Review of Rough, Soft, Fuzzy, and Functorial SuperHyperStructures

1. Preliminaries

In this section we collect the notation and basic notions used later. Unless explicitly stated otherwise, all underlying sets are finite.

1.1. SuperHyperStructure

A Classical Structure refers to a mathematical or real-world construct—such as logic, probability, statistics, algebra, geometry, graph theory, or automata. A HyperStructure enlarges this setting by replacing a base set S with its powerset $\mathcal{P}\left(S\right)$ and by allowing hyperoperations that take subsets to subsets, thereby expressing higher–order interactions [3,4,5]. Definitions are stated below.

Definition 1

(Classical Structure). AClassical Structure$\mathcal{C}$ is a mathematical object drawn from domains such as set theory, logic, probability, statistics, algebra, geometry, graph theory, automata theory, or game theory. Formally,

$$\mathcal{C}=\left(H,{\left\{{\#}^{\left(m\right)}\right\}}_{m\in \mathcal{I}}\right),$$

where:

• H is a nonemptycarrier set;
• for each $m\in \mathcal{I}\subseteq {\mathbb{Z}}_{>0}$, there is an m-ary operation ${\#}^{\left(m\right)}:H^m\to H$, subject to the axioms appropriate to the particular structure (e.g., associativity, commutativity, identities).

We call $\mathcal{C}$ a structure oftype$\{{\#}^{\left(m\right)}:m\in \mathcal{I}\}$. Typical instances include:

• Set: $(S,⌀)$, a carrier possibly equipped with distinguished elements or relations [6].
• Logic: $(L,\wedge ,\vee ,\neg )$ with binary $\wedge ,\vee$ and unary ¬, satisfying the usual logical laws [7].
• Probability: $(\Omega ,\mathcal{F},P)$, where $P:\mathcal{F}\to [0,1]$ is a measure on a σ-algebra $\mathcal{F}\subseteq \mathcal{P}\left(\Omega \right)$ [8].
• Statistics: $(X,\mathcal{A},\theta )$, with θ mapping data to parameters [9].
• Algebra:

  -

  Group$(G,*)$ with $*:G\times G\to G$ and the group axioms [10,11];

  -

  Ring$(R,+,\times )$ with two binary operations satisfying the ring axioms [12,13];

  -

  Vector space$(V,+,·)$ over a field $\mathbb{F}$ with scalar multiplication $·:\mathbb{F}\times V\to V$ [14,15].

• Geometry: $(X,\mathrm{dist})$ with a metric $\mathrm{dist}:X\times X\to \mathbb{R}$ [16,17].
• Graph: $(V,E)$ with $E\subseteq \left\{\right\{u,v\}:u,v\in V\}$ (undirected) or $E\subseteq V\times V$ (directed); adjacency and incidence are defined as usual [18,19].
• Automaton: $(Q,\Sigma ,\delta ,q_0,F)$ with finite state set Q, input alphabet Σ, transition function $\delta :Q\times \Sigma \to Q$, start state $q_0\in Q$, and accept set $F\subseteq Q$ [20,21].
• Game: $(N,{\left\{A_i\right\}}_{i\in N},{\left\{u_i\right\}}_{i\in N})$, where N is the player set, $A_i$ the action set of player i, and $u_i:{\prod }_{j\in N}{A}_j\to \mathbb{R}$ the payoff of player i [22,23].

Definition 2

(Powerset). (cf. [24,25])For a set S, thepowerset$\mathcal{P}\left(S\right)$ is the family of all subsets of S, including ∅ and S:

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

Definition 3

(Hyperoperation). (cf. [26,27])Ahyperoperationon S is a binary rule whose output is a subset of S rather than a single element; formally,

$$\circ :S\times S⟶\mathcal{P}\left(S\right).$$

Definition 4

(Hyperstructure). (cf. [28,29,30])AHyperstructureupgrades a classical structure by working on the powerset of the carrier. It is given by

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

where ∘ is a hyperoperation on subsets of S. In this setting, operations combine collections into new collections.

Example 1

(HyperStructure on ${\mathbb{Z}}_4$). Let $H={\mathbb{Z}}_4=\{0,1,2,3\}$ and define a binary hyperoperation

$$x⊞y:=\{x+y,x+y+1\}(mod4).$$

Concrete computations.

$$2⊞3=\{5,6\}(mod4)=\{1,2\},\{1,3\}⊞\left\{2\right\}=\left(1⊞2\right)\cup \left(3⊞2\right)=\{3,0\}\cup \{1,2\}=\{0,1,2,3\}.$$

Hence $(H,⊞)$ is a concreteHyperStructure.

A SuperHyperStructure pushes this idea further by iterating the powerset construction n times. Operations act on nested families of subsets, capturing hierarchical, multi-level interactions [2,31]. Related notions include SuperHyperAlgebra [32,33], SuperHyperGraph [34,35,36], and other super-level algebraic and combinatorial frameworks.

Definition 5

(n-th Powerset). ([31,37])For a set H, define inductively

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

Thus ${\mathcal{P}}^1\left(H\right)=\mathcal{P}\left(H\right)$ and ${\mathcal{P}}^2\left(H\right)=\mathcal{P}\left(\mathcal{P}\left(H\right)\right)$, etc. If one excludes the empty set at each stage, write ${\mathcal{P}}^{*}\left(X\right)=\mathcal{P}\left(X\right)\setminus \{\varnothing \}$ and set

$${\mathcal{P}}^{*1}\left(H\right)={\mathcal{P}}^{*}\left(H\right),{\mathcal{P}}^{*(k+1)}\left(H\right)={\mathcal{P}}^{*}\left({\mathcal{P}}^{*k}\left(H\right)\right).$$

Example 2

(n-th Powerset (Definition 5)). Let $H=\{a,b\}$. Then

$${\mathcal{P}}^0\left(H\right)=H=\{a,b\},{\mathcal{P}}^1\left(H\right)=\mathcal{P}\left(H\right)=\left\{\varnothing ,\left\{a\right\},\left\{b\right\},\{a,b\}\right\}.$$

The second iterated powerset is

$${\mathcal{P}}^2\left(H\right)=\mathcal{P}\left(\mathcal{P}\left(H\right)\right)=\left\{A|A\subseteq \{\varnothing ,\left\{a\right\},\left\{b\right\},\{a,b\}\}\right\}.$$

If we exclude ∅ at each stage, we obtain

$${\mathcal{P}}^{*1}\left(H\right)=\mathcal{P}\left(H\right)\setminus \{\varnothing \}=\left\{\left\{a\right\},\left\{b\right\},\{a,b\}\right\},$$

and

$${\mathcal{P}}^{*2}\left(H\right)={\mathcal{P}}^{*}\left({\mathcal{P}}^{*1}\left(H\right)\right)=\left\{A|\varnothing \ne A\subseteq \{\left\{a\right\},\left\{b\right\},\{a,b\}\}\right\}.$$

Definition 6

(SuperHyperOperations). (cf. [28])Let $H\ne \varnothing$ and define ${\mathcal{P}}^k\left(H\right)$ as above. An$(m,n)$-SuperHyperOperationis an m-ary map

$${\circ }^{(m,n)}:H^m⟶{\mathcal{P}}_{*}^n\left(H\right),$$

where ${\mathcal{P}}_{*}^n\left(H\right)$ denotes either the full n-th powerset ${\mathcal{P}}^n\left(H\right)$ or its nonempty variant ${\mathcal{P}}^n\left(H\right)\setminus \{\varnothing \}$. When the empty set is excluded, we speak of aclassical-typeSuperHyperOperation; when it is allowed, we refer to aneutrosophicSuperHyperOperation.

Example 3

(SuperHyperOperations). Let $H=\{0,1,2\}$. Define aunary$(m,n)=(1,2)$ superhyperoperation

$${\circ }^{(1,2)}:{\mathcal{P}}^1\left(H\right)=\mathcal{P}\left(H\right)⟶{\mathcal{P}}^2\left(H\right)=\mathcal{P}\left(\mathcal{P}\left(H\right)\right)$$

by

$${\circ }^{(1,2)}\left(A\right):=\left\{A,H\setminus A\right\}.$$

Then ${\circ }^{(1,2)}\left(A\right)$ is aset of subsetsof H, hence an element of ${\mathcal{P}}^2\left(H\right)$. For instance, with $A=\{0,2\}$,

$${\circ }^{(1,2)}\left(\{0,2\}\right)=\left\{\{0,2\},\left\{1\right\}\right\}\in {\mathcal{P}}^2\left(H\right).$$

This provides a concrete $(1,2)$-SuperHyperOperation.

Definition 7

(n-Superhyperstructure). (cf. [28,31,38,39])Ann-Superhyperstructureis a hyperstructure built on the n-fold iterated powerset of S:

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

with ∘ an operation on ${\mathcal{P}}^n\left(S\right)$. This yields a hierarchy in which operations act on increasingly nested collections.

Example 4

(n-Superhyperstructure). Let $S=\{x,y\}$ and take $n=2$. Consider the carrier ${\mathcal{P}}^2\left(S\right)=\mathcal{P}\left(\mathcal{P}\left(S\right)\right)$. Define

$$\odot :{\mathcal{P}}^2\left(S\right)\times {\mathcal{P}}^2\left(S\right)⟶{\mathcal{P}}^2\left(S\right),\odot (X,Y):=\{A\cup B\mid A\in X,B\in Y\}.$$

Since each $A\cup B$ is a subset of S, the right-hand side is a subset of $\mathcal{P}\left(S\right)$, i.e. an element of ${\mathcal{P}}^2\left(S\right)$. Hence

$${\mathcal{SH}}_2=\left({\mathcal{P}}^2\left(S\right),\odot \right)$$

is a valid 2-Superhyperstructure. Concretely, let

$$X=\left\{\left\{x\right\},\left\{y\right\}\right\},Y=\left\{\left\{y\right\},\{x,y\}\right\}.$$

Then

$$\odot (X,Y)=\left\{\left\{x\right\}\cup \left\{y\right\},\left\{x\right\}\cup \{x,y\},\left\{y\right\}\cup \left\{y\right\},\left\{y\right\}\cup \{x,y\}\right\}=\left\{\{x,y\},\{x,y\},\left\{y\right\},\{x,y\}\right\}=\left\{\left\{y\right\},\{x,y\}\right\}\in {\mathcal{P}}^2\left(S\right).$$

(1)

Definition 8

(SuperHyperStructure of order $(m,n)$). (cf. [2,40])Let $S\ne \varnothing$ and $m,n\ge 0$. A$(m,n)$-SuperHyperStructureof arity s is any map

$${\odot }^{(m,n)}:{\left({\mathcal{P}}^m\left(S\right)\right)}^s⟶{\mathcal{P}}^n\left(S\right).$$

Classical situations are recovered as special cases: $m=n=0$ yields ordinary s-ary operations; $m=0,n=1$ gives hyperoperations; and $s=1$ corresponds tosuperhyperoperations.

Example 5

(SuperHyperStructure of order $(m,n)$). Let $S=\{0,1,2\}$, set $(m,n)=(1,2)$ and arity $s=2$. Define

$${\odot }^{(1,2)}:{\left({\mathcal{P}}^1\left(S\right)\right)}^2⟶{\mathcal{P}}^2\left(S\right),{\odot }^{(1,2)}(A,B):=\left\{A,B,A\cup B\right\}.$$

By construction, ${\odot }^{(1,2)}(A,B)$ is a set of subsets of S, hence an element of ${\mathcal{P}}^2\left(S\right)$. For example, with $A=\{0,1\}$ and $B=\{1,2\}$,

$${\odot }^{(1,2)}(A,B)=\left\{\{0,1\},\{1,2\},\{0,1,2\}\right\}\in {\mathcal{P}}^2\left(S\right).$$

Thus $\left({\mathcal{P}}^1\left(S\right),{\odot }^{(1,2)}\right)$ realizes a concrete $(m,n)=(1,2)$ SuperHyperStructure of arity $s=2$.

1.2. Soft HyperStructure

A Soft HyperStructure consists of a parameter set A and a mapping $F:A\to \mathcal{P}\left(H\right)$, where each $a\in A$ yields a sub-hyperstructure $F\left(a\right)\subseteq H$ (cf.[41,42,43]).

Definition 9

(Soft HyperStructure). Let H be a hyperalgebra (hyperstructure) with signature $\Sigma =\{{★}_i:H^{n_i}\to {\mathcal{P}}^{*}\left(H\right)|i\in I\}$, where ${\mathcal{P}}^{*}\left(H\right)$ denotes the set of all nonempty subsets of H. A subset $K\subseteq H$ is asubhyperstructureof H if for every $i\in I$ and every $(x_1,\dots ,x_{n_i})\in K^{n_i}$ one has

$${★}_i(x_1,\dots ,x_{n_i})\subseteq K.$$

Let $(F,A)$ be a non-null soft set over H, i.e. $A\ne \varnothing$, $F:A\to {\mathcal{P}}^{*}\left(H\right)$ and the support $\mathrm{Supp}(F,A):=\{a\in A\mid F(a)\ne \varnothing \}\ne \varnothing$. Then $(F,A)$ is called aSoft HyperStructure over Hiff for every $a\in \mathrm{Supp}(F,A)$, the set $F\left(a\right)$ is a subhyperstructure of H.

Example 6

(Soft HyperStructure on ${\mathbb{Z}}_3$ with left-projection). Let $H={\mathbb{Z}}_3=\{0,1,2\}$ and define a (degenerate but valid) hyperoperation

$$x▹y:=\left\{x\right\}\left(\mathrm{left}\mathrm{projection}\right).$$

Take the parameter set $A=\{\mathsf{low},\mathsf{high}\}$ and define the soft set $F:A\to \mathcal{P}\left(H\right)$ by

$$F\left(\mathsf{low}\right)=\{0,1\},F\left(\mathsf{high}\right)=\{1,2\}.$$

Closure check (each parameter induces a subhyperstructure).If $x,y\in F\left(a\right)$, then $x▹y=\left\{x\right\}\subseteq F\left(a\right)$ because $x\in F\left(a\right)$. Hence $F\left(\mathsf{low}\right)$ and $F\left(\mathsf{high}\right)$ are both closed under ▹, so $(F,A)$ is a concreteSoft HyperStructureover $(H,▹)$.

1.3. Fuzzy HyperStructure

A Fuzzy HyperStructure assigns to each hyperoperation outcome a membership function $\mu :H\to [0,1]$, modeling graded belongingness of elements to hyperoperation results (cf.[44,45,46,47]).

Definition 10

(Fuzzy hyperoperation). (cf.[48,49]) Let H be a nonempty set and let $\mathcal{F}\left(H\right):=\{\mu :H\to [0,1\left]\right\}$ be the set of all fuzzy subsets of H. Afuzzy hyperoperationon H is a map

$$★:H\times H⟶\mathcal{F}\left(H\right),(x,y)⟼{\mu }_{x,y}(·),$$

so that for each $(x,y)\in H\times H$, ${\mu }_{x,y}:H\to [0,1]$ is a membership function on H. The pair $(H,★)$ is called afuzzy hypergroupoid.

Definition 11

(Fuzzy HyperStructure). AFuzzy HyperStructureis a pair $\left(H,{\left({★}_i\right)}_{i\in I}\right)$ where each ${★}_i$ is a fuzzy hyperoperation on H (possibly of specified arity). When $I=\left\{1\right\}$ we simply write $(H,★)$.

Remark 1

((Cut–set representation (and reconstruction)). For $\alpha \in (0,1]$, the α-cut of ★ induces a (crisp) hyperoperation

$$x{\circ }_{\alpha }y:=\{z\in H:{\mu }_{x,y}\left(z\right)\ge \alpha \}\in {\mathcal{P}}^{*}\left(H\right),$$

hence $(H,{\circ }_{\alpha })$ is a (crisp) hyperstructure for each α. Conversely, a family $\{{\circ }_{\alpha }:\alpha \in (0,1]\}$ of hyperoperations that is monotone in α (i.e. $\alpha \le \beta \Rightarrow x{\circ }_{\beta }y\subseteq x{\circ }_{\alpha }y$ for all $x,y$) uniquely determines ★ by

$${\mu }_{x,y}\left(z\right)=sup\{\alpha \in (0,1]:z\in x{\circ }_{\alpha }y\}.$$

Example 7

(Fuzzy HyperStructure on a three-element set). Let $S=\{x,y,z\}$. Define a fuzzy hyperoperation $★:S\times S\to {[0,1]}^S$ by specifying membership vectors $★(a,b)=({\mu }_{ab}\left(x\right),{\mu }_{ab}\left(y\right),{\mu }_{ab}\left(z\right))$. For instance,

$${\mu }_{xy}=(0.7,0.2,0.0),{\mu }_{xx}=(0.1,0.9,0.3),{\mu }_{yz}=(0.0,0.4,0.8).$$

$\alpha$-cut illustration.At level $\alpha =0.5$,

$${(x★y)}_{0.5}=\left\{x\right\},{(x★x)}_{0.5}=\left\{y\right\},{(y★z)}_{0.5}=\left\{z\right\}.$$

Thus $(S,★)$ is a concreteFuzzy HyperStructurewith explicit graded outputs.

1.4. Functorial Structure

A Functorial Structure is defined as a covariant functor $F:\mathcal{C}\to \mathbf{Set}$, assigning sets to objects and functions to morphisms, ensuring functoriality [50].

Definition 12

(Functorial Set). [50] Let $\mathcal{C}$ be a category and

$$F:\mathcal{C}⟶\mathbf{Set}$$

be a (covariant) endofunctor. For any object $X\in \mathrm{Ob}\left(\mathcal{C}\right)$, anF-set over Xis an element

$$s\in F\left(X\right).$$

We denote the collection of all F-sets over X simply by $F\left(X\right)$. A morphism $f:X\to Y$ in $\mathcal{C}$ induces apushforward

$$F\left(f\right):F\left(X\right)⟶F\left(Y\right),s\mapsto F\left(f\right)\left(s\right).$$

Example 8

(Concrete Example of a Functorial Set). Let $\mathcal{C}=\mathbf{FinSet}$, the category of finite sets and functions. Define the functor

$$F:\mathbf{FinSet}⟶\mathbf{Set}$$

by

$$F\left(X\right)=\mathcal{P}\left(X\right),F\left(f\right)\left(A\right)=f\left[A\right]=\left\{f\right(x)\mid x\in A\}$$

for any finite set X and function $f:X\to Y$.

Example instance.Take $X=\{1,2,3\}$ and $f:X\to Y=\{a,b\}$ defined by $f\left(1\right)=a$, $f\left(2\right)=b$, $f\left(3\right)=a$. Then

$$F\left(X\right)=\mathcal{P}\left(\right\{1,2,3\left\}\right)=\{\varnothing ,\{1\},\{2\},\{3\},\{1,2\},\dots ,\{1,2,3\left\}\right\}.$$

For $A=\{1,3\}\in F\left(X\right)$, the pushforward is

$$F\left(f\right)\left(A\right)=f\left[\right\{1,3\left\}\right]=\{a,a\}=\left\{a\right\}.$$

Thus A is an F-set over X, and $F\left(f\right)\left(A\right)$ is the corresponding F-set over Y.

Hence $(\mathbf{FinSet},F)$ provides a concrete example of aFunctorial Set.

Definition 13

(Functorial Structure). [50] Let $\mathcal{C}$ be a category. AFunctorial Structureon $\mathcal{C}$ is simply a covariant functor

$$F:\mathcal{C}⟶\mathbf{Set}.$$

For each object $X\in \mathrm{Ob}\left(\mathcal{C}\right)$, an element

$$s\in F\left(X\right)$$

is called anF-structure on X. Every morphism $f:X\to Y$ in $\mathcal{C}$ induces apushforward

$$F\left(f\right):F\left(X\right)⟶F\left(Y\right),s⟼F\left(f\right)\left(s\right),$$

and the usual functoriality conditions $F\left({id}_X\right)={id}_{F\left(X\right)}$ and $F(g\circ f)=F\left(g\right)\circ F\left(f\right)$ hold.

Example 9

(Functorial Structure: the list functor on FinSet)  Let $\mathcal{C}=\mathsf{FinSet}$ and define the functor $F:\mathcal{C}\to \mathbf{Set}$ by

$$F\left(X\right):=X^{*}\left(\mathrm{finite}\mathrm{words}\mathrm{over}X\right),F\left(f\right):X^{*}\to Y^{*},F\left(f\right)(x_1\dots x_k):=f\left(x_1\right)\dots f\left(x_k\right).$$

Functoriality (concrete check).Take $X=\{a,b\}$, $Y=\{0,1\}$, $Z=\{u,v\}$, with $f:X\to Y$, $f\left(a\right)=1$, $f\left(b\right)=0$, and $g:Y\to Z$, $g\left(1\right)=u$, $g\left(0\right)=v$. For $w=[a,b,a]\in X^{*}$,

$$F\left(f\right)\left(w\right)=[1,0,1],F\left(g\right)\left(F\left(f\right)\left(w\right)\right)=[u,v,u].$$

Since $(g\circ f)\left(a\right)=u$, $(g\circ f)\left(b\right)=v$, we get

$$F(g\circ f)\left(w\right)=[u,v,u]=F\left(g\right)\left(F\left(f\right)\left(w\right)\right),F\left({\mathrm{id}}_X\right)\left(w\right)=w.$$

Thus F is a (covariant) functor: a concreteFunctorial Structure.

2. Main Results

In this section, we examine and discuss newly introduced structures and related developments.

2.1. Rough HyperStructure

A Rough HyperStructure is a hyperstructure $(H,\circ )$ endowed with an indiscernibility relation R, so operations are defined on equivalence classes ${\left[x\right]}_R$ approximating uncertainty.

Notation 1.

Let $H\ne \varnothing$, $n\ge 2$, and ${\mathcal{P}}^{*}\left(H\right):=\mathcal{P}\left(H\right)\setminus \{\varnothing \}$. An n-ary hyperoperation is $f:H^n\to {\mathcal{P}}^{*}\left(H\right)$. For $A_1,\dots ,A_n\subseteq H$, its set–extension is

$$f(A_1,\dots ,A_n):=\bigcup \{f(a_1,\dots ,a_n):a_i\in A_i\}.$$

Fix a reflexive relation $R\subseteq H\times H$ and write $R\left(x\right):=\{y\in H:xRy\}$. Define rough approximations for $X\subseteq H$ by

$${\ell }_R\left(X\right):=\{x:R\left(x\right)\subseteq X\},u_R\left(X\right):=\{x:R\left(x\right)\cap X\ne \varnothing \}.$$

Note ${\ell }_R\left(X\right)\subseteq X\subseteq u_R\left(X\right)$, and ${\ell }_R,u_R$ are monotone and idempotent.

Notation 2

(Rough pairs and lifting). Let

$${\mathsf{RP}}_R\left(H\right):=\{(L,U)\in \mathcal{P}{\left(H\right)}^2:L\subseteq U,{\ell }_R\left(L\right)=L,u_R\left(U\right)=U\}.$$

For $(L_i,U_i)\in {\mathsf{RP}}_R\left(H\right)$ define the lifted hyperoperation

$$\widehat{f}\left((L_1,U_1),\dots ,(L_n,U_n)\right):=\left({\ell }_R\left(f(L_1,\dots ,L_n)\right),u_R\left(f(U_1,\dots ,U_n)\right)\right).$$

Definition 14

(Rough HyperStructure). The structure

$$\mathbf{RH}(H,f;R):=\left({\mathsf{RP}}_R\left(H\right),\widehat{f}\right)$$

is called theRough HyperStructuredetermined by $(H,f)$ and the reflexive relation R.

Example 10

(Rough HyperStructure via parity on ${\mathbb{Z}}_4$). Let $H={\mathbb{Z}}_4$ with the hyperoperation ⊞ above. Equip H with the equivalence (indiscernibility) relation R “same parity,” i.e.

$${\left[0\right]}_R=\{0,2\},{\left[1\right]}_R=\{1,3\}.$$

For $X\subseteq H$ define the rough approximations

$${\ell }_R\left(X\right):=\{x\in H:{\left[x\right]}_R\subseteq X\},u_R\left(X\right):=\{x\in H:{\left[x\right]}_R\cap X\ne \varnothing \}.$$

Concrete approximations.${\ell }_R\left(\{0,2\}\right)=\{0,2\}$ and $u_R\left(\{0,2\}\right)=\{0,2\}$, while ${\ell }_R\left(\left\{0\right\}\right)=\varnothing$ and $u_R\left(\left\{0\right\}\right)=\{0,2\}$.

Following the standard lift, define the rough-pair carrier

$${\mathsf{RP}}_R\left(H\right):=\{(L,U)\subseteq H\times H:L\subseteq U,{\ell }_R\left(L\right)=L,u_R\left(U\right)=U\}.$$

Lift ⊞ by

$$\widehat{⊞}\left((L_1,U_1),(L_2,U_2)\right):=\left({\ell }_R\left(L_1⊞L_2\right),u_R\left(U_1⊞U_2\right)\right),$$

where $A⊞B:={\bigcup }_{a\in A,b\in B}(a⊞b)$.Concrete computation.With $(L_1,U_1)=(\{0,2\},\{0,2\})$ and $(L_2,U_2)=(\{0,2\},\{0,2\})$,

$$L_1⊞L_2=\{0,1,2,3\}=H\Rightarrow \widehat{⊞}\left((L_1,U_1),(L_2,U_2)\right)=({\ell }_R\left(H\right),u_R\left(H\right))=(H,H).$$

Thus $\left({\mathsf{RP}}_R\left(H\right),\widehat{⊞}\right)$ is a concrete Rough HyperStructure.

Proposition 1

(Closure (well-definedness)). $\widehat{f}$ maps ${\mathsf{RP}}_R{\left(H\right)}^n$ into ${\mathsf{RP}}_R\left(H\right)$; hence it is an n-ary hyperoperation on ${\mathsf{RP}}_R\left(H\right)$.

Proof. 

Monotonicity of f (set–extension) gives $f(L_1,\dots ,L_n)\subseteq f(U_1,\dots ,U_n)$. Applying monotone ${\ell }_R,u_R$ yields ${\ell }_R\left(f(L_1,\dots ,L_n)\right)\subseteq u_R\left(f(U_1,\dots ,U_n)\right)$. Idempotence of ${\ell }_R,u_R$ ensures ${\ell }_R(·)$– and $u_R(·)$–closure. □

Theorem 

(Generalization of hyperstructures). Let $(H,f)$ be an n-ary hypergroupoid and take R to be equality on H. Define the embedding $\eta :H\to {\mathsf{RP}}_{=}\left(H\right)$ by $\eta \left(x\right):=\left(\right\{x\},\{x\left\}\right)$. Then for all $x_1,\dots ,x_n\in H$,

$$\widehat{f}\left(\eta \left(x_1\right),\dots ,\eta \left(x_n\right)\right)=\left(f(x_1,\dots ,x_n),f(x_1,\dots ,x_n)\right).$$

Consequently, collapsing exact pairs $(S,S)\mapsto S$ recovers $(H,f)$; hence $\mathbf{RH}(H,f;R)$ strictly generalizes $(H,f)$.

Proof. 

For $R==$, ${\ell }_{=}\left(X\right)=X=u_{=}\left(X\right)$. Using the set–extension with singletons gives $f(\left\{x_1\right\},\dots ,\left\{x_n\right\})=f(x_1,\dots ,x_n)$, whence the identity above. □

3. Fuzzy SuperHyperStructure

A Fuzzy SuperHyperStructure assigns fuzzy membership values in $[0,1]$ to superhyperoperations on iterated powersets, modeling hierarchical interactions with graded uncertainty semantics.

Notation 3

(Iterated powersets and fuzzy powersets). Let $S\ne \varnothing$. Define the iterated powersets

$${\mathcal{P}}^0\left(S\right):=S,{\mathcal{P}}^{k+1}\left(S\right):=\mathcal{P}\left({\mathcal{P}}^k\left(S\right)\right)(k\ge 0).$$

For any set X, itsfuzzy powersetis

$$\mathsf{Fuzz}\left(X\right):={[0,1]}^X=\{\mu :X\to [0,1]\}.$$

For $\mu \in {[0,1]}^X$ and $\alpha \in (0,1]$, the$\alpha$-cutis

$$X_{\alpha }\left(\mu \right):=\{x\in X:\mu \left(x\right)\ge \alpha \}.$$

Definition 15

(Typed fuzzy superhyperoperation). Fix integers $m,n\ge 0$ and an arity $s\in \mathbb{N}$. Afuzzy superhyperoperation of type $(m,n)$ and arity son S is a map

$${\tilde{\odot }}^{(m,n)}:{\left({\mathcal{P}}^m\left(S\right)\right)}^s⟶\mathsf{Fuzz}\left({\mathcal{P}}^n\left(S\right)\right).$$

For inputs $\mathbf{A}=(A_1,\dots ,A_s)\in {\left({\mathcal{P}}^m\left(S\right)\right)}^s$ we write ${\mu }_{\mathbf{A}}:={\tilde{\odot }}^{(m,n)}\left(\mathbf{A}\right)\in {[0,1]}^{{\mathcal{P}}^n\left(S\right)}.$

Definition 16

(Fuzzy SuperHyperStructure (FSuHS)). Let ${\left\{(m_j,n_j,s_j)\right\}}_{j\in J}$ be a finite signature of types. AFuzzy SuperHyperStructureon S is a family

$$\mathbf{FSuH}\left(S\right):=\left\{{\tilde{\odot }}_j^{(m_j,n_j)}:{\left({\mathcal{P}}^{m_j}\left(S\right)\right)}^{s_j}\to \mathsf{Fuzz}\left({\mathcal{P}}^{n_j}\left(S\right)\right)|j\in J\right\}.$$

Notation 4

($\alpha$-cut collapse and support). Given ${\tilde{\odot }}^{(m,n)}$ as above and $\alpha \in (0,1]$, define the$\alpha$-cut crispization

$${\odot }_{\alpha }^{(m,n)}:{\left({\mathcal{P}}^m\left(S\right)\right)}^s⟶\mathcal{P}\left({\mathcal{P}}^n\left(S\right)\right),{\odot }_{\alpha }^{(m,n)}\left(\mathbf{A}\right):=\{Y\in {\mathcal{P}}^n\left(S\right):{\mu }_{\mathbf{A}}\left(Y\right)\ge \alpha \}.$$

Itssupportis $supp\left({\mu }_{\mathbf{A}}\right):={\odot }_{0^{+}}^{(m,n)}\left(\mathbf{A}\right):=\{Y:{\mu }_{\mathbf{A}}\left(Y\right)>0\}$.

Example 11

(Fuzzy SuperHyperStructure on $S=\{x,y\}$). Work at type $(1,1)$ (inputs and outputs on ${\mathcal{P}}^{*}\left(S\right)$). Define a fuzzy superhyperoperation ${\tilde{\odot }}^{(1,1)}:{\left({\mathcal{P}}^{*}\left(S\right)\right)}^2\to {[0,1]}^{{\mathcal{P}}^{*}\left(S\right)}$ by

$${\mu }_{A,B}\left(Y\right):={\displaystyle \frac{|Y\cap (A\cup B\left)\right|}{|A\cup B|}}(A,B,Y\in {\mathcal{P}}^{*}\left(S\right)).$$

$\alpha$-cuts.For $\alpha =\frac{1}{2}$, if $A=\left\{x\right\}$ and $B=\left\{y\right\}$ then $A\cup B=\{x,y\}$ and

$${\left({\tilde{\odot }}^{(1,1)}(A,B)\right)}_{0.5}=\{\left\{x\right\},\left\{y\right\},\{x,y\}\}.$$

Thus $\left\{{\tilde{\odot }}^{(1,1)}\right\}$ is a concreteFuzzy SuperHyperStructure.

Lemma 1

($\alpha$-cut reconstruction). Let ${\tilde{\odot }}^{(m,n)}$ be a fuzzy superhyperoperation and fix $\mathbf{A}$. Then for every $Y\in {\mathcal{P}}^n\left(S\right)$,

$${\mu }_{\mathbf{A}}\left(Y\right)=sup\{\alpha \in (0,1]:Y\in {\odot }_{\alpha }^{(m,n)}\left(\mathbf{A}\right)\}.$$

Moreover, $\alpha \le \beta \Rightarrow {\odot }_{\beta }^{(m,n)}\left(\mathbf{A}\right)\subseteq {\odot }_{\alpha }^{(m,n)}\left(\mathbf{A}\right)$ (nested cuts).

Proof. 

By definition, $Y\in {\odot }_{\alpha }^{(m,n)}\left(\mathbf{A}\right)$ iff ${\mu }_{\mathbf{A}}\left(Y\right)\ge \alpha$. Hence $sup\{\alpha :Y\in {\odot }_{\alpha }^{(m,n)}\left(\mathbf{A}\right)\}={\mu }_{\mathbf{A}}\left(Y\right)$. If $\alpha \le \beta$ and ${\mu }_{\mathbf{A}}\left(Y\right)\ge \beta$ then ${\mu }_{\mathbf{A}}\left(Y\right)\ge \alpha$, proving the nesting. □

Definition 17

(Underlying crisp superoperations at level $\alpha$). For a FSuHS $\mathbf{FSuH}\left(S\right)$ and fixed $\alpha \in (0,1]$, the$\alpha$-cut superhyperstructureis

$${\mathbf{SH}}_{\alpha }\left(S\right):=\left\{{\odot }_{j,\alpha }^{(m_j,n_j)}:{\left({\mathcal{P}}^{m_j}\left(S\right)\right)}^{s_j}\to \mathcal{P}\left({\mathcal{P}}^{n_j}\left(S\right)\right)|j\in J\right\},$$

with ${\odot }_{j,\alpha }^{(m_j,n_j)}$ defined from ${\tilde{\odot }}_j^{(m_j,n_j)}$ as above.

Theorem 

(FSuHS generalizes SuperHyperStructure). Let $\mathbf{SH}\left(S\right)$ be a (crisp) superhyperstructure consisting of maps

$${\odot }_j^{(m_j,n_j)}:{\left({\mathcal{P}}^{m_j}\left(S\right)\right)}^{s_j}⟶{\mathcal{P}}^{n_j}\left(S\right).$$

Define an embedding $\mathcal{J}$ that sends each ${\odot }_j^{(m_j,n_j)}$ to the fuzzy superhyperoperation

$${\tilde{\odot }}_j^{(m_j,n_j)}\left(\mathbf{A}\right):={\mathbf{1}}_{{\odot }_j^{(m_j,n_j)}\left(\mathbf{A}\right)}\in {[0,1]}^{{\mathcal{P}}^{n_j}\left(S\right)},$$

i.e. ${\mu }_{\mathbf{A}}\left(Y\right)=1$ if $Y\in {\odot }_j^{(m_j,n_j)}\left(\mathbf{A}\right)$ and 0 otherwise. Then, for every $\alpha \in (0,1]$ and every input $\mathbf{A}$,

$${\odot }_{j,\alpha }^{(m_j,n_j)}\left(\mathbf{A}\right)={\odot }_j^{(m_j,n_j)}\left(\mathbf{A}\right),$$

so the α-cut of $\mathcal{J}\left(\mathbf{SH}\right(S\left)\right)$ recovers $\mathbf{SH}\left(S\right)$ exactly. In particular, $\mathcal{J}$ is injective on superoperations.

Proof. 

By construction, ${\mu }_{\mathbf{A}}\left(Y\right)\in \{0,1\}$ and $Y\in {\odot }_{j,\alpha }^{(m_j,n_j)}\left(\mathbf{A}\right)$ iff ${\mu }_{\mathbf{A}}\left(Y\right)\ge \alpha$. Since $\alpha \in (0,1]$, this is equivalent to ${\mu }_{\mathbf{A}}\left(Y\right)=1$, i.e. $Y\in {\odot }_j^{(m_j,n_j)}\left(\mathbf{A}\right)$. Thus ${\odot }_{j,\alpha }^{(m_j,n_j)}={\odot }_j^{(m_j,n_j)}$ for all $\alpha$. If two crisp operations differed on some input, their characteristic functions would differ at that input, proving injectivity. □

Definition 18

((Fuzzy HyperStructure (baseline)). AFuzzy HyperStructureon S with arity s is a map

$$★:S^s⟶{[0,1]}^S,$$

assigning to each $\mathbf{x}\in S^s$ a fuzzy subset $★\left(\mathbf{x}\right)$ of S.

Theorem 

(FSuHS generalizes Fuzzy HyperStructure). Let $(S,★)$ be a Fuzzy HyperStructure of arity s. Regard it as a special case of FSuHS by choosing $m=0$ and $n=0$ (so ${\mathcal{P}}^0\left(S\right)=S$) and defining

$${\tilde{\odot }}^{(0,0)}:S^s⟶{[0,1]}^S,{\tilde{\odot }}^{(0,0)}:=★.$$

Then $(S,★)$ and $\left\{{\tilde{\odot }}^{(0,0)}\right\}$ are the same data. Conversely, any FSuHS operation of type $(0,0)$ is precisely a fuzzy hyperoperation on S.

Proof. 

If $m=n=0$, the domain is $S^s$ and the codomain is $\mathsf{Fuzz}\left({\mathcal{P}}^0\left(S\right)\right)={[0,1]}^S$, which matches exactly the definition of ★. Thus the identifications are tautological in both directions. □

Proposition 2

(Crisp supports and thresholded superstructures). For any FSuHS and any $\alpha \in (0,1]$, the family ${\mathbf{SH}}_{\alpha }\left(S\right)$ is a (crisp) typed superhyperstructure. Moreover, for each input $\mathbf{A}$ and output Y,

$${\mu }_{\mathbf{A}}\left(Y\right)=sup\{\alpha \in (0,1]:Y\in {\odot }_{\alpha }^{(m,n)}\left(\mathbf{A}\right)\},$$

so the fuzzy data are uniquely determined by the tower of crisp α-cuts.

Proof. 

Each ${\odot }_{\alpha }^{(m,n)}$ maps inputs to subsets of the appropriate codomains by definition, thus forming crisp (super)operations of the same types; composition/typing constraints are inherited verbatim. The reconstruction formula is the $\alpha$-cut reconstruction lemma applied pointwise in Y. □

3.1. Rough SuperHyperStructure

A Rough SuperHyperStructure combines superhyperoperations with rough approximations, employing lower and upper operators on iterated powersets to handle uncertainty in multi-level systems.

Notation 5

(Levelwise rough approximation). For each level $m\in {\mathbb{N}}_0$ fix a reflexive relation $R^{\langle m\rangle }$ on $H^{\langle m\rangle }$ and write

$$R^{\langle m\rangle }\left(x\right):=\{y\in H^{\langle m\rangle }:x{R}^{\langle m\rangle }y\}.$$

For $X\subseteq H^{\langle m\rangle }$ define the lower/upper approximations

$${\ell }_m\left(X\right):=\{x:R^{\langle m\rangle }\left(x\right)\subseteq X\},u_m\left(X\right):=\{x:R^{\langle m\rangle }\left(x\right)\cap X\ne \varnothing \}.$$

Then ${\ell }_m,u_m$ are monotone, idempotent, and satisfy ${\ell }_m\left(X\right)\subseteq X\subseteq u_m\left(X\right)$.

Notation 6

(Rough pairs per level and lifting of operations). Let

$${\mathsf{RP}}_m:=\{(L,U)\in \mathcal{P}{\left(H^{\langle m\rangle }\right)}^2:L\subseteq U,{\ell }_m\left(L\right)=L,u_m\left(U\right)=U\}.$$

Given $F_j$ of type $\left({\ell }_{j,1},\dots ,{\ell }_{j,n_j}\right)\to r_j$, define the lifted operation

$${\widehat{F}}_j:\prod _{k=1}^{n_j}{\mathsf{RP}}_{{\ell }_{j,k}}⟶{\mathsf{RP}}_{r_j},$$

$${\widehat{F}}_j\left((L_1,U_1),\dots ,(L_{n_j},U_{n_j})\right):=\left({\ell }_{r_j}\left(F_j(L_1,\dots ,L_{n_j})\right),u_{r_j}\left(F_j(U_1,\dots ,U_{n_j})\right)\right).$$

Definition 19

(Rough SuperHyperStructure). The structure

$$\mathbf{RSH}(H,\mathcal{F};\mathcal{R}):=\left(\prod _{m\ge 0}{\mathsf{RP}}_m,{\left\{{\widehat{F}}_j\right\}}_{j\in J}\right)$$

with $\mathcal{R}={\left\{R^{\langle m\rangle }\right\}}_{m\ge 0}$ is called theRough SuperHyperStructuredetermined by the superoperations $\mathcal{F}$ and the levelwise rough data $\mathcal{R}$.

Example 12

(Rough SuperHyperStructure on ${\mathbb{Z}}_4$ at level 1). Let $H={\mathbb{Z}}_4=\{0,1,2,3\}$ and $R^{\langle 0\rangle }$ be “same parity”: $\left[0\right]=\{0,2\}$, $\left[1\right]=\{1,3\}$. Lift to level 1 by $R^{\langle 1\rangle }\left(A\right):=\{B\subseteq H:\exists x\in A,\exists y\in B,x{R}^{\langle 0\rangle }y\}$. Define lower/upper approximations on level 1:

$${\ell }_1\left(\mathcal{X}\right)=\{A:R^{\langle 1\rangle }\left(A\right)\subseteq \mathcal{X}\},u_1\left(\mathcal{X}\right)=\{A:R^{\langle 1\rangle }\left(A\right)\cap \mathcal{X}\ne \varnothing \}.$$

Let the (crisp) superoperation at level 1 be $F(A,B):=A▵B$ (symmetric difference). Form rough pairs ${\mathsf{RP}}_1=\{(L,U):L\subseteq U,{\ell }_1\left(L\right)=L,u_1\left(U\right)=U\}$ and lift

$$\widehat{F}\left((L_1,U_1),(L_2,U_2)\right):=\left({\ell }_1\left(F(L_1,L_2)\right),u_1\left(F(U_1,U_2)\right)\right).$$

Concrete check.Take $L_1=U_1=\{\left\{0\right\},\left\{2\right\}\}$ and $L_2=U_2=\{\left\{1\right\},\left\{3\right\}\}$. Then $F(L_1,L_2)=\{\{0,1\},\{0,3\},\{1,2\},\{2,3\}\}$, which is $R^{\langle 1\rangle }$-saturated; hence $\widehat{F}\left((L_1,U_1),(L_2,U_2)\right)=(F(L_1,L_2),F(L_1,L_2)).$ Thus $\left({\mathsf{RP}}_1,\widehat{F}\right)$ yields aRough SuperHyperStructure.

Proposition 3

(Closure/well-definedness). For each $j\in J$, the output of ${\widehat{F}}_j$ lies in ${\mathsf{RP}}_{r_j}$; hence every ${\widehat{F}}_j$ is a well-defined superhyperoperation on rough pairs.

Proof. 

Let $(L_k,U_k)\in {\mathsf{RP}}_{{\ell }_{j,k}}$. By monotonicity of the set–extension, $F_j(L_1,\dots ,L_{n_j})\subseteq F_j(U_1,\dots ,U_{n_j})$. Applying monotone ${\ell }_{r_j},u_{r_j}$ gives

$${\ell }_{r_j}\left(F_j(L_1,\dots ,L_{n_j})\right)\subseteq u_{r_j}\left(F_j(U_1,\dots ,U_{n_j})\right).$$

Idempotence yields ${\ell }_{r_j}(·)$– and $u_{r_j}(·)$–closure, i.e., the pair belongs to ${\mathsf{RP}}_{r_j}$. □

Theorem 

(Rough SuperHyperStructure generalizes SuperHyperStructure). Assume $R^{\langle m\rangle }$ is equality on $H^{\langle m\rangle }$ for all m. Define the levelwise embedding

$${\eta }_m:H^{\langle m\rangle }⟶{\mathsf{RP}}_m,{\eta }_m\left(x\right):=(\left\{x\right\},\left\{x\right\}).$$

Then for every $j\in J$ and $x_k\in H^{\langle {\ell }_{j,k}\rangle }$,

$${\widehat{F}}_j\left({\eta }_{{\ell }_{j,1}}\left(x_1\right),\dots ,{\eta }_{{\ell }_{j,n_j}}\left(x_{n_j}\right)\right)=\left(F_j(x_1,\dots ,x_{n_j}),F_j(x_1,\dots ,x_{n_j})\right).$$

Collapsing exact pairs $(S,S)\mapsto S$ recovers $\mathbf{SH}(H,\mathcal{F})$ from $\mathbf{RSH}(H,\mathcal{F};\mathcal{R})$.

Proof. 

For equality, ${\ell }_m=u_m=\mathrm{id}$. Using the set–extension on singletons, $F_j(\left\{x_1\right\},\dots ,\left\{x_{n_j}\right\})=F_j(x_1,\dots ,x_{n_j})$. Hence the displayed identity holds, and the collapse map gives back the original outputs of $F_j$. □

Theorem 

(Rough SuperHyperStructure generalizes Rough HyperStructure). Let $(H,f)$ be an n-ary hypergroupoid (one-level hyperstructure). Regard it as a superstructure with a single operation F of type $(0,\dots ,0)\to 0$. Fix a reflexive relation $R^{\langle 0\rangle }=:R$ on H and ignore higher levels (or set $R^{\langle m\rangle }$ arbitrary). Then the level-0 component of $\mathbf{RSH}(H,\{F\};\mathcal{R})$ is canonically isomorphic to the Rough HyperStructure

$$\mathbf{RH}(H,f;R)=\left({\mathsf{RP}}_R\left(H\right),\widehat{f}\right),$$

with the identification ${\mathsf{RP}}_0\equiv {\mathsf{RP}}_R\left(H\right)$ and

$$\widehat{F}\left((L_1,U_1),\dots ,(L_n,U_n)\right)=\left({\ell }_0\left(f(L_1,\dots ,L_n)\right),u_0\left(f(U_1,\dots ,U_n)\right)\right)=\widehat{f}(\dots ).$$

Proof. 

By construction, $H^{\langle 0\rangle }=H$, ${\ell }_0={\ell }_R$, $u_0=u_R$, and the set–extension of F coincides with that of f. Therefore the lifted operation at level 0 matches $\widehat{f}$ exactly, yielding the stated identification. □

3.2. Soft SuperHyperStructure

A Soft SuperHyperStructure equips a superhyperstructure with parameterized families of sub-superhyperstructures, where each parameter selects context-dependent subsets closed under superhyperoperations.

Notation 7

(Soft sets over the levelled universe). Let A be a nonempty parameter set. Asoft set over the levelled universeis a map

$$S:A⟶\prod _{m\ge 0}\mathcal{P}\left(H^{\langle m\rangle }\right),a⟼{\left(S_a^{\langle m\rangle }\right)}_{m\ge 0},$$

withsupport$\mathrm{Supp}\left(S\right):=\{a\in A:\exists m,S_a^{\langle m\rangle }\ne \varnothing \}$. We assume $\mathrm{Supp}\left(S\right)\ne \varnothing$ and, for all a, that every $S_a^{\langle m\rangle }$ is either empty or nonempty as specified below.

Definition 20

(Soft SuperHyperStructure). Let $\mathbf{SH}(H,\mathcal{F})$ be as above. A soft set S over the levelled universe is aSoft SuperHyperStructure over $\mathbf{SH}(H,\mathcal{F})$if for every $a\in \mathrm{Supp}\left(S\right)$ and every operation $F_j\in \mathcal{F}$ one has the levelwise closure condition

$$F_j\left(S_a^{\langle {\ell }_{j,1}\rangle },\dots ,S_a^{\langle {\ell }_{j,n_j}\rangle }\right)\subseteq S_a^{\langle r_j\rangle }.$$

(2)

Equivalently, for each a, the family ${\left(S_a^{\langle m\rangle }\right)}_{m\ge 0}$ defines asub-superhyperstructureof $\mathbf{SH}(H,\mathcal{F})$ under the restrictions of all $F_j$. We write $\mathbf{SSH}(H,\mathcal{F};S)$ for such a soft structure.

Remark 2.

Only levels appearing in the types $\{{\ell }_{j,k},r_j\}$ are relevant; the definition is independent of arbitrary values $S_a^{\langle m\rangle }$ at unused levels.

Example 13

(Soft SuperHyperStructure on ${\mathbb{Z}}_3$). Let $H={\mathbb{Z}}_3=\{0,1,2\}$ and work at level 1 with the superoperation

$$\odot (A,B):=A\cup B(A,B\in {\mathcal{P}}^{*}\left(H\right)).$$

Let $A_{\mathrm{par}}=\{\mathsf{low},\mathsf{high}\}$ and define the soft selection

$$S_{\mathsf{low}}^{\langle 1\rangle }=\left\{\left\{0\right\},\left\{1\right\},\{0,1\}\right\},S_{\mathsf{high}}^{\langle 1\rangle }=\left\{\left\{1\right\},\left\{2\right\},\{1,2\}\right\}.$$

Closure.For each parameter $p\in A_{\mathrm{par}}$ and $A,B\in S_p^{\langle 1\rangle }$, $\odot (A,B)=A\cup B\in S_p^{\langle 1\rangle }$ by construction; hence every $S_p^{\langle 1\rangle }$ is a sub-superhyperstructure. Therefore $(A_{\mathrm{par}},S^{\langle 1\rangle })$ is aSoft SuperHyperStructureover $({\mathcal{P}}^{*}\left(H\right),\odot )$.

Proposition 4

(Basic closure). If S satisfies (2), then for each $a\in \mathrm{Supp}\left(S\right)$ and all $j\in J$,

$$\forall X_k\subseteq S_a^{\langle {\ell }_{j,k}\rangle }{F}_j(X_1,\dots ,X_{n_j})\subseteq S_a^{\langle r_j\rangle }.$$

Proof. 

By monotonicity of the set–extension (Notation), $X_k\subseteq S_a^{\langle {\ell }_{j,k}\rangle }$ implies $F_j(X_1,\dots ,X_{n_j})\subseteq F_j(S_a^{\langle {\ell }_{j,1}\rangle },\dots ,S_a^{\langle {\ell }_{j,n_j}\rangle })\subseteq S_a^{\langle r_j\rangle }$ by (2). □

Theorem 

(Soft SuperHyperStructure generalizes Soft HyperStructure). Suppose all operations act at level 0, i.e. ${\ell }_{j,k}=r_j=0$ for all $j,k$. Let $f_j:=F_j{\upharpoonright }_{H^{\langle 0\rangle }}:H^{n_j}\to {\mathcal{P}}^{*}\left(H\right)$ and $(H,{\left\{f_j\right\}}_{j\in J})$ be the underlying hyperstructure. Then S is a Soft SuperHyperStructure over $\mathbf{SH}(H,\mathcal{F})$ iff the projected soft set $S^{\left(0\right)}:A\to \mathcal{P}\left(H\right)$ given by $S^{\left(0\right)}\left(a\right):=S_a^{\langle 0\rangle }$ is aSoft HyperStructureover $(H,\left\{f_j\right\})$, i.e.

$$\forall a\in \mathrm{Supp}\left(S\right),\forall j\in J:f_j\left(S^{\left(0\right)}\left(a\right),\dots ,S^{\left(0\right)}\left(a\right)\right)\subseteq S^{\left(0\right)}\left(a\right).$$

Proof 

(⇒) With ${\ell }_{j,k}=r_j=0$, (2) gives $F_j\left(S_a^{\langle 0\rangle },\dots ,S_a^{\langle 0\rangle }\right)\subseteq S_a^{\langle 0\rangle }$. Since $F_j$ on level 0 equals $f_j$ (by definition), this is the soft hyperstructure closure for $S^{\left(0\right)}$.

(⇐) Conversely, assuming $S^{\left(0\right)}$ satisfies the soft hyperstructure closure for all $f_j$, we have $f_j\left(S_a^{\langle 0\rangle },\dots ,S_a^{\langle 0\rangle }\right)\subseteq S_a^{\langle 0\rangle }$, which is precisely (2) in the present (level-0) typing. Thus S is a Soft SuperHyperStructure. □

Theorem 

(SuperHyperStructures embed into Soft SuperHyperStructures). Let $\mathbf{SH}(H,\mathcal{F})$ be a SuperHyperStructure. Fix a singleton parameter set $A=\{•\}$ and define

$$S{(•)}^{\langle m\rangle }:=H^{\langle m\rangle }\left(\mathrm{for}\mathrm{all}m\right).$$

Then S is a Soft SuperHyperStructure over $\mathbf{SH}(H,\mathcal{F})$. Moreover, thecollapsefunctor

$$\mathfrak{C}:\mathbf{SSH}(H,\mathcal{F};S)⟶\mathbf{SH}(H,\mathcal{F}),\mathfrak{C}``\mathrm{forgets}A\mathrm{and}\mathrm{keeps}\mathrm{the}\mathrm{underlying}\mathrm{operations}’’,$$

recovers $\mathbf{SH}(H,\mathcal{F})$ (i.e. is identity-on-operations).

Proof. 

For every $j\in J$ and all inputs $X_k\subseteq S{(•)}^{\langle {\ell }_{j,k}\rangle }=H^{\langle {\ell }_{j,k}\rangle }$, the output satisfies $F_j(X_1,\dots ,X_{n_j})\subseteq H^{\langle r_j\rangle }=S{(•)}^{\langle r_j\rangle }$, so (2) holds. The collapse functor simply discards the (trivial) soft parameterization and leaves the operations $\left\{F_j\right\}$ untouched, yielding the original $\mathbf{SH}(H,\mathcal{F})$. □

3.3. Functorial HyperStructure

A Functorial HyperStructure encodes hyperoperations as natural transformations between functors $U^{\times n}$ and $V\circ U$, preserving categorical structure and morphism compatibility.

Notation 8

(Underlying and value functors). Let $\mathcal{C}$ be a category. Fix a (covariant) functor

$$U:\mathcal{C}⟶\mathbf{Set}$$

that assigns to every object X its underlying set $U\left(X\right)$. Let $V:\mathbf{Set}\to \mathbf{Set}$ be another (covariant) endofunctor (thevalue functor). For $n\in \mathbb{N}$, write $U^{\times n}:\mathcal{C}\to \mathbf{Set}$ for the functor $X\mapsto U{\left(X\right)}^n$.

Definition 21

(Functorial hyperoperation and Functorial HyperStructure). AFunctorial n-ary hyperoperation over $(U,V)$is a natural transformation

$$\Phi :U^{\times n}⟹V\circ U,$$

i.e. for each $X\in \mathrm{Ob}\left(\mathcal{C}\right)$ a map ${\Phi }_X:U{\left(X\right)}^n\to V\left(U\left(X\right)\right)$ such that for every $f:X\to Y$ in $\mathcal{C}$ the naturality square

$$\begin{array}{ccc}U{\left(X\right)}^n& \stackrel{{\Phi }_X}{\to }& V\left(U\right(X\left)\right)\\ U{\left(f\right)}^{\times n}\downarrow \phantom{U{\left(f\right)}^{\times n}}& & \downarrow & & \left(U\right(f\left)\right)V\\ U{\left(Y\right)}^n& \stackrel{{\Phi }_Y}{\to }& V\left(U\right(Y\left)\right)\end{array}$$

commutes. AFunctorial HyperStructure (FHS)on $(\mathcal{C};U,V)$ is a family $\mathbf{F}={\{{\Phi }_j:U^{\times n_j}\Rightarrow V\circ U\}}_{j\in J}$ of such natural transformations (finite signature allowed).

Remark 3

(Reading the outputs). When $V={\mathcal{P}}^{*}$ (nonempty powerset), ${\Phi }_X\left(\overrightarrow{x}\right)\in {\mathcal{P}}^{*}\left(U\left(X\right)\right)$ is a genuinehyperoutput; when $V={\mathrm{Id}}_{\mathbf{Set}}$, ${\Phi }_X\left(\overrightarrow{x}\right)\in U\left(X\right)$ is a usual (single-valued) operation. Other choices of V encode fuzzy, rough, or soft outputs (see Theorems below).

Example 14

(Functorial HyperStructure on FinSet). Let $\mathcal{C}=\mathsf{FinSet}$, $U={\mathrm{Id}}_{\mathcal{C}}$, $V={\mathcal{P}}^{*}$. Define a natural transformation (functorial hyperoperation)

$$\Phi :U^{\times 2}\Rightarrow V\circ U,{\Phi }_X(x,y)=\{x,y\}\in {\mathcal{P}}^{*}\left(X\right).$$

Naturality.For $f:X\to Y$,

$$\left(VU\left(f\right)\right){\Phi }_X(x,y)=f\left[\{x,y\}\right]=\{f\left(x\right),f\left(y\right)\}={\Phi }_Y\left(U\left(f\right)\left(x\right),U\left(f\right)\left(y\right)\right).$$

Hence $(\mathcal{C};U,V,\Phi )$ is aFunctorial HyperStructure.

Theorem 

(FHS generalizes Functorial Structure). Given a functor $F:\mathcal{C}\to \mathbf{Set}$ (Definition 13), there is a canonical FHS whose “structures over X” are exactly elements of $F\left(X\right)$.

Proof. 

Take $U:={\mathrm{Id}}_{\mathcal{C}}$ and $V:=F$. A choice of $s\in F\left(X\right)$ is the same as a component of a nullary natural transformation $\sigma :\mathbf{1}\Rightarrow F$ (where $\mathbf{1}$ is the terminal functor), with ${\sigma }_X(*)=s$. Thus a Functorial HyperStructure with a single nullary operation $\sigma$ reproduces the notion of an F-structure on X. □

Theorem 

(FHS generalizes hyperstructures). Let $(H,\{f_j:H^{n_j}\to {\mathcal{P}}^{*}\left(H\right)\})$ be a hyperstructure. Set $\mathcal{C}:=\mathbf{1}$ (one-object category), choose $U(*)=H$, and $V:={\mathcal{P}}^{*}$. For each j, define ${\Phi }_j$ by the single component ${\Phi }_{j,*}:H^{n_j}\to {\mathcal{P}}^{*}\left(H\right),{\Phi }_{j,*}=f_j.$ Then ${\left\{{\Phi }_j\right\}}_{j\in J}$ is an FHS, and this assignment is an isomorphism of data.

Proof. 

With $\mathcal{C}=\mathbf{1}$ there are no nontrivial morphisms, so naturality is automatic. The componentwise identification ${\Phi }_{j,*}=f_j$ gives a bijection between signatures. □

Notation 9

(Rough-pair functor). For a set Y equipped with a fixed reflexive relation $R_Y\subseteq Y\times Y$, write $\mathsf{RP}\left(Y\right):=\{(L,U)\mid L\subseteq U,{\ell }_{R_Y}\left(L\right)=L,u_{R_Y}\left(U\right)=U\}$. For a map $g:Y\to Y^{\prime }$ define

$$g_{♯}:\mathsf{RP}\left(Y\right)⟶\mathsf{RP}\left(Y^{\prime }\right),(L,U)⟼\left({\ell }_{R_{Y^{\prime }}}\left(g\left[L\right]\right),u_{R_{Y^{\prime }}}\left(g\left[U\right]\right)\right).$$

This makes $V_{\mathrm{rough}}:\mathbf{Set}\to \mathbf{Set}$, $Y\mapsto \mathsf{RP}\left(Y\right)$, a functor (naturality follows from functoriality of direct image and idempotence/monotonicity of $\ell ,u$).

Theorem 

(FHS generalizes Rough HyperStructure). Let $(H,\left\{{\widehat{f}}_j\right\})$ be a rough hyperstructure on the rough pairs of H (e.g. as constructed from a base hyperstructure via lower/upper approximation). Set $\mathcal{C}:=\mathbf{1}$, $U(*)=H$, and $V:=V_{\mathrm{rough}}$. Defining ${\Phi }_{j,*}:H^{n_j}\to \mathsf{RP}\left(H\right)$ by ${\Phi }_{j,*}={\widehat{f}}_j$ yields an FHS that is equivalent to the given Rough HyperStructure.

Proof. 

Again, with $\mathcal{C}=\mathbf{1}$, naturality is vacuous. The component ${\Phi }_{j,*}$ is exactly the lifted rough operation ${\widehat{f}}_j$, so the structures coincide. □

Notation 10

(Fuzzy-set functor). Let $V_{\mathrm{fuz}}:\mathbf{Set}\to \mathbf{Set}$ be the functor $Y\mapsto {[0,1]}^Y$, and for $g:Y\to Y^{\prime }$ define the Zadeh pushforward $\left(g_{♯}\mu \right)\left(y^{\prime }\right):=sup\{\mu \left(y\right):g\left(y\right)=y^{\prime }\}$.

Theorem 

(FHS generalizes Fuzzy HyperStructure). Given a fuzzy hyperstructure $(H,\left\{{★}_j\right\})$ with ${★}_j:H^{n_j}\to {[0,1]}^H$, choose $\mathcal{C}:=\mathbf{1}$, $U(*)=H$, and $V:=V_{\mathrm{fuz}}$. Set ${\Phi }_{j,*}={★}_j$. Then $\left\{{\Phi }_j\right\}$ is an FHS equivalent to the given fuzzy hyperstructure.

Proof. 

Identical to Theorem 9, replacing ${\mathcal{P}}^{*}$ by ${[0,1]}^{(-)}$. □

Notation 11

((Soft-set functor (fixed parameter set $A\ne \varnothing$)). Define $V_{\mathrm{soft}}\left(Y\right):={\left({\mathcal{P}}^{*}\left(Y\right)\right)}^A$ and for $g:Y\to Y^{\prime }$ set $\left(g_{♯}F\right)\left(a\right):=g\left[F\left(a\right)\right]$ ($a\in A$).

Theorem 

(FHS captures Soft HyperStructure). Let $(H,\left\{f_j\right\})$ be a hyperstructure and $(F,A)$ a soft set with each $F\left(a\right)\subseteq H$ a subhyperstructure. Consider $\mathcal{C}:=\mathbf{1}$, $U(*)=H$, $V:=V_{\mathrm{soft}}$, and the FHS consisting of:

• the original hyperoperations encoded as ${\Phi }_j^{\mathrm{hyp}}:H^{n_j}\to {\mathcal{P}}^{*}\left(H\right)$ via $V={\mathcal{P}}^{*}$ (as in Theorem 9); and
• for each $a\in A$, anullarynatural transformation ${\sigma }_a:\mathbf{1}\Rightarrow V_{\mathrm{soft}}\circ U$ with ${\sigma }_{a,*}(*)=S_a$, where $S_a\in {\left({\mathcal{P}}^{*}\left(H\right)\right)}^A$ is the “Dirac” soft set selecting $F\left(a\right)$ in the a-th coordinate and H elsewhere.

Then the soft-closure condition “$F\left(a\right)$ is a subhyperstructure for all a” is equivalent to the family of FHS-axioms

$${\Phi }_{j,*}^{\mathrm{hyp}}\left(F\left(a\right),\dots ,F\left(a\right)\right)\subseteq F\left(a\right)\mathrm{for}\mathrm{all}j\in J,a\in A.$$

Proof 

(⇒) If $F\left(a\right)$ is a subhyperstructure, closure under each $f_j$ gives the displayed inclusion, which is exactly the compatibility between ${\Phi }_j^{\mathrm{hyp}}$ and the constant soft selections ${\sigma }_a$. (⇐) Conversely, the inclusions state precisely that each $F\left(a\right)$ is closed under all $f_j$, i.e. each $F\left(a\right)$ is a subhyperstructure. □

Remark 4

(Unifying view). Definitions 21 and Theorems 8–12 show that by an appropriate choice of the value functor V and of nullary/finite-arity natural transformations, Functorial HyperStructuressubsume:

• ordinary functorial structures ($V=F$, nullary σ);
• crisp hyperstructures ($V={\mathcal{P}}^{*}$);
• rough hyperstructures ($V=\mathsf{RP}(-)$);
• fuzzy hyperstructures ($V={[0,1]}^{(-)}$);
• soft hyperstructures ($V={\left({\mathcal{P}}^{*}(-)\right)}^A$ plus nullary selectors).

Functoriality packages the pushforward of structure along morphisms of $\mathcal{C}$ as the naturality of the operations.

3.4. Functorial SuperHyperStructure

A Functorial SuperHyperStructure is a family of multi-level hyperoperations encoded as natural transformations between functors, unifying hierarchical and categorical structure.

Notation 12

(Level functors and value functors). Let $\mathcal{C}$ be a category. Alevel systemon $\mathcal{C}$ is a family of covariant functors

$$\mathcal{U}:=\left\{U^{\langle m\rangle }:\mathcal{C}\to \mathbf{Set}|m\in {\mathbb{N}}_0\right\},$$

and avalue systemis a family of endofunctors on $\mathbf{Set}$

$$\mathcal{V}:=\left\{V^{\langle m\rangle }:\mathbf{Set}\to \mathbf{Set}|m\in {\mathbb{N}}_0\right\}.$$

For a multiindex $\tau =({\ell }_1,\dots ,{\ell }_n\mid r)$ (input levels ${\ell }_k$ and output level r), write

$$U^{\tau }:=\prod _{k=1}^n{U}^{\langle {\ell }_k\rangle }\mathrm{and}{W}^{\tau }:=V^{\langle r\rangle }\circ U^{\langle r\rangle }.$$

Definition 22

(Typed functorial superhyperoperation). Fix a type $\tau =({\ell }_1,\dots ,{\ell }_n\mid r)$. Atyped functorial superhyperoperation of type $\tau$is a natural transformation

$$\Phi :U^{\tau }⟹W^{\tau },$$

i.e. for every $X\in \mathrm{Ob}\left(\mathcal{C}\right)$ a function ${\Phi }_X:{\prod }_{k=1}^n{U}^{\langle {\ell }_k\rangle }\left(X\right)\to V^{\langle r\rangle }\left(U^{\langle r\rangle }\left(X\right)\right)$ such that for every $f:X\to Y$ in $\mathcal{C}$, the square

$$\begin{array}{ccc}\prod _k{U}^{\langle {\ell }_k\rangle }\left(X\right)& \stackrel{{\Phi }_X}{\to }& V^{\langle r\rangle }\left(U^{\langle r\rangle }\left(X\right)\right)\\ {\prod }_k{U}^{\langle {\ell }_k\rangle }\left(f\right)\downarrow \phantom{{\prod }_k{U}^{\langle {\ell }_k\rangle }\left(f\right)}& & \downarrow & & \langle r\rangle \left(U^{\langle r\rangle }\left(f\right)\right)V\\ \prod _k{U}^{\langle {\ell }_k\rangle }\left(Y\right)& \stackrel{{\Phi }_Y}{\to }& V^{\langle r\rangle }\left(U^{\langle r\rangle }\left(Y\right)\right)\end{array}$$

commutes.

Definition 23

(Functorial SuperHyperStructure (FSS)). AFunctorial SuperHyperStructureon $(\mathcal{C};\mathcal{U},\mathcal{V})$ is a finite signature $\Sigma ={\left\{{\tau }_j\right\}}_{j\in J}$ of types together with a family

$$\mathbf{FSH}:=\{{\Phi }_j:U^{{\tau }_j}\Rightarrow W^{{\tau }_j}\mid j\in J\}$$

of typed functorial superhyperoperations. For each $X\in \mathrm{Ob}\left(\mathcal{C}\right)$, thefiber at Xconsists of the components $\{{\left({\Phi }_j\right)}_X:j\in J\}$.

Remark 5

(Reading classical cases from $\mathcal{V}$). Typical choices of value functors $V^{\langle m\rangle }$ include:

• Crisp hyper:$V^{\langle m\rangle }={\mathcal{P}}^{*}$ (nonempty powerset).
• Fuzzy:$V^{\langle m\rangle }\left(Y\right)={[0,1]}^Y$ with pushforward $\left(g_{♯}\mu \right)\left(y^{\prime }\right)=sup\{\mu \left(y\right):g\left(y\right)=y^{\prime }\}$.
• Rough:$V^{\langle m\rangle }\left(Y\right)={\mathsf{RP}}_{R_Y^{\langle m\rangle }}\left(Y\right)$ (rough pairs for a fixed reflexive $R_Y^{\langle m\rangle }$) with $g_{♯}(L,U):=\left({\ell }_{R_{Y^{\prime }}^{\langle m\rangle }}\left(g\left[L\right]\right),u_{R_{Y^{\prime }}^{\langle m\rangle }}\left(g\left[U\right]\right)\right)$.
• Soft (fixed $A\ne \varnothing$):$V^{\langle m\rangle }\left(Y\right)={\left({\mathcal{P}}^{*}\left(Y\right)\right)}^A$ with $\left(g_{♯}F\right)\left(a\right):=g\left[F\left(a\right)\right]$.

All are functorial (composition and identities are preserved).

Example 15

(Functorial SuperHyperStructure on FinSet). Let $\mathcal{C}=\mathsf{FinSet}$, $U^{\langle 0\rangle }={\mathrm{Id}}_{\mathcal{C}}$ and $U^{\langle 1\rangle }={\mathcal{P}}^{*}$ (nonempty powerset functor). Set $V^{\langle 1\rangle }={\mathcal{P}}^{*}$ and consider the type $\tau =(1,1\mid 1)$. Define a natural transformation (typed functorial superhyperoperation)

$$\Phi :U^{\langle 1\rangle }\times U^{\langle 1\rangle }⟹V^{\langle 1\rangle }\circ U^{\langle 1\rangle },{\Phi }_X(A,B):=\{A\cup B\}.$$

Naturality.For $f:X\to Y$,

$$\left(V^{\langle 1\rangle }{U}^{\langle 1\rangle }\left(f\right)\right){\Phi }_X(A,B)=\left\{f[A\cup B]\right\}=\{f\left[A\right]\cup f\left[B\right]\}={\Phi }_Y\left(U^{\langle 1\rangle }\left(f\right)\left(A\right),U^{\langle 1\rangle }\left(f\right)\left(B\right)\right).$$

Thus $(\mathcal{C};\left\{U^{\langle m\rangle }\right\},\left\{V^{\langle m\rangle }\right\},\left\{\Phi \right\})$ is aFunctorial SuperHyperStructure.

Theorem 

(Reduction to Functorial Structure). Let $F:\mathcal{C}\to \mathbf{Set}$ be a covariant functor (Definition 13). Choose one level and no operations:

$$\mathcal{U}=\{U^{\langle 0\rangle }:={\mathrm{Id}}_{\mathcal{C}}\},\mathcal{V}=\{V^{\langle 0\rangle }:=F\},\Sigma =⌀.$$

Then the FSS data $(\mathcal{C};\mathcal{U},\mathcal{V},\Sigma )$ carries exactly the same information as the functor F; in particular, the fiber at X is $F\left(X\right)$.

Proof. 

By definition $W^{\langle 0\rangle }=V^{\langle 0\rangle }\circ U^{\langle 0\rangle }=F$. With $\Sigma =⌀$ (no operations), the only remaining data is the functor F. □

Theorem 

(Reduction to SuperHyperStructure). Let $\mathbf{SH}(H,\mathcal{F})$ be a (typed) SuperHyperStructure on a single set H, with levels $H^{\langle 0\rangle }=H$ and $H^{\langle m+1\rangle }={\mathcal{P}}^{*}\left(H^{\langle m\rangle }\right)$, and operations $F_j:{\prod }_k{H}^{\langle {\ell }_{j,k}\rangle }\to {\mathcal{P}}^{*}\left(H^{\langle r_j\rangle }\right)$. Take $\mathcal{C}:=\mathbf{1}$ (one-object category), define

$$U^{\langle m\rangle }(*)=H^{\langle m\rangle },V^{\langle m\rangle }:={\mathcal{P}}^{*},$$

and let ${\Phi }_j$ be the unique natural transformations with components ${\left({\Phi }_j\right)}_{*}=F_j$. Then $(\mathcal{C};\mathcal{U},\mathcal{V},\left\{{\Phi }_j\right\})$ is an FSS whose fiber at * is exactly $\mathbf{SH}(H,\mathcal{F})$.

Proof. 

Naturality is automatic in $\mathbf{1}$. The components reproduce the given $F_j$. □

Theorem 

(Reduction to Rough SuperHyperStructure). Let $\mathbf{RSH}(H,\mathcal{F};\mathcal{R})$ be a Rough SuperHyperStructure with levelwise reflexive relations $\mathcal{R}=\left\{R^{\langle m\rangle }\right\}$ and lifted operations ${\widehat{F}}_j$ on rough pairs. For $\mathcal{C}:=\mathbf{1}$ set

$$U^{\langle m\rangle }(*)=H^{\langle m\rangle },V^{\langle m\rangle }\left(Y\right)={\mathsf{RP}}_{R_Y^{\langle m\rangle }}\left(Y\right),$$

and define ${\Phi }_j$ by ${\left({\Phi }_j\right)}_{*}={\widehat{F}}_j$. Then the resulting FSS is (componentwise) identical to $\mathbf{RSH}(H,\mathcal{F};\mathcal{R})$.

Proof. 

Same as Theorem 14, using the rough-pair value functors. □

Theorem 

(Reduction to Fuzzy SuperHyperStructure). Let $\mathbf{FuSH}(H,\mathcal{F})$ be a fuzzy superhyperstructure with operations $F_j^{\mathrm{fuz}}:{\prod }_k{H}^{\langle {\ell }_{j,k}\rangle }\to {[0,1]}^{H^{\langle r_j\rangle }}$. With $\mathcal{C}:=\mathbf{1}$, take $U^{\langle m\rangle }(*)=H^{\langle m\rangle }$ and $V^{\langle m\rangle }\left(Y\right)={[0,1]}^Y$, and set ${\left({\Phi }_j\right)}_{*}=F_j^{\mathrm{fuz}}$. Then we obtain an FSS whose fiber equals $\mathbf{FuSH}(H,\mathcal{F})$.

Proof. 

Identical to Theorem 14, now with fuzzy value functors. □

Theorem 

(Reduction to Functorial HyperStructure). Let $\mathbf{FH}(\mathcal{C};U,V,\left\{{\Phi }_j\right\})$ be a Functorial HyperStructure (single level) with ${\Phi }_j:U^{\times n_j}\Rightarrow V\circ U$. Choose the one-level systems $\mathcal{U}=\{U^{\langle 0\rangle }:=U\}$ and $\mathcal{V}=\{V^{\langle 0\rangle }:=V\}$, and keep the same family $\left\{{\Phi }_j\right\}$. Then $(\mathcal{C};\mathcal{U},\mathcal{V},\left\{{\Phi }_j\right\})$ is an FSS whose data coincide with $\mathbf{FH}$.

Proof. 

This is Definition 22 specialized to a single level. □

Theorem 

(Reduction to Soft SuperHyperStructure). Let $\mathbf{SSH}(H,\mathcal{F};S)$ be a Soft SuperHyperStructure with parameter set $A\ne \varnothing$ and per-parameter sub-superhyperstructures $S_a^{\langle m\rangle }\subseteq H^{\langle m\rangle }$ satisfying the closure condition

$$F_j\left(S_a^{\langle {\ell }_{j,1}\rangle },\dots ,S_a^{\langle {\ell }_{j,n_j}\rangle }\right)\subseteq S_a^{\langle r_j\rangle }(\forall j,\forall a\in A).$$

Fix $\mathcal{C}:=\mathbf{1}$, set $U^{\langle m\rangle }(*)=H^{\langle m\rangle }$ and $V^{\langle m\rangle }\left(Y\right)={\left({\mathcal{P}}^{*}\left(Y\right)\right)}^A$, and let ${\left({\Phi }_j\right)}_{*}$ be the coordinatewise images of $F_j$ under the inclusion into the A-indexed product. For each $a\in A$ introduce anullarynatural transformation ${\sigma }_a:\mathbf{1}\Rightarrow V^{\langle m\rangle }\circ U^{\langle m\rangle }$ with component ${\left({\sigma }_a\right)}_{*}(*)={\left(S_a^{\langle m\rangle }\right)}_{m\ge 0}$. Then $(\mathcal{C};\mathcal{U},\mathcal{V},\left\{{\Phi }_j\right\}\cup {\left\{{\sigma }_a\right\}}_{a\in A})$ is an FSS, and the soft-closure condition above is equivalent to the FSS equations

$${\left({\Phi }_j\right)}_{*}\left(\underset{n_j}{\underbrace{{\sigma }_a,\dots ,{\sigma }_a}}\right)⪯{\sigma }_a(\mathrm{coordinatewise}\mathrm{in}A\mathrm{and}\mathrm{levelwise}\mathrm{in}m).$$

Consequently, $\mathbf{SSH}(H,\mathcal{F};S)$ is (componentwise) recovered from this FSS.

Proof. 

In $\mathbf{1}$, naturality is trivial. The inequality displayed is exactly the statement that applying $F_j$ to the selected subsets $S_a^{\langle {\ell }_{j,k}\rangle }$ lands inside $S_a^{\langle r_j\rangle }$ for each a, which is the definition of soft closure. Conversely, those inclusions reproduce the soft structure. □

4. Conclusion

This paper has examined several extended variants of the classical SuperHyperStructure, including Rough, Soft, Fuzzy, and Functorial SuperHyperStructures. It is our hope that future work will investigate practical applications of these concepts in real-world contexts, as well as their deeper mathematical properties, accompanied by rigorous quantitative analyses. We also envision further studies on possible extensions employing the framework of Plithogenic Sets [51,52,53], thereby enriching the theoretical landscape and expanding potential applications.