function pCF to capture attribute

conflicts.
(h, k)-ary (m,n)-SuperHyperSoft Set Extends soft sets with higher-order parameters and multi-

outputs.
(h, k)-ary (m,n)-SuperHyperRough Set Provides hierarchical lower and upper approximations.

• Designing efficient algorithms for constructing and manipulating these SuperHype-
rUncertain Sets;

• Investigating in detail the mathematical properties and structural features of the
proposed frameworks;

• Exploring potential applications for modeling hierarchical uncertainty in practical
systems.

Furthermore, it is our hope that the concepts presented in this work will stimulate
further research into their applications in areas such as Graphs, HyperFunctions [70, 71],
HyperGraphs [30, 72, 73], and SuperHyperGraphs [32, 74].

Acknowledgmeents

We wish to thank everyone whose guidance, ideas, and assistance contributed to this
research. Our appreciation goes to the readers for their interest and to the scholars
whose publications provided the groundwork for this study. We are also indebted to the
individuals and institutions that supplied the resources and infrastructure necessary for
completing and disseminating this paper. Lastly, we extend our gratitude to all who
offered their support in various capacities.

Data Availability

This paper is purely theoretical and does not involve any empirical data. We welcome
future empirical studies that build upon and test the concepts presented here.

Ethical Approval

As this work is entirely conceptual and involves no human or animal subjects, ethical
approval was not required.



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 64 of 69

Conflicts of Interest

The authors declare no conflicts of interest in connection with this study or its publi-
cation.

Funding

No external or organizational funding supported this work.

Research Integrity

The authors affirm that, to the best of their knowledge, this manuscript represents
their original research. It has not been previously published in any journal, nor is it
currently being considered for publication elsewhere.

Disclaimer on Computational Tools

No computer-based tools—such as symbolic computation systems, automated theorem
provers, or proof assistants (e.g., Mathematica, SageMath, Coq)—were employed in the
development, analysis, or verification of the results contained in this paper. All derivations
and proofs were conducted manually through analytical methods by the authors.

Code Availability

No code or software was developed for this study.

Clinical Trial

This study did not involve any clinical trials.

Consent to Participate

Not applicable.

Disclaimer on Scope and Accuracy

The theoretical models and concepts proposed in this manuscript have not yet under-
gone empirical testing or practical deployment. Future work may investigate their utility
in applied or experimental contexts. While the authors have taken care to maintain ac-
curacy and provide appropriate citations, inadvertent errors or omissions may remain.
Readers are encouraged to consult original references for confirmation and further study.

The authors assert that all mathematical results and justifications included in this
work have been carefully reviewed and are believed to be correct. Should any inaccuracies



T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (4) (2025), 6834 65 of 69

or ambiguities be discovered, the authors welcome constructive feedback and will provide
clarification upon request.

The conclusions presented are valid only within the specific theoretical framework
and assumptions described in the text. Generalizing these results to other mathematical
contexts may require further investigation. All opinions and interpretations expressed
herein are solely those of the authors and do not necessarily reflect the views of their
respective institutions.

Use of Generative AI and AI-Assisted Tools

I use generative AI and AI-assisted tools for tasks such as English grammar checking,
and I do not employ them in any way that violates ethical standards.

Consent to Publish

All authors have given their consent for submission of this manuscript to the journal.

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