Superhypermagma, Lie Superhypergroup, Quotient Superhypergroups, and Reduced Superhypergroups

Abstract

Classical algebraic structures—such as magmas, groups, and Lie groups—are characterized by increasingly strong requirements in binary operation, ranging from no additional constraints to associativity, identity, inverses, and smooth-manifold structures. The hyperstructure paradigm extends these notions by allowing the operation to return subsets of elements, giving rise to hypermagmas, hypergroups, and Lie hypergroups, along with their variants such as quotient, reduced, and fuzzy hypergroups. In this work, we introduce the concept of superhyperstructures, obtained by iterating the powerset construction, and develop the theory of superhypermagmas and Lie superhypergroups. We further define and analyze quotient superhypergroups, reduced superhypergroups, and fuzzy superhypergroups, exploring their algebraic properties and interrelationships.

1. Introduction

1.1. Hyperstructures and Superhyperstructures

A (finite) hyperstructure generalizes a classical algebraic structure by allowing its binary operation to return a subset of elements instead of a single element [1,2,3,4]. Prominent examples include hypergraphs [5,6], hypergroups [7,8], hyperalgebras [9], hyper-uncertain sets [10,11], and hypertopologies [12], all of which have played significant roles in both pure and applied mathematics.

By iterating the powerset construction n times, one obtains an n-superhyperstructure [1], which provides a natural framework for modeling hierarchical and multi-layered systems. Instances of these include superhypergraphs [13,14,15,16], superhypergroups [17], superhyperalgebras [18,19,20], superhypertopology [21,22,23], superhypergame [24], superhyperalgorithms [25], superhyper-uncertain sets [26,27,28], and superhyperfunctions [29,30,31]; their formal theory and applications are only now beginning to be explored.

An overview of (finite) classical structures, hyperstructures, and superhyperstructures is presented in

Table 1. Overview of (finite) classical structures, hyperstructures, and superhyperstructures.

Structure	Underlying Set	Operation Signature
Classical structure	H	${\#}_0:H^m\to H$
Hyperstructure	$\mathcal{P}\left(H\right)$ 1	$\circ :H\times H\to \mathcal{P}\left(H\right)$
n-Superhyperstructure	${\mathcal{P}}_n\left(H\right)$ 2	$\circ :{\mathcal{P}}_n\left(H\right)\times {\mathcal{P}}_n\left(H\right)\to {\mathcal{P}}_n\left(H\right)$

${}^1\mathcal{P}\left(H\right)$: the powerset of H, i.e., the set of all subsets of H. ${}^2{\mathcal{P}}_n\left(H\right)$: the n-th iterated powerset, defined by ${\mathcal{P}}_1\left(H\right)=\mathcal{P}\left(H\right)$ and ${\mathcal{P}}_{k+1}\left(H\right)=\mathcal{P}\left({\mathcal{P}}_k\left(H\right)\right)(k\ge 1)$.

. Unless otherwise noted, all sets and algebraic structures (including groups) in this paper are assumed to be finite, and n denotes a positive integer.

1.2. Groups and Related Concepts

Fundamental (finite) algebraic structures such as magmas and groups underpin much of modern mathematics. A magma is a set with a binary operation without further constraints [32,33,34]. A (finite) classical group refines this by requiring associativity, an identity element, and inverses for all elements [35,36]. A (finite) Lie group further endows a group with a smooth-manifold structure in which multiplication and inversion are smooth maps [37,38,39,40]. These classical notions extend naturally within the hyperstructure paradigm, yielding hypermagmas, hypergroups [41,42,43], and Lie hypergroups. Moreover, specialized variants—quotient hypergroups (via compatible equivalence relations), reduced hypergroups (with trivial essential indistinguishability), and fuzzy hypergroups (with fuzzy hyperoperations) [44,45,46,47]—further enrich the theory.

A comparison of groups, hypergroups, and superhypergroups is presented in

Table 2. Comparison of (finite) groups, hypergroups, and superhypergroups.

Structure	Underlying Set	Operation	Key Properties
Group	G	$·:G\times G\to G$	Associative, Identity, Inverses
Hypergroup	$\mathcal{P}\left(G\right)$	$★:G\times G\to \mathcal{P}\left(G\right)$	Associative (union-based), Identity, Reversibility
Superhypergroup	${\mathcal{P}}_n\left(G\right)$	$\circ :G\times G\to {\mathcal{P}}_n\left(G\right)$	Strong superhyperassociative, Identity, Inverses

★ hyperoperation in a hypergroup. ∘ superhyperoperation in a superhypergroup.

.

1.3. Our Contribution

Given the foundational importance of hyperstructures and the recent emergence of superhyperstructures, definitions and investigations of superhypergroups have attracted attention. However, superextensions of hypermagmas, Lie hypergroups, and related constructs such as quotient, reduced, and fuzzy hypergroups remain largely uncharted. In this paper, we introduce and develop the following new notions:

• Superhypermagmas and Lie superhypergroups, extending magmas and Lie groups via n-th powerset operations.
• Quotient superhypergroups and reduced superhypergroups obtained from quotient and reduction processes on superhypergroups.
• Fuzzy superhypergroups, combining fuzzification with superhyperstructure.

We establish their algebraic properties.

1.4. Organization of the Paper

The remainder of this paper is organized as follows. In Section 2, we review the necessary background, including the definitions of the powerset and nth-powerset, hyperstructures and superhyperstructures, groups, quotient hypergroups, reduced hypergroups, and fuzzy hypergroups. Section 3 introduces the new concepts developed in this work: superhypermagmas and Lie superhypergroups; quotient superhypergroups and reduced superhypergroups; and fuzzy superhypergroups. Finally, Section 4 offers concluding remarks and outlines directions for future research.

2. Preliminaries

This section provides an introduction to the foundational concepts and definitions required for the discussions in this paper.

2.1. Powerset and nth Powerset

We begin by recalling the notions of the powerset and the nth powerset, which form the foundation for our subsequent constructions. Unless otherwise noted, all sets and algebraic structures (including groups) in this paper are assumed to be finite, and n denotes a positive integer. For further details on these operations and their properties, the reader is referred to the standard literature.

Definition 1

(Set [48]). A (finite) set is a collection of distinct objects, called elements, which are unambiguously defined. If A is a set and x is an element of A, we write $x\in A$. Sets are usually denoted by enclosing their elements in curly braces.

Definition 2

(Subset [48]). For any two (finite) sets A and B, A is said to be a subset of B (written $A\subseteq B$) if every element of A is also an element of B:

$$\forall x\in A,x\in B.$$

If additionally $A\ne B$, then A is called a proper subset of B, which is denoted $A\subset B$.

Definition 3

(Empty Set [48]). The empty set, denoted ∅, is the unique set that contains no elements:

$$\forall x,x\notin \varnothing .$$

It follows that ∅ is a subset of every set.

Definition 4

(Base Set). A (finite) base set S is the underlying set from which more elaborate structures, such as powersets and hyperstructures, are constructed. It is defined by

$$S=\{x\mid x\mathit{belongs}\mathit{to}\mathit{a}\mathit{specified}\mathit{domain}\}.$$

All elements appearing in constructions like $\mathcal{P}\left(S\right)$ or ${\mathcal{P}}_n\left(S\right)$ are drawn from S.

Definition 5

(Powerset). The (finite) powerset of a set S, denoted $\mathcal{P}\left(S\right)$, is the collection of all subsets of S, including both ∅ and S itself:

$$\mathcal{P}\left(S\right)=\{A\mid A\subseteq S\}.$$

Example 1

(Real-life Powerset). Let

$$S=\{\mathit{sprinkles},\mathit{chocolate}\mathit{chips},\mathit{nuts}\}.$$

Then the powerset

$$\mathcal{P}\left(S\right)=\{\varnothing ,\{\mathit{sprinkles}\},\{\mathit{chocolate}\mathit{chips}\},\{\mathit{nuts}\},$$

$$\{\mathit{sprinkles},\mathit{chocolate}\mathit{chips}\},\{\mathit{sprinkles},\mathit{nuts}\},$$

$$\{\mathit{chocolate}\mathit{chips},\mathit{nuts}\},\{\mathit{sprinkles},\mathit{chocolate}\mathit{chips},\mathit{nuts}\}\}.$$

In practice, this models all possible topping combinations you could put on your ice-cream.

Definition 6

(n-th Powerset cf. [1,19]). The (finite) n-th powerset of a set H, denoted $P_n\left(H\right)$, is defined recursively by

$$P_1\left(H\right)=\mathcal{P}\left(H\right),P_{n+1}\left(H\right)=\mathcal{P}\left(P_n\left(H\right)\right)\mathrm{for}n\ge 1.$$

Similarly, the n-th nonempty powerset, denoted $P_n^{\ast }\left(H\right)$, is given by

$$P_1^{\ast }\left(H\right)={\mathcal{P}}^{\ast }\left(H\right),P_{n+1}^{\ast }\left(H\right)={\mathcal{P}}^{\ast }\left(P_n^{\ast }\left(H\right)\right),$$

where ${\mathcal{P}}^{\ast }\left(H\right)$ denotes the powerset of H with the empty set omitted.

Example 2

(Nested Committees in an Organization). Let $H=\{\mathit{Ayano},\mathit{Shinya},\mathit{Hiroko}\}$ be the set of employees in a small department. Then,

$$P_1\left(H\right)=\mathcal{P}\left(H\right)=\{⌀,\left\{\mathit{Ayano}\right\},\left\{\mathit{Shinya}\right\},\left\{\mathit{Hiroko}\right\},$$

$$\{\mathit{Ayano},\mathit{Shinya}\},\{\mathit{Ayano},\mathit{Hiroko}\},$$

$$\{\mathit{Shinya},\mathit{Hiroko}\},\{\mathit{Ayano},\mathrm{Shinya},\mathit{Hiroko}\}\},$$

which represents every possible committee. Next,

$$P_2\left(H\right)=\mathcal{P}\left(P_1\left(H\right)\right)$$

is the set of all collections of committees (for example, a project may require two or more committees working in parallel). Similarly,

$$P_3\left(H\right)=\mathcal{P}\left(P_2\left(H\right)\right)$$

models all the ways to choose sets of committees—collections, and so on. In practice, $P_2\left(H\right)$ can describe how an organization assigns multiple committees to different tasks, while $P_3\left(H\right)$ could represent various deployment plans of those committee assignments across several projects.

2.2. Hyperstructure and Superhyperstructure

To establish a robust theoretical foundation for hyperstructures [2,3,4,49] and superhyperstructures [50,51,52], we introduce key definitions that formalize their properties.

Definition 7

(Classical Structure cf. [1,53]). A (finite) classical structure is a mathematical system based on a (finite) nonempty set H that is endowed with one or more (finite) classical operations satisfying a prescribed set of axioms. A classical operation is a mapping

$${\#}_0:H^m\to H,$$

where $m\ge 1$ and $H^m$ denotes the m-fold Cartesian product of H. Typical examples include operations like addition or multiplication in algebraic systems such as groups, rings, or fields.

Definition 8

(Hyperoperation). (cf. [54,55])

• A (finite) hyperoperation is a generalization of a binary operation in which the combination of two elements yields a set of elements rather than a single element. Formally, for a set S, a hyperoperation ∘ is defined by

  $$\circ :S\times S\to \mathcal{P}\left(S\right),$$

  where $\mathcal{P}\left(S\right)$ represents the powerset of S.

Definition 9

(Hyperstructure). (cf. [1,56])

• A (finite) hyperstructure is an extension of a classical structure where the operations are defined on the powerset of a base set. It is formally given by

  $$\mathcal{H}=\left(\mathcal{P}\right(S),\circ ),$$

  with S as the base set and ∘ acting on subsets of S.

Example 3

(Social-Network Mutual Friends). Let

$$S=\{\mathit{Tae},\mathit{Rei},\mathit{Masafumi},\mathit{Ichiro}\}$$

be the set of users in a social network. Define the hyperoperation

$$\circ :S\times S⟶\mathcal{P}\left(S\right),x\circ y=\left\{\mathit{all}\mathit{users}\mathit{who}\mathit{are}\mathit{friends}\mathit{with}\mathit{both}x\mathit{and}y\right\}.$$

For example, if Masafumi and Ichiro are both friends of Tae and of Rei, then

$$\mathit{Tae}\circ \mathit{Rei}=\{\mathit{Masafumi},\mathit{Ichiro}\}.$$

This shows how, in practice, combining two people can naturally produce a set of results rather than a single one.

Definition 10

(Superhyperoperations [1]). Let H be a (finite) nonempty set and $P\left(H\right)$ its (finite) powerset. Define the n-th powerset $P^n\left(H\right)$ recursively by

$$P^0\left(H\right)=H,P^{k+1}\left(H\right)=P(P^k\left(H\right))\mathrm{for}k\ge 0.$$

A superhyperoperation of order $(m,n)$ is an m-ary operation

$${\circ }^{(m,n)}:H^m\to P_{\ast }^n\left(H\right),$$

where $P_{\ast }^n\left(H\right)$ denotes the n-th powerset of H (either excluding the empty set, which we refer to as a classical-type operation, or including it, known as a neutrosophic-type operation). These operations serve as higher-order generalizations of hyperoperations by capturing multi-level complexity through iterative powerset constructions.

Definition 11

(n-Superhyperstructure cf. [1]). A (finite) n-superhyperstructure generalizes a (finite) hyperstructure by incorporating the n-th powerset of a base set. It is defined as

$${\mathcal{SH}}_n=({\mathcal{P}}_n\left(S\right),\circ ),$$

where S is the base set, ${\mathcal{P}}_n\left(S\right)$ is its n-th powerset, and ∘ is an operation on elements of ${\mathcal{P}}_n\left(S\right)$.

Example 4

(Multi-Level Team Coordination). Let

$$S=\{\mathit{Ayano},\mathit{Shinya},\mathit{Hiroko},\mathit{Yutaka}\}$$

be the set of staff in a small department. Then

$$P_1\left(S\right)=\mathcal{P}\left(S\right),P_2\left(S\right)=\mathcal{P}\left(P_1\left(S\right)\right),P_3\left(S\right)=\mathcal{P}\left(P_2\left(S\right)\right),$$

and so on. An element of $P_2\left(S\right)$ might be a set of committees, for example $\left\{\right\{\mathit{Ayano},\mathit{Shinya}\}$, $\{\mathit{Hiroko},\mathit{Yutaka}\}\}$, representing two project teams. We define the merge operation

$$X\circ Y=X\cup Y,X,Y\subseteq P_n\left(S\right),$$

which combines different collections of committees. Hence, $\langle P_n\left(S\right),\circ \rangle$ is an n-superhyperstructure that naturally models hierarchical team formation and coordination across multiple organizational levels.

Example 5

(Hierarchical System Management). Let

$$S=\{\mathit{WebServer},\mathit{Database},\mathit{Cache},\mathit{LoadBalancer}\}$$

be the set of core system components. Then,

$$P_1\left(S\right)=\mathcal{P}\left(S\right)$$

represents all possible component groups (e.g., $\{\mathit{WebServer},\mathit{Database}\}$ as a front-end cluster). Next,

$$P_2\left(S\right)=\mathcal{P}\left(P_1\left(S\right)\right)$$

is the set of all collections of clusters—such as $\left\{\right\{\mathit{WebServer},\mathit{Database}\},\{\mathit{Cache},\mathit{LoadBalancer}\left\}\right\}$, modeling separate operational clusters. Similarly,

$$P_3\left(S\right)=\mathcal{P}\left(P_2\left(S\right)\right)$$

captures choices of deployment plans, grouping cluster collections for different environments (e.g., staging, production).

Define the merge operation

$$X\circ Y=X\cup Y,X,Y\subseteq P_n\left(S\right),$$

which unifies sets of clusters or deployment plans. Hence, $〈P_n\left(S\right),\circ 〉$ forms an n-superhyperstructure that naturally models multi-level system management from individual servers to full deployment architectures.

2.3. Hypermagma

A magma is a nonempty set equipped with a binary operation without any further requirements [32,33,34]. By contrast, a group is a magma whose operation is associative, which possesses an identity element, and in which every element has an inverse. The precise definition of a magma is given below.

Definition 12

(Magma [32,33,34]). A magma is an ordered pair $(M,·)$ where the following apply:

1. 

M is a nonempty set.

2. 

$·:M\times M\to M$ is a binary operation on M.

That is, for every $a,b\in M$, the product $a·b$ is a uniquely defined element of M.

Definition 13

(Group cf. [35,36]). A group is a magma $(G,·)$ that satisfies the following three axioms:

1. 

Associativity: For all $a,b,c\in G$,

$$(a·b)·c=a·(b·c).$$

2. 

Identity: There exists an element $e\in G$ (called the identity) such that for all $a\in G$,

$$e·a=a·e=a.$$

3. 

Inverses: For each $a\in G$, there exists an element $a^{-1}\in G$ (called the inverse of a) such that

$$a·a^{-1}=a^{-1}·a=e.$$

A hypermagma is a set with a hyperoperation mapping each pair to a subset, generalizing binary operations to set operations. The definition of a hypermagma is described as follows [57,58,59].

Definition 14

(Hypermagma [59]). A hypermagma is a pair $(H,★)$ where the following apply:

1. 

H is a nonempty set.

2. 

$★:H\times H\to \mathcal{P}\left(H\right)$ is a hyperoperation; that is, for each ordered pair $(x,y)\in H\times H$, the value

$$x★y\subseteq H$$

is a (possibly empty) subset of H.

Often one requires that the hyperoperation be total, meaning

$$x★y\ne ⌀\mathit{for}\mathit{all}x,y\in H.$$

This generalizes the notion of a binary operation by allowing the “product” of two elements to be a set of elements rather than a single element.

Example 6

(A Simple Hypermagma). Let

$$H=\{a,b\},$$

and define the hyperoperation $★:H\times H\to \mathcal{P}\left(H\right)$ by the table

$$\begin{array}{ccc}★& a& b\\ a& \left\{a\right\}& \{a,b\}\\ b& \{a,b\}& \left\{b\right\}\end{array}$$

That is,

$$a★a=\left\{a\right\},a★b=b★a=\{a,b\},b★b=\left\{b\right\}.$$

Since some products produce two-element subsets, $(H,★)$ is a hypermagma but not an ordinary magma.

Definition 15

(Hypergroup [41,42,43]). A hypergroup is a triple $(H,★,e)$ where the following apply:

1. 

H is a nonempty set.

2. 

$★:H\times H\to \mathcal{P}\left(H\right)$ is a total hyperoperation (i.e., $x★y\ne ⌀$ for all $x,y\in H$).

3. 

$e\in H$ is an element (called the identity) satisfying the following conditions:

(a) 

Associativity: For all $x,y,z\in H$,

$$x★(y★z)=(x★y)★z,$$

where the operation on subsets is defined by

$$x★(y★z):=\bigcup _{w\in y★z}x★wand(x★y)★z:=\bigcup _{w\in x★y}w★z.$$

(b) 

Identity: For all $x\in H$,

$$x\in e★xandx\in x★e.$$

(In some treatments, the stronger condition

$$e★x=\left\{x\right\}andx★e=\left\{x\right\}$$

is assumed.)

(c) 

Reversibility (Inverse Property): For every $x\in H$, there exists an element $x^{-1}\in H$ (called the inverse of x) such that

$$e\in x★x^{-1}ande\in x^{-1}★x.$$

This definition follows the classical idea (after Marty) of a hypergroup, which generalizes groups by replacing the binary operation with a set-valued hyperoperation.

Example 7

(A Simple Hypergroup). Let

$$H=\{0,1\},$$

and define the hyperoperation $★:H\times H\to \mathcal{P}\left(H\right)$ by

$$\begin{array}{ccc}★& 0& 1\\ 0& \left\{0\right\}& \{0,1\}\\ 1& \{0,1\}& \left\{1\right\}\end{array}$$

One checks the following:

• Associativity: $x★(y★z)=(x★y)★z$ under the subset-union definition.
• Identity: 0 satisfies $x\in 0★x$ and $x\in x★0$ for both $x=0,1$.
• Reversibility: Each $x\in H$ has itself as an inverse since $0\in x★x$.

Thus, $(H,★,0)$ is a hypergroup.

2.4. Superhypergroup

A superhypergroup is equipped with an n-th hyperoperation assigning each pair an n-th powerset element, obeying closure, associativity, identity, inverses. The definition of a superhypergroup is described as follows.

Definition 16

(Superhypergroup [3,17]). A superhypergroup is a structure $(H,\circ )$ where the following apply:

1. 

H is a nonempty set.

2. 

$\circ :H\times H\to {\mathcal{P}}_n\left(H\right)$ is a superhyperoperation (with fixed $n\ge 1$) satisfying the following:

(a) 

(Closure) For all $x,y\in H$, the set $\circ (x,y)$ is nonempty.

(b) 

(Strong Superhyperassociativity) For all $x,y,z\in H$,

$$\circ \left(x,\circ (y,z)\right)=\circ \left(\circ (x,y),z\right),$$

where for any nonempty subsets $A,B\subseteq H$, we define

$$A\circ B=\bigcup _{a\in A,b\in B}\circ (a,b).$$

(c) 

(Identity) There exists an element $e\in H$ such that for all $x\in H$,

$$\circ (e,x)=\circ (x,e)=\left\{x\right\}.$$

(d) 

(Inverses) For each $x\in H$, there exists an element $x^{-1}\in H$ such that

$$e\in \circ (x,x^{-1})ande\in \circ (x^{-1},x).$$

Example 8

(A Trivial Superhypergroup (n = 1)). Let

$$H=\{e,a\},P_1\left(H\right)=\mathcal{P}\left(H\right)=\{⌀,\left\{e\right\},\left\{a\right\},\{e,a\}\}.$$

Define the superhyperoperation $\circ :H\times H\to P_1\left(H\right)$ by

$$\begin{array}{ccc}\circ & e& a\\ e& \left\{e\right\}& \left\{a\right\}\\ a& \left\{a\right\}& \{e,a\}\end{array}$$

That is,

$$e\circ e=\left\{e\right\},e\circ a=\left\{a\right\},a\circ e=\left\{a\right\},a\circ a=\{e,a\}.$$

One easily verifies the following:

• Closure: Each $\circ (x,y)$ is nonempty.
• Strong Superhyperassociativity: Follows from $\left\{x\right\}\cup \{y\cup z\}=\{x\cup y\}\cup \left\{z\right\}$.
• Identity: e satisfies $\circ (e,x)=\left\{x\right\}=\circ (x,e)$.
• Inverses: e is its own inverse and a has inverse a since $e\in a\circ a$.

Hence, $\left(H,\circ \right)$ is a superhypergroup for $n=1$.

Example 9

(A superhypergroup for $n=2$). Let

$$H=\{e,a\}.$$

Then,

$$P_1\left(H\right)=\mathcal{P}\left(H\right)=\{⌀,\left\{e\right\},\left\{a\right\},\{e,a\}\},$$

and

$$P_2\left(H\right)=\mathcal{P}\left(P_1\left(H\right)\right).$$

Define the superhyperoperation

$$\circ :H\times H⟶P_2\left(H\right)$$

by the table

$$\begin{array}{ccc}\circ & e& a\\ e& \left\{\right\{e\},\{e,a\left\}\right\}& \left\{\right\{a\left\}\right\}\\ a& \left\{\right\{a\left\}\right\}& \left\{\right\{e\},\{a\left\}\right\}\end{array}$$

That is,

$$e\circ e=\left\{\right\{e\},\{e,a\left\}\right\},e\circ a=\left\{\right\{a\left\}\right\},a\circ e=\left\{\right\{a\left\}\right\},a\circ a=\left\{\right\{e\},\{a\left\}\right\}.$$

One verifies the following:

• Closure: Each $\circ (x,y)\subseteq P_1\left(H\right)$ is nonempty.
• Strong Superhyperassociativity: $\circ \left(x,\circ (y,z)\right)=\circ \left(\circ (x,y),z\right)$ for all $x,y,z\in H$.
• Identity: With e as identity, $\circ (e,x)=\left\{\right\{x\left\}\right\}=\circ (x,e)$.
• Inverses: Each x has itself as inverse since $e\in {\bigcup }_{b\in x\circ x}b$.

Hence, $\left(H,\circ \right)$ is a superhypergroup for $n=2$.

Theorem 1.

Let $(H,\circ )$ be a superhypergroup with superhyperoperation $\circ :H\times H\to {\mathcal{P}}_n\left(H\right)$. Define an operation on ${\mathcal{P}}_n\left(H\right)$ for any nonempty subsets $A,B\in {\mathcal{P}}_n\left(H\right)$ by

$$A\circ B=\bigcup _{a\in A,b\in B}\circ (a,b).$$

Then, the structure

$${\mathcal{SH}}_n=\left({\mathcal{P}}_n\left(H\right),\circ \right)$$

forms an n-superhyperstructure.

Proof. 

We prove the claim by extending the superhyperoperation recursively to the n-th powerset of H.

Step 1. For $n=1$, we have ${\mathcal{P}}_1\left(H\right)=\mathcal{P}\left(H\right)$. For any nonempty subsets $A,B\subseteq H$, define

$$A\circ B=\bigcup _{a\in A,b\in B}\circ (a,b).$$

Since $(H,\circ )$ is a superhypergroup, each $\circ (a,b)$ is nonempty, and the union is taken over nonempty sets; hence, $A\circ B$ is nonempty.

Step 2. Suppose that for some $k\ge 1$, the operation has been extended to ${\mathcal{P}}_k\left(H\right)$ so that $({\mathcal{P}}_k\left(H\right),\circ )$ satisfies the axioms corresponding to a superhyperstructure. Define the $(k+1)$-th powerset by

$${\mathcal{P}}_{k+1}\left(H\right)=\mathcal{P}\left({\mathcal{P}}_k\left(H\right)\right),$$

and for $A,B\in {\mathcal{P}}_{k+1}\left(H\right)$, define

$$A\circ B=\bigcup _{X\in A,Y\in B}\left(X\circ Y\right),$$

where $X\circ Y$ is given by the induction hypothesis.

Step 3. The strong superhyperassociativity of the operation on H ensures that for all $x,y,z\in H$,

$$\circ \left(x,\circ (y,z)\right)=\circ \left(\circ (x,y),z\right).$$

Since the extension to subsets is defined via unions, the associativity property lifts to ${\mathcal{P}}_{k+1}\left(H\right)$; that is, for any nonempty $A,B,C\in {\mathcal{P}}_{k+1}\left(H\right)$, we have

$$A\circ (B\circ C)=(A\circ B)\circ C.$$

Moreover, the identity element $e\in H$ satisfies

$$\circ (e,x)=\left\{x\right\}\mathrm{and}\circ (x,e)=\left\{x\right\}$$

for all $x\in H$, so that for any nonempty subset $A\in {\mathcal{P}}_{k+1}\left(H\right)$, we obtain

$$\left\{e\right\}\circ A=A\mathrm{and}A\circ \left\{e\right\}=A.$$

Similarly, the inverse property in H guarantees that the inverse relation is preserved when passing to the power set.

Thus, by induction, the operation ∘ extends to ${\mathcal{P}}_n\left(H\right)$ in such a way that all the axioms of an n-superhyperstructure (as given in the definition) are satisfied.

Therefore, ${\mathcal{SH}}_n=\left({\mathcal{P}}_n\left(H\right),\circ \right)$ is an n-superhyperstructure. □

2.5. Lie Groups and Lie Hypergroups

A Lie group is a smooth manifold endowed with a group structure such that multiplication and inversion are smooth maps [37,38,39,40]. The definition of a Lie group is described as follows.

Definition 17

(Lie Group [37,38,39,40]). A Lie group is a group G that is also a smooth (i.e., $C^{\infty }$) manifold such that the following maps are smooth:

1. 

The multiplication map

$$m:G\times G\to G,(x,y)\mapsto xy;$$

2. 

The inversion map

$$i:G\to G,x\mapsto x^{-1}.$$

A Lie hypergroup is a smooth manifold with a set-valued hyperoperation, smoothly dependent on inputs, satisfying identity and inverse conditions. The definition of a Lie hypergroup is described as follows.

Definition 18

(Lie Hypergroup [60,61,62,63]). A Lie hypergroup is a hypergroup $(P,\circ )$ that satisfies the following properties:

1. 

Smooth Manifold: The underlying set P is a smooth manifold.

2. 

Smooth Hyperoperation: The hyperoperation

$$\circ :P\times P\to {\mathcal{P}}^{\ast }\left(P\right)$$

(where ${\mathcal{P}}^{\ast }\left(P\right)$ denotes the collection of all nonempty subsets of P) is such that for every $x,y\in P$, the sets $x\circ y$ depend smoothly on x and y. In other words, there exist natural smooth structures on the appropriate hyperspaces so that the maps

$$L_x:P\to {\mathcal{P}}^{\ast }\left(P\right),y\mapsto x\circ y,$$

and

$$R_y:P\to {\mathcal{P}}^{\ast }\left(P\right),x\mapsto x\circ y,$$

are smooth.

3. 

Identity and Inversion: There exists an element $e\in P$ (called the identity) such that for all $x\in P$,

$$x\in e\circ xandx\in x\circ e.$$

Moreover, for every $x\in P$, there exists an inverse $x^{-1}\in P$ such that

$$e\in x\circ x^{-1}ande\in x^{-1}\circ x,$$

and the inversion map $x\mapsto x^{-1}$ is smooth.

4. 

Compatible Lie Group Action: There exists a Lie group G that acts smoothly on P via a map

$$\phi :P\times G\to P,$$

and this action is compatible with the hyperoperation in the sense that for every $x,y\in P$ and every $g\in G$,

$$\phi (x\circ y,g)\subseteq \phi (x,g)\circ \phi (y,g).$$

Example 10

(Interval Hypergroup on $\mathbb{R}$). Let

$$P=\mathbb{R},$$

a smooth 1-dimensional manifold, and fix $\epsilon >0$. Define

$$\circ :P\times P⟶{\mathcal{P}}^{\ast }\left(P\right),x\circ y=[x+y-\epsilon ,x+y+\epsilon ].$$

Then, the following apply:

• Smooth manifold: $P=\mathbb{R}$ is $C^{\infty }$.
• Smooth hyperoperation: The endpoints $x+y\pm \epsilon$ depend smoothly on $(x,y)$, so $x\circ y$ varies smoothly in the Hausdorff topology on nonempty closed subsets of $\mathbb{R}$.
• Identity: With $e=0$, we have $0\circ x=[x-\epsilon ,x+\epsilon ]\ni x$ and similarly $x\circ 0\ni x$.
• Inverses: For each $x\in \mathbb{R}$, its inverse is $x^{-1}=-x$, since

  $$x\circ (-x)=[-\epsilon ,\epsilon ]\ni 0,(-x)\circ x=[-\epsilon ,\epsilon ]\ni 0.$$
• Associativity: One checks

  $$(x\circ y)\circ z=\bigcup _{u\in [x+y-\epsilon ,x+y+\epsilon ]}[u+z-\epsilon ,u+z+\epsilon ]=[x+y+z-2\epsilon ,x+y+z+2\epsilon ]$$

  and similarly $x\circ (y\circ z)$, so associativity holds.
• Compatible action: The Lie group $(\mathbb{R},+)$ acts by translation:

  $$\phi (u,g)=u+g,\phi (x\circ y,g)=[x+y-\epsilon +g,x+y+\epsilon +g]=\phi (x,g)\circ \phi (y,g).$$

Hence, $\left(\mathbb{R},\circ \right)$ is a Lie hypergroup.

2.6. Quotient Hypergroup

A quotient hypergroup is formed by partitioning a hypergroup with a compatible equivalence relation, inheriting a hyperoperation on equivalence classes. The definition of a quotient hypergroup is described as follows.

Definition 19

(Quotient Hypergroup [64,65,66,67]). Let $(H,\circ )$ be a hypergroup and let ∼ be an equivalence relation on H that is compatible with the hyperoperation, meaning that for any $x,x^{\prime },y,y^{\prime }\in H$ with $x\sim x^{\prime }$ and $y\sim y^{\prime }$, it holds that

$$x\circ y\subseteq \{z\in H\mid \exists z^{\prime }\in x^{\prime }\circ y^{\prime }\mathrm{with}z\sim z^{\prime }\}.$$

Then, the quotient set

$$H/\sim =\left\{\right[x]\mid x\in H\},$$

together with the operation defined by

$$\left[x\right]\circ \left[y\right]=\left\{\right[z]\mid z\in x\circ y\},$$

for all $x,y\in H$, is called a quotient hypergroup.

Example 11

(Quotient Hypergroup). Let $(H,★)$ be the hypergroup with

$$H=\{e,a,b\}$$

and hyperoperation ★ defined by

$$\begin{array}{cccc}★& e& a& b\\ e& \left\{e\right\}& \left\{a\right\}& \left\{b\right\}\\ a& \left\{a\right\}& \{e,a\}& \{a,b\}\\ b& \left\{b\right\}& \{a,b\}& \{e,b\}\end{array}$$

Define an equivalence relation ∼ on H by

$$e\sim e,a\sim b,b\sim a.$$

Then, the quotient set is

$$H/\sim =\left\{\right[e],[a\left]\right\},$$

and the induced operation on equivalence classes,

$$\left[x\right]\circ \left[y\right]=\left\{\right[z]\mid z\in x★y\},$$

is given by

$$\begin{array}{ccc}\circ & \left[e\right]& \left[a\right]\\ \left[e\right]& \left\{\right[e\left]\right\}& \left\{\right[a\left]\right\}\\ \left[a\right]& \left\{\right[a\left]\right\}& \left\{\right[e],[a\left]\right\}\end{array}$$

with identity class $\left[e\right]$. Hence, $(H/\sim ,\circ ,[e\left]\right)$ is a quotient hypergroup.

2.7. Reduced Hypergroup

A reduced hypergroup is one whose essential indistinguishability relation is trivial; all equivalence classes are singletons, yielding a reduced form. The definition of a reduced hypergroup is described as follows.

Definition 20

(Reduced Hypergroup [68,69,70]). Let $(H,\circ )$ be a hypergroup and consider the essential indistinguishability relation ${\sim }_e$ on H defined by

$$x{\sim }_{e}y⟺x{\sim }_{o}yandx{\sim }_{i}y,$$

where ${\sim }_o$ (the operational equivalence) and ${\sim }_i$ (the inseparability relation) are equivalence relations on H that capture, respectively, the indistinguishability of elements with respect to left and right hyperoperations. The hypergroup $(H,\circ )$ is said to be reduced if for every $x\in H$, the equivalence class

$${\left[x\right]}_{{\sim }_e}=\left\{x\right\}.$$

In this case, the quotient hypergroup $H/{\sim }_e$ is called the reduced form of $(H,\circ )$.

Example 12

(A Reduced Hypergroup). Let

$$H=\{e,a\},$$

and define the hyperoperation $★:H\times H\to \mathcal{P}\left(H\right)$ by

$$\begin{array}{ccc}★& e& a\\ e& \left\{e\right\}& \left\{a\right\}\\ a& \left\{a\right\}& \{e,a\}\end{array}$$

with identity e. One computes the following:

• Operational equivalence ${\sim }_o$: $x{\sim }_{o}y$ if $x★z=y★z$ for all z. Here, $e★e=\left\{e\right\}\ne \left\{a\right\}=a★e$, so only $e{\sim }_{o}e$ and $a{\sim }_{o}a$.
• Inseparability ${\sim }_i$: $x{\sim }_{i}y$ if $z★x=z★y$ for all z. Similarly, only identical pairs satisfy this.

Hence, the essential indistinguishability ${\sim }_e={\sim }_o\cap {\sim }_i$ is the identity relation, so each class ${\left[x\right]}_{{\sim }_e}=\left\{x\right\}$. Therefore, $(H,★)$ is a reduced hypergroup.

2.8. Fuzzy Sets, Fuzzy Groups, and Fuzzy Hypergroups

A fuzzy set on universe H assigns every element a real-number-valued membership degree in $[0,1]$, representing its partial uncertain belonging. The definition of a fuzzy set is described as follows [71,72].

Definition 21

(Fuzzy Set [72]). Let H be a nonempty set. A fuzzy set on H is a function

$$\mu :H\to [0,1],$$

where for each $x\in H$, the value $\mu \left(x\right)$ represents the degree of membership of x in the fuzzy set.

The definition of a fuzzy group is described as follows.

Definition 22

(Fuzzy Group cf. [11,73,74,75]). Let G be a group. A fuzzy set $\mu :G\to [0,1]$ is called a fuzzy subgroup (or fuzzy group) if for all $x,y\in G$, the following condition holds:

$$\mu \left(x{y}^{-1}\right)\ge min\{\mu \left(x\right),\mu \left(y\right)\}.$$

Equivalently, each level set

$$\{x\in G\mid \mu (x)\ge t\},t\in [0,1],$$

is a subgroup of G.

Example 13

(A Simple Fuzzy Subgroup of ${\mathbb{Z}}_2$). Let

$$G={\mathbb{Z}}_2=\{0,1\}$$

be the group under addition mod 2. Define a fuzzy set $\mu :G\to [0,1]$ by

$$\mu \left(0\right)=1,\mu \left(1\right)=0.6.$$

Since in ${\mathbb{Z}}_2$ we have

$$x-y\equiv x+y(mod2),$$

one checks for all $x,y\in G$:

$$\mu (x-y)=\mu (x+y)\ge min\left\{\mu \right(x),\mu (y\left)\right\}.$$

Hence μ satisfies the fuzzy-subgroup inequality and is a fuzzy subgroup of G.

A fuzzy hypergroup is a set equipped with an associative, reproductive fuzzy hyperoperation that maps each pair of elements to a fuzzy subset. The definition of a fuzzy hypergroup is described as follows.

Definition 23

(Fuzzy Hypergroup cf. [44,45,46,47]). Let H be a nonempty set and denote by

$$F\left(H\right)=\{\mu :H\to [0,1\left]\right\}$$

the set of all fuzzy subsets of H. A fuzzy hyperoperation on H is a mapping

$$\tilde{\circ }:H\times H\to F\left(H\right).$$

The pair $\langle H,\tilde{\circ }\rangle$ is called a fuzzy hypergroupoid. For any two nonempty fuzzy subsets $A,B\in F\left(H\right)$, the hyperoperation is extended by defining the fuzzy set

$$\left(A\tilde{\circ }B\right)\left(y\right)=\underset{a,b\in H}{sup}\{A\left(a\right)\wedge B\left(b\right)\wedge \left(a\tilde{\circ }b\right)\left(y\right)\},\forall y\in H.$$

A fuzzy hypergroupoid is said to be a fuzzy semihypergroup if the associative law holds; that is, for all $x,y,z\in H$,

$$\left(x\tilde{\circ }y\right)\tilde{\circ }z=x\tilde{\circ }\left(y\tilde{\circ }z\right).$$

It is called a fuzzy quasihypergroup if it satisfies the reproductive law: for every $x\in H$,

$$x\tilde{\circ }H=H\mathrm{and}H\tilde{\circ }x=H,$$

where H is identified with its characteristic fuzzy set (i.e., $H\left(x\right)=1$ for all $x\in H$). Finally, a fuzzy hypergroup is defined to be a fuzzy semihypergroup that is also a fuzzy quasihypergroup.

Example 14

(A Simple Fuzzy Hypergroup). Let

$$H=\{a,b\},$$

and define the fuzzy hyperoperation $\tilde{\circ }:H\times H\to F\left(H\right)$ by the membership functions

$$\begin{array}{cc}\hfill \left(a\tilde{\circ }a\right)\left(a\right)& =1,\left(a\tilde{\circ }a\right)\left(b\right)=0.5,\hfill \\ \hfill \left(a\tilde{\circ }b\right)\left(a\right)& =0.4,\left(a\tilde{\circ }b\right)\left(b\right)=1,\hfill \\ \hfill \left(b\tilde{\circ }a\right)\left(a\right)& =0.4,\left(b\tilde{\circ }a\right)\left(b\right)=1,\hfill \\ \hfill \left(b\tilde{\circ }b\right)\left(a\right)& =0.3,\left(b\tilde{\circ }b\right)\left(b\right)=1.\hfill \end{array}$$

For any fuzzy subsets $A,B\in F\left(H\right)$, the convolution

$$\left(A\tilde{\circ }B\right)\left(y\right)=\underset{x,z\in H}{sup}\left\{A\left(x\right)\wedge B\left(z\right)\wedge \left(x\tilde{\circ }z\right)\left(y\right)\right\}$$

is easily computed. One checks the following:

• Associativity: $\left(A\tilde{\circ }B\right)\tilde{\circ }C=A\tilde{\circ }\left(B\tilde{\circ }C\right)$ for all $A,B,C$.
• Reproductivity: If H denotes its characteristic fuzzy set ($H\left(x\right)=1$ for all x), then

  $$x\tilde{\circ }H=H\mathrm{and}H\tilde{\circ }x=H\mathrm{for}x=a,b.$$

Therefore, $〈H,\tilde{\circ }〉$ is a fuzzy hypergroup.

3. Main Results

As summarized in the Introduction, this section presents the principal contributions of the paper. We introduce and develop the following new structures:

• Superhypermagmas and Lie superhypergroups, extending classical magmas and Lie groups via nth-powerset constructions.
• Quotient superhypergroups and reduced superhypergroups, arising from natural quotient and reduction processes on superhypergroups.
• Fuzzy superhypergroups, which integrate fuzzification into the superhyperstructure framework.

3.1. SuperHyperMagma

A superhypermagma generalizes hypermagmas by mapping each element pair to a nonempty subset in the nth powerset, and it ensures closure. The definition of a superhypermagma is described as follows.

Definition 24

(Superhypermagma). Let H be a nonempty set and let $n\ge 1$ be a fixed positive integer. A superhyperoperation on H is a function

$$\circ :H\times H\to {\mathcal{P}}_n\left(H\right),$$

where ${\mathcal{P}}_n\left(H\right)$ denotes the n-th powerset of H (defined recursively by ${\mathcal{P}}_1\left(H\right)=\mathcal{P}\left(H\right)$ and ${\mathcal{P}}_{k+1}\left(H\right)=\mathcal{P}\left({\mathcal{P}}_k\left(H\right)\right)$ for $k\ge 1$). A pair $(H,\circ )$ is called a superhypermagma. Notice that when $n=1$, we recover the definition of a hypermagma. In this way, the notion of superhypermagma generalizes that of a hypermagma.

Example 15

(A Simple Superhypermagma (n = 2)). Let

$$H=\{a,b\}.$$

Then,

$$P_1\left(H\right)=\mathcal{P}\left(H\right)=\left\{⌀,\left\{a\right\},\left\{b\right\},\{a,b\}\right\},$$

and

$$P_2\left(H\right)=\mathcal{P}\left(P_1\left(H\right)\right).$$

Define the superhyperoperation

$$\circ :H\times H\to P_2\left(H\right)$$

by the table

$$\begin{array}{ccc}\circ & a& b\\ a& \left\{\left\{a\right\},\left\{b\right\}\right\}& \left\{\{a,b\}\right\}\\ b& \left\{\{a,b\}\right\}& \left\{\left\{a\right\},\left\{b\right\},\{a,b\}\right\}\end{array}$$

That is,

$$a\circ a=\left\{\right\{a\},\{b\left\}\right\},a\circ b=b\circ a=\left\{\right\{a,b\left\}\right\},b\circ b=\left\{\right\{a\},\{b\},\{a,b\left\}\right\}.$$

Since $\circ (x,y)\in P_2\left(H\right)$ for all $x,y\in H$, $\langle H,\circ \rangle$ is a superhypermagma.

Theorem 2

(Superhypermagma Generalizes Hypermagma). Every hypermagma $(H,★)$ (with $★:H\times H\to \mathcal{P}\left(H\right)$) is a superhypermagma with $n=1$; that is, by setting

$$\circ =★and{\mathcal{P}}_1\left(H\right)=\mathcal{P}\left(H\right),$$

the structure $(H,\circ )$ is a superhypermagma. Hence, the notion of superhypermagma generalizes that of hypermagma.

Proof. 

By definition, a hypermagma is a pair $(H,★)$ where ★ maps into $\mathcal{P}\left(H\right)$. Since $\mathcal{P}\left(H\right)={\mathcal{P}}_1\left(H\right)$, the mapping

$$\circ :H\times H\to {\mathcal{P}}_1\left(H\right)$$

defined by $\circ (x,y)=★(x,y)$ exactly meets the definition of a superhyperoperation for $n=1$. Therefore, every hypermagma is a special case of a superhypermagma. □

Theorem 3

(Superhypergroup Yields an n-Superhyperstructure). Let $(H,\circ )$ be a superhypergroup with superhyperoperation

$$\circ :H\times H\to {\mathcal{P}}_n\left(H\right).$$

For any nonempty subsets $A,B\in {\mathcal{P}}_n\left(H\right)$, define

$$A\circ B=\bigcup _{a\in A,b\in B}\circ (a,b).$$

Then, the structure

$${\mathcal{SH}}_n=\left({\mathcal{P}}_n\left(H\right),\circ \right)$$

forms an n-superhyperstructure.

Proof. 

We prove the claim by induction on n.

Base Case ($n=1$): For $n=1$, ${\mathcal{P}}_1\left(H\right)=\mathcal{P}\left(H\right)$. The operation is defined for any nonempty subsets $A,B\subseteq H$ by

$$A\circ B=\bigcup _{a\in A,b\in B}\circ (a,b).$$

Since $(H,\circ )$ is a superhypergroup, each $\circ (a,b)$ is nonempty and the union is over nonempty sets; hence, $A\circ B$ is nonempty. The strong superhyperassociativity of ∘ on H ensures that the associativity condition lifts to $\mathcal{P}\left(H\right)$. Moreover, the identity element $e\in H$ satisfies

$$\circ (e,x)=\left\{x\right\}\mathrm{and}\circ (x,e)=\left\{x\right\}$$

for all $x\in H$, so that $\left\{e\right\}$ acts as the identity on $\mathcal{P}\left(H\right)$. Thus, $\left(\mathcal{P}\right(H),\circ )$ is a 1-superhyperstructure.

Inductive Step: Assume that for some $k\ge 1$, the structure

$${\mathcal{SH}}_k=\left({\mathcal{P}}_k\left(H\right),\circ \right)$$

forms a k-superhyperstructure. Define the $(k+1)$th powerset by

$${\mathcal{P}}_{k+1}\left(H\right)=\mathcal{P}\left({\mathcal{P}}_k\left(H\right)\right),$$

and extend the operation to ${\mathcal{P}}_{k+1}\left(H\right)$ by setting, for any $A,B\in {\mathcal{P}}_{k+1}\left(H\right)$,

$$A\circ B=\bigcup _{X\in A,Y\in B}(X\circ Y),$$

where $X\circ Y$ is defined by the induction hypothesis. The strong superhyperassociativity on H guarantees that this extended operation is associative on ${\mathcal{P}}_{k+1}\left(H\right)$. The identity and inverse properties similarly extend by considering the singleton $\left\{e\right\}$ (and appropriate inverses) at each level.

Thus, by induction, ${\mathcal{SH}}_n=\left({\mathcal{P}}_n\left(H\right),\circ \right)$ is an n-superhyperstructure. □

3.2. Lie Superhypergroup

A Lie superhypergroup is a smooth manifold with an n-th powerset-valued, smooth, associative superhyperoperation, identity element, and smooth inverse structure. The definition of a Lie superhypergroup is described as follows.

Definition 25

(Lie Superhypergroup). A Lie superhypergroup is a structure $(P,\circ )$ satisfying the following conditions:

1. 

Smooth Manifold: The underlying set P is a smooth manifold.

2. 

Superhyperoperation: There exists a fixed positive integer $n\ge 1$ such that the operation

$$\circ :P\times P\to {\mathcal{P}}_n\left(P\right)$$

is defined, where ${\mathcal{P}}_n\left(P\right)$ denotes the nth powerset of P (defined recursively by ${\mathcal{P}}_1\left(P\right)=\mathcal{P}\left(P\right)$ and ${\mathcal{P}}_{k+1}\left(P\right)=\mathcal{P}\left({\mathcal{P}}_k\left(P\right)\right)$ for $k\ge 1$). In this way, the Lie superhypergroup generalizes the notion of a Lie hypergroup (which is the case $n=1$).

3. 

Smoothness of the Operation: The superhyperoperation ∘ is smooth in the sense that for every $x,y\in P$, the element $\circ (x,y)$ in ${\mathcal{P}}_n\left(P\right)$ depends smoothly on x and y (with respect to a suitable smooth structure on ${\mathcal{P}}_n\left(P\right)$).

4. 

Strong Superhyperassociativity: For all $x,y,z\in P$,

$$\circ \left(x,\circ (y,z)\right)=\circ \left(\circ (x,y),z\right),$$

where the extension of ∘ to nonempty subsets $A,B\subseteq P$ is given by

$$A\circ B=\bigcup _{a\in A,b\in B}\circ (a,b).$$

5. 

Identity: There exists an element $e\in P$ such that for every $x\in P$,

$$\circ (e,x)=\circ (x,e)=\left\{x\right\}.$$

6. 

Inverses: For every $x\in P$, there exists an element $x^{-1}\in P$ such that

$$e\in \circ (x,x^{-1})ande\in \circ (x^{-1},x).$$

Example 16

(Lie Superhypergroup on $\mathbb{R}$ with $n=2$). Let $P=\mathbb{R}$, a smooth one-dimensional manifold, and fix $\epsilon >0$. Then,

$${\mathcal{P}}_1\left(P\right)=\mathcal{P}\left(\mathbb{R}\right),{\mathcal{P}}_2\left(P\right)=\mathcal{P}\left({\mathcal{P}}_1\left(P\right)\right).$$

Define

$$\circ :P\times P⟶{\mathcal{P}}_2\left(P\right),x\circ y=\left\{[x+y-\epsilon ,x+y+\epsilon ]\right\},$$

where $[x+y-\epsilon ,x+y+\epsilon ]\subset \mathbb{R}$. Then, the following apply:

• Smoothness: The map $(x,y)\mapsto \left\{\right[x+y-\epsilon ,x+y+\epsilon \left]\right\}$ is smooth with respect to the natural hyperspace structure on ${\mathcal{P}}_2\left(P\right)$.
• Strong Superhyperassociativity: For all $x,y,z$,

  $$x\circ (y\circ z)=\left\{[x+(y+z)-2\epsilon ,x+(y+z)+2\epsilon ]\right\}=(x\circ y)\circ z.$$
• Identity: With $e=0$, we have $0\circ x=\left\{\right[x-\epsilon ,x+\epsilon \left]\right\}\ni \left\{x\right\}$ and similarly $x\circ 0\ni \left\{x\right\}$.
• Inverses: Each x has inverse $-x$ since

  $$x\circ (-x)=\left\{\right[-\epsilon ,\epsilon \left]\right\}\ni \left\{0\right\},(-x)\circ x=\left\{\right[-\epsilon ,\epsilon \left]\right\}\ni \left\{0\right\}.$$

Hence, $\left(\mathbb{R},\circ \right)$ is a Lie superhypergroup for $n=2$.

Theorem 4.

Let $(P,\circ )$ be a Lie superhypergroup with superhyperoperation

$$\circ :P\times P\to {\mathcal{P}}_n\left(P\right).$$

Define an operation on ${\mathcal{P}}_n\left(P\right)$ for any nonempty subsets $A,B\in {\mathcal{P}}_n\left(P\right)$ by

$$A\circ B=\bigcup _{a\in A,b\in B}\circ (a,b).$$

Then, the structure

$${\mathcal{SH}}_n=\left({\mathcal{P}}_n\left(P\right),\circ \right)$$

forms an n-superhyperstructure.

Proof. 

We prove the claim by induction on n.

Base Case ($n=1$): For $n=1$, ${\mathcal{P}}_1\left(P\right)=\mathcal{P}\left(P\right)$. For any nonempty subsets $A,B\subseteq P$, define

$$A\circ B=\bigcup _{a\in A,b\in B}\circ (a,b).$$

Since $(P,\circ )$ is a Lie superhypergroup, each $\circ (a,b)$ is nonempty, and the union is taken over nonempty sets. The strong superhyperassociativity, the identity, and the inverse properties in P ensure that the corresponding properties hold for $\left(\mathcal{P}\right(P),\circ )$. Hence, $\left(\mathcal{P}\right(P),\circ )$ is a 1-superhyperstructure.

Inductive Step: Assume that for some $k\ge 1$, the structure

$${\mathcal{SH}}_k=\left({\mathcal{P}}_k\left(P\right),\circ \right)$$

forms a k-superhyperstructure. Define the $(k+1)$th powerset by

$${\mathcal{P}}_{k+1}\left(P\right)=\mathcal{P}\left({\mathcal{P}}_k\left(P\right)\right).$$

For any $A,B\in {\mathcal{P}}_{k+1}\left(P\right)$, define

$$A\circ B=\bigcup _{X\in A,Y\in B}\left(X\circ Y\right),$$

where the operation $X\circ Y$ is defined via the induction hypothesis. The strong superhyperassociativity on P guarantees that the associativity property lifts to ${\mathcal{P}}_{k+1}\left(P\right)$. Likewise, the identity element $e\in P$ yields the singleton $\left\{e\right\}$ as the identity in ${\mathcal{P}}_{k+1}\left(P\right)$, and the inverse property extends appropriately.

Thus, by induction, the structure

$${\mathcal{SH}}_n=\left({\mathcal{P}}_n\left(P\right),\circ \right)$$

satisfies all the axioms of an n-superhyperstructure. □

3.3. Quotient Superhypergroups

A quotient superhypergroup partitions a superhypergroup by a compatible equivalence, naturally inheriting a resultant n-th powerset-valued superhyperoperation on equivalence classes. The definition of a quotient superhypergroup is described as follows.

Definition 26

(Quotient Superhypergroup). Let $(H,\circ )$ be a superhypergroup with superhyperoperation

$$\circ :H\times H\to {\mathcal{P}}_n\left(H\right),$$

and let ∼ be an equivalence relation on H that is compatible with ∘ (i.e., if $x\sim x^{\prime }$ and $y\sim y^{\prime }$, then

$$x\circ y\subseteq \{z\in H\mid \exists z^{\prime }\in x^{\prime }\circ y^{\prime }withz\sim z^{\prime }\}).$$

Then, the quotient set

$$H/\sim =\left\{\right[x]\mid x\in H\}$$

endowed with the induced operation

$$\left[x\right]\circ \left[y\right]=\left\{\right[z]\mid z\in x\circ y\},$$

for all $x,y\in H$, is called a quotient superhypergroup.

Example 17

(Quotient superhypergroup for $n=1$). Let

$$H=\{e,a,b\},$$

with superhyperoperation $\circ :H\times H\to {\mathcal{P}}_1\left(H\right)=\mathcal{P}\left(H\right)$ given by

$$\begin{array}{cccc}\circ & e& a& b\\ e& \left\{e\right\}& \left\{a\right\}& \left\{b\right\}\\ a& \left\{a\right\}& \{e,a\}& \{a,b\}\\ b& \left\{b\right\}& \{a,b\}& \{e,b\}\end{array}$$

Define the equivalence relation ∼ on H by

$$e\sim e,a\sim b,b\sim a.$$

Then, the quotient set is

$$H/\sim =\left\{\right[e],[a\left]\right\},$$

and the induced operation $\left[x\right]\circ \left[y\right]=\left\{\right[z]\mid z\in x\circ y\}$ yields

$$\begin{array}{ccc}\circ & \left[e\right]& \left[a\right]\\ \left[e\right]& \left\{\right[e\left]\right\}& \left\{\right[a\left]\right\}\\ \left[a\right]& \left\{\right[a\left]\right\}& \left\{\right[e],[a\left]\right\}\end{array}$$

with identity class $\left[e\right]$. Therefore, $\left(H/\sim ,\circ ,\left[e\right]\right)$ is a quotient superhypergroup.

Example 18

(Quotient Superhypergroup for $n=2$). Let

$$H=\{e,a\},$$

and form

$$P_1\left(H\right)=\mathcal{P}\left(H\right)=\{⌀,\left\{e\right\},\left\{a\right\},\{e,a\}\},P_2\left(H\right)=\mathcal{P}\left(P_1\left(H\right)\right).$$

Define the superhyperoperation $\circ :H\times H\to P_2\left(H\right)$ by

$$\begin{array}{ccc}\circ & e& a\\ e& \left\{\right\{e\left\}\right\}& \left\{\right\{a\left\}\right\}\\ a& \left\{\right\{a\left\}\right\}& \left\{\right\{e,a\left\}\right\}\end{array}$$

where, for instance, $e\circ a=\left\{\left\{a\right\}\right\}\subset P_1\left(H\right)$. Now, let ∼ be the trivial (identity) relation on H, so that

$$H/\sim =\left\{\right[e],[a\left]\right\}.$$

The induced operation on classes is

$$\left[x\right]\circ \left[y\right]=\left\{\right[z]\mid z\in SforsomeS\in x\circ y\},$$

giving the table

$$\begin{array}{ccc}\circ & \left[e\right]& \left[a\right]\\ \left[e\right]& \left\{\right[e\left]\right\}& \left\{\right[a\left]\right\}\\ \left[a\right]& \left\{\right[a\left]\right\}& \left\{\right[e],[a\left]\right\}\end{array}$$

with identity class $\left[e\right]$. Hence, $\left(H/\sim ,\circ ,\left[e\right]\right)$ is a quotient superhypergroup for $n=2$.

Theorem 5

(Quotient Superhypergroup Possesses an n-Superhyperstructure). Let $(H,\circ )$ be a superhypergroup with superhyperoperation $\circ :H\times H\to {\mathcal{P}}_n\left(H\right)$ and let ∼ be a compatible equivalence relation on H. Then, the quotient superhypergroup $H/\sim$, with the operation

$$\left[x\right]\circ \left[y\right]=\left\{\right[z]\mid z\in x\circ y\},$$

forms an n-superhyperstructure.

Proof. 

We must show that the induced operation on $H/\sim$ is well defined and satisfies the axioms of an n-superhyperstructure.

Well-definedness: Since ∼ is compatible with ∘, if $x\sim x^{\prime }$ and $y\sim y^{\prime }$, then the elements of $x\circ y$ are ∼-related to those of $x^{\prime }\circ y^{\prime }$. Hence, the set

$$\left[x\right]\circ \left[y\right]=\left\{\right[z]\mid z\in x\circ y\}$$

is independent of the choice of representatives.

Axioms: Because $\circ :H\times H\to {\mathcal{P}}_n\left(H\right)$ makes H into a superhypergroup, the operation ∘ on H satisfies the following:

• Strong Superhyperassociativity: For all $x,y,z\in H$,

  $$\circ \left(x,\circ (y,z)\right)=\circ \left(\circ (x,y),z\right).$$
• Existence of Identity and Inverses: There exists $e\in H$ such that $\circ (e,x)=\circ (x,e)=\left\{x\right\}$ and for each $x\in H$ there is $x^{-1}$ with $e\in \circ (x,x^{-1})$ and $e\in \circ (x^{-1},x)$.

These properties extend to unions and hence to the induced operation on the quotient $H/\sim$. In particular, the operation on $H/\sim$ is associative (in the superhyper sense), has an identity $\left[e\right]$, and inverses are inherited by taking the equivalence classes.

Thus, $H/\sim$ satisfies the axioms of an n-superhyperstructure. □

3.4. Reduced Superhypergroups

A reduced superhypergroup is a superhypergroup whose essential indistinguishability relation is trivial, so every element’s equivalence class is a singleton. The definition of a reduced superhypergroup is described as follows.

Definition 27

(Reduced Superhypergroup). Let $(H,\circ )$ be a superhypergroup. Define an essential indistinguishability relation ${\sim }_e$ on H (often defined as the conjunction of an operational equivalence and an inseparability relation). The superhypergroup $(H,\circ )$ is said to be reduced if for every $x\in H$,

$${\left[x\right]}_{{\sim }_e}=\left\{x\right\}.$$

In this case, the quotient superhypergroup $H/{\sim }_e$ is called the reduced form of $(H,\circ )$.

Example 19

(A Reduced Superhypergroup for $n=1$). Let

$$H=\{e,a\},$$

and consider the superhyperoperation $\circ :H\times H\to {\mathcal{P}}_1\left(H\right)=\mathcal{P}\left(H\right)$ defined by

$$\begin{array}{ccc}\circ & e& a\\ e& \left\{e\right\}& \left\{a\right\}\\ a& \left\{a\right\}& \{e,a\}\end{array}$$

with identity e. One checks the following:

• The operational equivalence and inseparability relations each identify only identical elements, so their conjunction, the essential indistinguishability ${\sim }_e$, is trivial.
• Consequently, for every $x\in H$, the equivalence class ${\left[x\right]}_{{\sim }_e}=\left\{x\right\}$.

Therefore, $(H,\circ )$ is already reduced as a superhypergroup, and its reduced form $H/{\sim }_e$ coincides with H itself.

Example 20

(A Reduced Superhypergroup for $n=2$). Let

$$H=\{e,a\},P_1\left(H\right)=\mathcal{P}\left(H\right)=\{⌀,\left\{e\right\},\left\{a\right\},\{e,a\}\},P_2\left(H\right)=\mathcal{P}\left(P_1\left(H\right)\right).$$

Define the superhyperoperation

$$\circ :H\times H⟶P_2\left(H\right)$$

by

$$\begin{array}{ccc}\circ & e& a\\ e& \left\{\right\{e\left\}\right\}& \left\{\right\{a\left\}\right\}\\ a& \left\{\right\{a\left\}\right\}& \left\{\right\{e,a\left\}\right\}\end{array}$$

so that, for instance, $a\circ a=\left\{\right\{e,a\left\}\right\}$. One checks the following:

• Operational equivalence ${\sim }_o$: $x{\sim }_{o}y$ if $\circ (x,z)=\circ (y,z)$ for all z. Here, $\circ (e,e)=\left\{\right\{e\left\}\right\}\ne \left\{\right\{a\left\}\right\}=\circ (a,e)$, so only identical pairs are equivalent.
• Inseparability ${\sim }_i$: $x{\sim }_{i}y$ if $\circ (z,x)=\circ (z,y)$ for all z, likewise trivial.

Their conjunction ${\sim }_e={\sim }_o\cap {\sim }_i$ is the identity relation; hence, each equivalence class ${\left[x\right]}_{{\sim }_e}=\left\{x\right\}$. Therefore, $(H,\circ )$ is already reduced, and its reduced form $H/{\sim }_e$ coincides with H itself.

Theorem 6

(Reduced Superhypergroup Possesses an n-Superhyperstructure). Let $(H,\circ )$ be a superhypergroup with superhyperoperation $\circ :H\times H\to {\mathcal{P}}_n\left(H\right)$ and let ${\sim }_e$ be the essential indistinguishability relation on H. Then, the quotient superhypergroup

$$H/{\sim }_e=\{\left[x\right]\mid x\in H\},$$

with the induced operation

$$\left[x\right]\circ \left[y\right]=\left\{\right[z]\mid z\in x\circ y\},$$

forms an n-superhyperstructure. Moreover, if $(H,\circ )$ is reduced (i.e., ${\left[x\right]}_{{\sim }_e}=\left\{x\right\}$ for all $x\in H$), then the quotient is isomorphic to H, and every equivalence class is a singleton.

Proof. 

The proof is analogous to that for the quotient superhypergroup.

Well-definedness: Since ${\sim }_e$ is an equivalence relation compatible with the superhyperoperation, the induced operation on $H/{\sim }_e$ is well defined.

Axioms: The axioms (strong superhyperassociativity, identity, and inverses) hold for H by hypothesis. When passing to the quotient, these properties are preserved by the same argument used in the previous theorem. In particular, the induced operation on $H/{\sim }_e$ makes it an n-superhyperstructure.

Reduction: By definition, $(H,\circ )$ is reduced if every equivalence class ${\left[x\right]}_{{\sim }_e}$ is trivial, i.e., ${\left[x\right]}_{{\sim }_e}=\left\{x\right\}$. Thus, in the quotient $H/{\sim }_e$, each element corresponds uniquely to an element of H and the structure is said to be in its reduced form. □

3.5. Fuzzy Superhypergroup

A fuzzy superhypergroup assigns to each ordered pair an element of the nth fuzzy powerset, satisfying associative and reproductive laws. The definition of a fuzzy superhypergroup is described as follows.

Notation 1

(nth fuzzy powerset). We denote by

$$F\left(H\right)=\{\mu :H\to [0,1\left]\right\}$$

the collection of all fuzzy subsets of H. For a fixed positive integer $n\ge 1$, we define recursively the nth fuzzy powerset of H by

$$F_1\left(H\right)=F\left(H\right)and{F}_{k+1}\left(H\right)=\{\mu :F_k\left(H\right)\to [0,1]\}fork\ge 1.$$

Definition 28

(Fuzzy Superhypergroup). Let H be a nonempty set and let $n\ge 1$ be a fixed positive integer. A fuzzy superhyperoperation on H is a mapping

$$\tilde{\circ }:H\times H\to F_n\left(H\right),$$

where $F_n\left(H\right)$ is the nth fuzzy powerset of H defined above. The pair $\langle H,\tilde{\circ }\rangle$ is called a fuzzy superhypergroupoid. It is said to be a fuzzy superhypergroup if the following apply:

1. 

(Associativity) For all $x,y,z\in H$,

$$\left(x\tilde{\circ }y\right)\tilde{\circ }z=x\tilde{\circ }\left(y\tilde{\circ }z\right),$$

where the extension of $\tilde{\circ }$ to fuzzy subsets in $F_n\left(H\right)$ is defined naturally (by taking supremum over membership degrees).

2. 

(Reproductive Law) For every $x\in H$,

$$x\tilde{\circ }H=HandH\tilde{\circ }x=H,$$

where H is identified with its characteristic fuzzy set in $F_1\left(H\right)$ and then embedded in $F_n\left(H\right)$.

Note that when $n=1$, a fuzzy superhypergroup coincides with a fuzzy hypergroup. Hence, fuzzy superhypergroups generalize fuzzy hypergroups.

Example 21

(A Simple Fuzzy Superhypergroup (n = 2)). Let

$$H=\{a,b\},$$

and let ${\delta }_a,{\delta }_b\in F_1\left(H\right)$ be the characteristic fuzzy sets

$${\delta }_a\left(a\right)=1,{\delta }_a\left(b\right)=0,{\delta }_b\left(a\right)=0,{\delta }_b\left(b\right)=1.$$

Then,

$$F_2\left(H\right)=\{\mu :\{{\delta }_a,{\delta }_b\}\to [0,1]\}$$

consists of all pairs $\left(\mu \left({\delta }_a\right),\mu \left({\delta }_b\right)\right)$. Define

$$\tilde{\circ }:H\times H\to F_2\left(H\right)$$

by

$$\begin{array}{ccc}\tilde{\circ }& a& b\\ a& (1,0)& (0.5,1)\\ b& (0.5,1)& (0,1)\end{array}$$

meaning, for example,

$$\left(a\tilde{\circ }b\right)\left({\delta }_a\right)=0.5,\left(a\tilde{\circ }b\right)\left({\delta }_b\right)=1.$$

One checks the following:

• Associativity: $\left(x\tilde{\circ }y\right)\tilde{\circ }z=x\tilde{\circ }\left(y\tilde{\circ }z\right)$ holds for all $x,y,z\in H$.
• Reproductive law: Identifying H with its characteristic fuzzy set in $F_1\left(H\right)$, one verifies

  $$x\tilde{\circ }H=HandH\tilde{\circ }x=H\forall x\in H.$$

Therefore, $〈H,\tilde{\circ }〉$ is a fuzzy superhypergroup for $n=2$.

Theorem 7

(Fuzzy Superhypergroup Yields an n-Superhyperstructure). Let $\langle H,\tilde{\circ }\rangle$ be a fuzzy superhypergroup with fuzzy superhyperoperation

$$\tilde{\circ }:H\times H\to F_n\left(H\right).$$

For any two nonempty fuzzy subsets $A,B\in F_n\left(H\right)$, define the extended operation by

$$\left(A\tilde{\circ }B\right)\left(y\right)=\underset{a,b\in H}{sup}\{A\left(a\right)\wedge B\left(b\right)\wedge \left(\tilde{\circ }(a,b)\right)\left(y\right)\},\forall y\in H.$$

Then, the structure

$${\mathcal{FSH}}_n=\left(F_n\left(H\right),\tilde{\circ }\right)$$

forms an n-superhyperstructure.

Proof. 

We prove the claim by induction on n.

Base Case ($n=1$): For $n=1$, $F_1\left(H\right)=F\left(H\right)$. By the definition of a fuzzy hypergroup, the operation $\tilde{\circ }:H\times H\to F\left(H\right)$ extends to fuzzy subsets in such a way that associativity (in the fuzzy sense) and the reproductive law hold. Hence, ${\mathcal{FSH}}_1=\left(F\left(H\right),\tilde{\circ }\right)$ is a 1-superhyperstructure.

Inductive Step: Assume that for some $k\ge 1$, the structure

$${\mathcal{FSH}}_k=\left(F_k\left(H\right),\tilde{\circ }\right)$$

forms a k-superhyperstructure. Define the $(k+1)$th fuzzy powerset by

$$F_{k+1}\left(H\right)=\{\mu :F_k\left(H\right)\to [0,1]\}.$$

Extend the fuzzy superhyperoperation to $F_{k+1}\left(H\right)$ by setting, for any $A,B\in F_{k+1}\left(H\right)$ and for all $y\in H$,

$$\left(A\tilde{\circ }B\right)\left(y\right)=\underset{X,Y\in F_k\left(H\right)}{sup}\{A\left(X\right)\wedge B\left(Y\right)\wedge \left(X\tilde{\circ }Y\right)\left(y\right)\},$$

where $X\tilde{\circ }Y$ is defined by the induction hypothesis. The strong associativity of $\tilde{\circ }$ on H ensures that this extension is associative on $F_{k+1}\left(H\right)$, and the identity and reproductive laws extend naturally (using the characteristic fuzzy set corresponding to H).

Thus, by induction, ${\mathcal{FSH}}_n=\left(F_n\left(H\right),\tilde{\circ }\right)$ satisfies the axioms of an n-superhyperstructure.

□

4. Conclusions and Future Work

In this paper, we introduced the notion of superhyperstructures by iterating the powerset construction and developed the theories of superhypermagmas and Lie superhypergroups. We also defined and examined quotient superhypergroups, reduced superhypergroups, and fuzzy superhypergroups, highlighting their algebraic properties and relationships.

Looking ahead, we plan to extend the superhypergroup framework by incorporating additional uncertainty-modeling paradigms—such as intuitionistic fuzzy sets [76], vague sets [77], picture fuzzy sets [78,79], bipolar fuzzy sets [80], soft sets [81,82], rough sets [83], hesitant fuzzy sets [84], neutrosophic sets [85,86], quadripartitioned neutrosophic sets [87,88], and plithogenic sets [89,90]—to further enrich and generalize the theory. Building upon these extensions, we also aim to investigate their mathematical structures and explore potential real-world applications.