**SuperHyperStructure & Neutrosophic
SuperHyperStructure**

** ** The SuperHyperStructure and
Neutrosophic SuperHyperStructure [15], together with their particular cases such
as: SuperHyperAlgebra and
Neutrosophic SuperHyperAlgebra (endowed with SuperHyperOperations and
SuperHyperAxioms) [2016, 2022], SuperHyperGraph and Neutrosophic SuperHyperGraph
[2019 - 2022], SuperHyperSoft Set,
SuperHyperFunction and Neutrosophic SuperHyperFunction [2022],
SuperHyperTopology and Neutrosophic SuperHyperTopology [2022] were founded by
Smarandache [2] and developed between 2016 - 2024.

All the above
structures are built on the *n-th PowerSet of a Set H,* as in our real
world {because a set (or system) *H* (that may be a set of items, an
organization, country, etc.) is composed by *sub-sets* that are parts of *
P(H)*, which in their turn are organized in *sub-sub-systems* that are
parts of *P(P(H)) = P*^{2}(H), and so on}.

** 1.
"Hyper" and "Super" prefixes**

The prefix “__Hyper__” [Marty [1], 1934] stand for the codomain of
the functions and operations to be *P(H),* or the powerset of the set *H*.
While the prefix “__Super__” [Smarandache [2], 2016] stands for using the *P*^{n}(H),
n ≥ 2, or the *n-th* PowerSet of the Set *H* {because the *set*
(or *system*) *H* (that
may be a set of items, a company, institution, country, region, etc.) is
organized in *sub-systems *that are part* of P(H)*, which in their turn are organized in *
sub-sub-systems*, that are part of *P(P(H))* = *P*^{2}(H) and so on} in the domain and/or codomain of the functions
and operations and axioms.

** **

** ** **
2. SuperHyperStructure**

A SuperHyperStructure
[2, 2016] is a structure built on the n-th powerset
P_{*}^{n}(H) of a non-empty set H, for integer n ≥
1, whose SuperHyperOperators are defined as follows:

*#*_{SHS}_{ }:
(P_{*}^{r}(H)*)*^{m} ―> P_{*}^{n}(H),

where P_{*}^{r}(H)
is the r-powerset of H, for
integer r ≥ 1,
while similarly P_{*}^{n}(H) is the
n-th powerset of H,

and the SuperHyperAxioms
act on it*.*

Indeterminacy is __not__ allowed on this structure.

**
3. Neutrosophic SuperHyperStructure**

A Neutrosophic SuperHyperStructure is
a structure built on the n-th powerset P^{n}(H) of a non-empty set H, for integer n ≥
1, whose Neutrosophic SuperHyperOperators are defined as follows:

*#*_{SHS}_{ }:
(P^{r}(H)*)*^{m} ―> P^{n}(H),

where P^{r}(H)
is the r-powerset of H, for
integer r ≥ 1, while similarly P^{n}(H) is the
n-th powerset of H,

and the SuperHyperAxioms
act on it*.*

Indeterminacy is allowed on this structure.

4. History:
From classical Structures and HyperStructures to SuperHyperStructures

(i)** Classical Structure**

A classical **Structure** is built on a non-empty set *H*, whose
classical **Operations** #_{S} are defined as:

*#*_{S}_{ }:
H^{m} ―> H, for integer m
≥ 1,

and with classical **Axioms** acting on it.

(ii) Classical **HyperStructure**

A classical HyperStructure (Marty, 1934) is built on a non-empty set *H*, whose
**HyperOperations** are defined as:

*#*_{HS}_{ }:
H^{m} ―> P_{*}(H),
where P_{*}(H) is the powerset of *H, *without the empty set.

and the **HyperAxioms** acting on *it.*

(iii)
Neutrosophic HyperStructure

The Neutrosophic SuperHyperStructure is an extension of the Classical
HyperStructure, because it allows indeterminacy (uncertainty, unknown), denoted
by the empty set ( *θ* ), into the the *
powerset of H*, that it is denoted by *P(H), *without _{*}

(iv) **SuperHyperStructure**

A SuperHyperStructure is built on n-th powerset P^{n}(H) of a non-empty set H, for integer n ≥
1, whose **SuperHyperOperators** are defined as follows:

*#*_{SHS}_{ }:
(P^{r}(H)*)*^{m} ―> P^{n}(H),

where P^{r}(H) is the r-powerset of H, for
integer r ≥ 1, while similarly P^{n}(H) is the
n-th powerset of H,

and the SuperHyperAxioms
act on it*.*

Indeterminacy is not allowed in this structure.

(iv) **Neutrosophic SuperHyperStructure**

A
Neutrosophic SuperHyperStructure is built on n-th powerset P^{n}(H) of a non-empty set H, for integer n ≥
1, whose **SuperHyperOperators** are defined as follows:

*#*_{SHS}_{ }:
(P^{r}(H)*)*^{m} ―> P^{n}(H),

where P^{r}(H) is the r-powerset of H, for
integer r ≥ 1, while similarly P^{n}(H) is the
n-th powerset of H,

and the SuperHyperAxioms
act on it*.*

Indeterminacy (denoted by the *empty set θ*
) is allowed in this structure.

(v) **The
n-th powerset of a set H with
Indeterminacy** (denoted by the empty
set θ)

(vi) **The n-th powerset
of a set H without Indeterminacy**

The **n-th PowerSet ***P*^{n}(H)
and** ***P*_{*}^{n}(H) of a non-empty set *H*, were
introduced by Smarandache [2] in 2016.

**References**

[1] F. Marty, Sur une généralisation de la Notion de Groupe,
8th Congress Math. Scandinaves, Stockholm, Sweden, (1934), 45–49.

[2] F. Smarandache, SuperHyperAlgebra and Neutrosophic
SuperHyperAlgebra, Section into the authors book Nidus Idearum. Scilogs, II: de
rerum consectatione, Second Edition, (2016), pp. 107– 108,
https://fs.unm.edu/NidusIdearum2-ed2.pdf

[3] Mohammad Hamidi, Florentin Smarandache, Mohadeseh
Taghinezhad: Decision
Making Based on Valued Fuzzy

Superhypergraphs. *Computer Modeling in
Engineering & Sciences*, vol. 138, no. 2, 1907-1923, 2023.

[4] Abdullah Kargın, Florentin Smarandache, Memet Şahin: New
Type Hyper Groups, New Type SuperHyper

Groups and Neutro-Type SuperHyper
Groups. Chapter One in Florentin Smarandache, Memet Şahin, Derya

Bakbak, Vakkas Uluçay & Abdullah Kargın (Editors) - *Neutrosophic SuperHyperAlgebra
And New Types of*

*Topologies*, Global Knowledge, pp. 10-24, 2023.

[5] Florentin Smarandache, Extension
of HyperGraph to n-SuperHyperGraph and to Plithogenic n-SuperHyperGraph, and
Extension of HyperAlgebra to n-ary (Classical-/Neutro-/Anti-) HyperAlgebra, Neutrosophic Sets
and Systems, vol. 33, 2020, pp. 290-296. DOI: 10.5281/zenodo.3783103

[6]
S Santhakumar, I R Sumathi and J Mahalakshmi J, A
Novel Approach to the Algebraic Structure of Neutrosophic SuperHyper Algebra ,
Neutrosophic Sets and Systems, Vol. 60, 2023, pp. 593-602. DOI: 10.5281/zenodo.10236475

[7] Sirus Jahanpanah and Roohallah
Daneshpayeh, On
Derived SuperHyper BE-Algebras, Neutrosophic
Sets and Systems, Vol. 57, 2023, pp. 318-327. DOI: __10.5281/zenodo.8271390__

[8] Marzieh Rahmati and Mohammad Hamidi, Extension
of G-Algebras to SuperHyper G-Algebras, Neutrosophic
Sets and Systems, Vol. 55, 2023, pp. 557-567. __DOI__: __10.5281/zenodo.7879543__

[9]
Mohammad Hamidi, On
Superhyper BCK-Algebras, Neutrosophic
Sets and Systems, Vol. 53, 2023, pp. 580-588. DOI: __10.5281/zenodo.7536091__

[10] Huda E. Khali^{ },^{ }Gonca
D. GÜNGÖR, Muslim A. Noah Zainal^{, }Neutrosophic
SuperHyper Bi-Topological Spaces: Original Notions and New Insights,
Neutrosophic Sets and Systems, Vol. 51, 2022, pp. 33-45. DOI: __10.5281/zenodo.7135241__

[11] Pairote
Yiarayong, On
2-SuperHyperLeftAlmostSemihyp regroups, Neutrosophic Sets and
Systems, Vol. 51, 2022, pp. 516-524. DOI: __10.5281/zenodo.713536__

[12] Florentin
Smarandache The
SuperHyperFunction and the Neutrosophic SuperHyperFunction, Neutrosophic Sets
and Systems, Vol. 49, 2022, pp. 594-600. DOI: __10.5281/zenodo.6466524__

[13]
F. Smarandache, Introduction
to the n-SuperHyperGraph - the most general form of graph today, Neutrosophic
Sets and Systems, Vol. 48, 2022, pp. 483-485, DOI: __10.5281/zenodo.6096894__

**
[14] Florentin Smarandache, Foundation
of Revolutionary Topologies: An Overview, Examples, Trend**

**
Analysis,
Research Issues, Challenges, and Future Directions, Neutrosophic
Systems with Applications, **

**
pp. 45-66, Vol. 13,
2024. **

[15] F. Smarandache,
SuperHyperStructure and Neutrosophic SuperHyperStructure, Neutrosophic Sets and

Systems, Vol. 63,
pp. 367-381, 2024.

**
[16]
M. Hamidi, M. Taghinezhad (2024): ****Application
of SuperHyperGraphs-Based Domination Number in**

**
Real World****.** *Journal
of Mahani Mathematical Research* 13(1),
211-228; DOI:

**
10.22103/jmmr.2023.21203.1415**

**
[17]
Huda E. Khalid, Gonca Durmaz Gungor, Muslim A. Noah (2024): ****Neutrosophic
SuperHyper Bi-**

**
Topological Spaces: Extra Topics****. ***Neutrosophic
Optimization and Intelligent Systems* 2, 43-

**
55; ****https://doi.org/10.61356/j.nois.2024.2217**