Multiary Hyperstructure and Multiary Superhyperstructure

1. Preliminaries

This section gathers the basic notions and notation used throughout the paper. Unless explicitly stated otherwise, we work in the finite setting. By convention, the empty set is treated as an element of every set.

1.1. Hyperstructure and Superhyperstructure

A Hyperstructure is organized around the powerset and serves as a vehicle for modeling relations among elements of a set [1,2,3,4,5,6]. Owing to its flexibility, the hyperstructure framework has been investigated across several areas, including mathematics and chemistry [7,8,9,10]. A Superhyperstructure advances this idea by utilizing the n-th powerset to encode multi-layered hierarchical interactions, thereby enabling deeper abstraction and greater structural complexity [11,12,13,14]. Because of this wide scope, superhyperstructures have likewise been explored in mathematics, chemistry, and related disciplines [11,15,16,17]. Prominent instances include constructs such as the SuperHyperGraph [18,19]. We next record the n-th powerset, which underpins these structures.

Definition 1

(Base Set). A base set S is the underlying collection from which higher-level constructions—powersets and (super)hyperstructures—are built. Formally,

$$S=\{x\mid x\mathit{is}\mathit{an}\mathit{element}\mathit{of}a\mathit{specified}\mathit{domain}\}.$$

All elements appearing in $\mathcal{P}\left(S\right)$ or in the iterated powersets ${\mathcal{P}}_n\left(S\right)$ ultimately arise from members of S.

Definition 2

(Powerset). [17,20,21] The powerset of a set S, denoted $\mathcal{P}\left(S\right)$, is the family of all subsets of S, including ∅ and S itself:

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

Definition 3

(n-th Powerset). (cf. [14,22]) For a set H, the n-th powerset ${\mathcal{P}}_n\left(H\right)$ is defined recursively by

$${\mathcal{P}}_1\left(H\right)=\mathcal{P}\left(H\right),{\mathcal{P}}_{n+1}\left(H\right)=\mathcal{P}\left({\mathcal{P}}_n\left(H\right)\right),n\ge 1.$$

The nonempty version ${\mathcal{P}}_n^{*}\left(H\right)$ is given by

$${\mathcal{P}}_1^{*}\left(H\right)={\mathcal{P}}^{*}\left(H\right),{\mathcal{P}}_{n+1}^{*}\left(H\right)={\mathcal{P}}^{*}\left({\mathcal{P}}_n^{*}\left(H\right)\right),$$

where ${\mathcal{P}}^{*}\left(H\right)=\mathcal{P}\left(H\right)\setminus \{\varnothing \}$.

Example 1

(n-th powerset — a fully explicit small instance). Let $H=\{a,b\}$. Then the first powerset is

$${\mathcal{P}}_1\left(H\right)=\mathcal{P}\left(H\right)=\left\{⌀,\left\{a\right\},\left\{b\right\},\{a,b\}\right\}.$$

The second powerset is the powerset of this 4-element set, hence it has $2^4=16$ elements:

$$\begin{array}{cc}\hfill {\mathcal{P}}_2\left(H\right)=\mathcal{P}\left(\mathcal{P}\left(H\right)\right)=\{& ⌀,\{⌀\},\left\{\right\{a\left\}\right\},\left\{\right\{b\left\}\right\},\left\{\right\{a,b\left\}\right\},\hfill \\ & \{⌀,\{a\left\}\right\},\{⌀,\{b\left\}\right\},\{⌀,\{a,b\left\}\right\},\left\{\right\{a\},\{b\left\}\right\},\left\{\right\{a\},\{a,b\left\}\right\},\left\{\right\{b\},\{a,b\left\}\right\},\hfill \\ & \{⌀,\{a\},\{b\left\}\right\},\{⌀,\{a\},\{a,b\left\}\right\},\{⌀,\{b\},\{a,b\left\}\right\},\left\{\right\{a\},\{b\},\{a,b\left\}\right\},\hfill \\ & \{⌀,\{a\},\{b\},\{a,b\left\}\right\}\}.\hfill \end{array}$$

(Here the last element equals ${\mathcal{P}}_1\left(H\right)$ itself, viewed as a single member of ${\mathcal{P}}_2\left(H\right)$.) The nonempty versions are ${\mathcal{P}}_1^{*}\left(H\right)=\mathcal{P}\left(H\right)\setminus \{⌀\}=\left\{\left\{a\right\},\left\{b\right\},\{a,b\}\right\}$ and ${\mathcal{P}}_2^{*}\left(H\right)={\mathcal{P}}_2\left(H\right)\setminus \{⌀\}$.

To provide a self-contained foundation for hyperstructures and superhyperstructures, we recall the following standard notions.

Definition 4

(Classical Structure). (cf. [14,22,23]) A Classical Structure consists of a nonempty set H together with one or more classical operations satisfying specified axioms. A classical m-ary operation has the form

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

with $m\ge 1$. Familiar examples include the operations defining groups, rings, and fields.

Definition 5

(Hyperoperation). (cf. [24,25,26,27]) A hyperoperation on a set S is a map

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

so that combining two inputs returns a set of outcomes (not necessarily a singleton).

Example 2

(Hyperoperation — combining travel legs with uncertainty (real life)). Let $S=\{0,1,\dots ,120\}$ encode travel durations in minutes. Define a hyperoperation

$$\circ :S\times S⟶\mathcal{P}\left(S\right),x\circ y:=\{t\in S\mid |t-(x+y)|\le 5\}.$$

Meaning. Two trip segments of nominal lengths x and y minutes are combined, but real traffic adds a $\pm 5$ minute variability, so the result is a set of plausible totals.

Concrete calculation. With $x=20$ and $y=35$,

$$x+y=55,x\circ y=\{50,51,52,53,54,55,56,57,58,59,60\}\subseteq S.$$

Definition 6

(Hyperstructure). (cf. [14,22,28,29]) A Hyperstructure augments a base set S by operating on its powerset. Formally,

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

where ∘ acts on subsets of S.

Example 3

(Hyperstructure — playlist growth from two seed sets (real life)). Let $S=\{{\mathrm{s}}_1,{\mathrm{s}}_2,{\mathrm{s}}_3,{\mathrm{s}}_4,{\mathrm{s}}_5\}$ be songs and consider the hyperstructure

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

where $\circ :\mathcal{P}\left(S\right)\times \mathcal{P}\left(S\right)\to \mathcal{P}\left(S\right)$ acts by

$$A\circ B:=(A\cup B)\cup R(A,B),R(A,B):=\left\{\begin{array}{cc}\left\{{\mathrm{s}}_4\right\},\hfill & \{{\mathrm{s}}_1,{\mathrm{s}}_2\}\subseteq A\cup B,\hfill \\ ⌀,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Meaning. Merging two seed playlists $A,B$ returns the union plus one data-driven recommendation ${\mathrm{s}}_4$ if both ${\mathrm{s}}_1$ and ${\mathrm{s}}_2$ are present.

Concrete calculation. Take $A=\{{\mathrm{s}}_1,{\mathrm{s}}_3\}$ and $B=\left\{{\mathrm{s}}_2\right\}$. Then $A\cup B=\{{\mathrm{s}}_1,{\mathrm{s}}_2,{\mathrm{s}}_3\}$ and $R(A,B)=\left\{{\mathrm{s}}_4\right\}$, so

$$A\circ B=\{{\mathrm{s}}_1,{\mathrm{s}}_2,{\mathrm{s}}_3,{\mathrm{s}}_4\}\in \mathcal{P}\left(S\right).$$

Definition 7

(SuperHyperOperation). [14] Let H be nonempty. Define recursively, for $k\ge 0$,

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

For fixed $m,n\ge 0$ and arity $s\ge 1$, an $(m,n)$-SuperHyperOperation is a map

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

If the codomain may include ∅, we obtain the neutrosophic variant; otherwise we are in the classical case.

Example 4

(SuperHyperOperation $(m,n)$ — bundle suggestions from two baskets (real life)). Let $H=\{\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}\}$ be products, fix $m=1$, $n=2$, and arity $s=2$. Thus inputs live in ${\mathcal{P}}^1\left(H\right)=\mathcal{P}\left(H\right)$ and outputs in ${\mathcal{P}}^2\left(H\right)=\mathcal{P}\left(\mathcal{P}\left(H\right)\right)$. Define

$${\odot }^{(1,2)}(X,Y):=\left\{B\subseteq X\cup Y|2\le |B|\le 3\right\}.$$

Meaning. Given two customer baskets $X,Y$, return a family of candidate bundles (size 2 or 3) drawn from the union of what they viewed/bought.

Concrete calculation. If $X=\{\mathrm{A},\mathrm{C}\}$ and $Y=\{\mathrm{B},\mathrm{C}\}$, then $X\cup Y=\{\mathrm{A},\mathrm{B},\mathrm{C}\}$ and

$${\odot }^{(1,2)}(X,Y)=\left\{\{\mathrm{A},\mathrm{B}\},\{\mathrm{A},\mathrm{C}\},\{\mathrm{B},\mathrm{C}\},\{\mathrm{A},\mathrm{B},\mathrm{C}\}\right\}\in {\mathcal{P}}^2\left(H\right).$$

Definition 8

(n-Superhyperstructure). (cf. [13,14,22]) Ann-Superhyperstructuregeneralizes hyperstructures by acting on the n-th powerset:

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

with ∘ defined on ${\mathcal{P}}_n\left(S\right)$.

Example 5

(n-Superhyperstructure — families of task-lists with closure (real life)). Let $S=\{\mathrm{a},\mathrm{b},\mathrm{c}\}$ denote atomic tasks and take $n=2$, so the universe is ${\mathcal{P}}_2\left(S\right)=\mathcal{P}\left(\mathcal{P}\left(S\right)\right)$, i.e., families of task-sets (plans). Define

$$\circ :{\mathcal{P}}_2\left(S\right)\times {\mathcal{P}}_2\left(S\right)⟶{\mathcal{P}}_2\left(S\right),\circ (F,G):=F\cup G\cup \{X\cup Y\mid X\in F,Y\in G\}.$$

Meaning. Combining two plan-families keeps all existing plans and adds every pairwise union (a simple “closure” under merging).

Concrete calculation. With $F=\left\{\right\{\mathrm{a}\},\{\mathrm{b}\left\}\right\}$ and $G=\left\{\right\{\mathrm{b}\},\{\mathrm{c}\left\}\right\}$,

$$\circ (F,G)=\left\{\left\{\mathrm{a}\right\},\left\{\mathrm{b}\right\},\left\{\mathrm{c}\right\},\{\mathrm{a},\mathrm{b}\},\{\mathrm{a},\mathrm{c}\},\{\mathrm{b},\mathrm{c}\}\right\}\in {\mathcal{P}}_2\left(S\right).$$

Thus ${\mathcal{SH}}_2=({\mathcal{P}}_2\left(S\right),\circ )$ is an n-Superhyperstructure.

Definition 9

(SuperHyperStructure of order $(m,n)$). (cf. [11,30,31,32]) Let S be nonempty and $m,n\ge 0$. A $(m,n)$-SuperHyperStructure of arity s is any choice of

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

The special cases recover standard settings: $m=n=0$ gives ordinary s-ary operations; $m=0,n=1$ yields hyperoperations; and $s=1$ corresponds to superhyperfunctions.

Example 6

(SuperHyperStructure of order $(m,n)$ — photo pairing generator (real life)). Let

$$S=\{{\mathrm{p}}_1,{\mathrm{p}}_2,{\mathrm{p}}_3,{\mathrm{p}}_4\}$$

be photos, and fix $m=1$, $n=2$, arity $s=1$. Define the superhyperoperation (a superhyperfunction)

$${\odot }^{(1,2)}:\mathcal{P}\left(S\right)⟶\mathcal{P}\left(\mathcal{P}\left(S\right)\right),$$

$${\odot }^{(1,2)}\left(X\right):=\left\{\begin{array}{cc}\{B\subseteq X\mid |B|=2\},\hfill & \left|X\right|\ge 2,\hfill \\ \left\{X\right\},\hfill & \left|X\right|\le 1.\hfill \end{array}\right.$$

Meaning. From one album X, return the family of all two-photo pairings for layout exploration (or the album itself if too small).

Concrete calculation. For $X=\{{\mathrm{p}}_1,{\mathrm{p}}_2,{\mathrm{p}}_3\}$,

$${\odot }^{(1,2)}\left(X\right)=\left\{\{{\mathrm{p}}_1,{\mathrm{p}}_2\},\{{\mathrm{p}}_1,{\mathrm{p}}_3\},\{{\mathrm{p}}_2,{\mathrm{p}}_3\}\right\}\in {\mathcal{P}}^2\left(S\right).$$

Hence $({\mathcal{P}}^1\left(S\right);{\odot }^{(1,2)})$ is a SuperHyperStructure of order $(1,2)$.

Definition 10

($(h,k)$-ary SuperHyperstructure). Let S be a nonempty base set. For each $i\in \{1,2,\dots ,h\}$ choose a nonempty subset $A_i\subseteq S$ and fix an integer $m_i\ge 0$. Denote by

$${\mathcal{P}}_{m_i}\left(A_i\right)$$

the $m_i$-th iterated powerset of $A_i$. Similarly, for each $j\in \{1,2,\dots ,k\}$ choose a nonempty subset $B_j\subseteq S$ and fix an integer $n_j\ge 0$, and denote by

$${\mathcal{P}}_{n_j}\left(B_j\right)$$

the $n_j$-th iterated powerset of $B_j$.

Define the domain D and codomain C as

$$D={\mathcal{P}}_{m_1}\left(A_1\right)\times {\mathcal{P}}_{m_2}\left(A_2\right)\times \dots \times {\mathcal{P}}_{m_h}\left(A_h\right),$$

$$C={\mathcal{P}}_{n_1}\left(B_1\right)\times {\mathcal{P}}_{n_2}\left(B_2\right)\times \dots \times {\mathcal{P}}_{n_k}\left(B_k\right).$$

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

$$\mathcal{SH}=\left(D,C,{\left\{{\circ }_{\alpha }\right\}}_{\alpha \in I}\right),$$

where ${\left\{{\circ }_{\alpha }\right\}}_{\alpha \in I}$ is an indexed family of SuperHyperoperations

$${\circ }_{\alpha }:D⟶C,\alpha \in I.$$

For each $(X_1,\dots ,X_h)\in D$, one has

$${\circ }_{\alpha }(X_1,\dots ,X_h)=(Y_1,\dots ,Y_k)\in C,$$

with $Y_j\in {\mathcal{P}}_{n_j}\left(B_j\right)$ for all j. The structural properties imposed on the maps ${\circ }_{\alpha }$—such as associativity, commutativity, or distributivity—are specified according to the algebraic framework adopted, thereby extending classical algebraic systems into a higher-order superhyperstructural setting.

Example 7

($(h,k)$-ary SuperHyperstructure — basket and context to recommendations and coupon (real life)). Let the base set

$$S=\{\mathrm{pA},\mathrm{pB},\mathrm{pC}\}\cup \left\{\mathrm{tagSummer}\right\}\cup \left\{\mathrm{coupX}\right\}$$

contain products, a context tag, and a coupon token. Choose $h=2$ inputs with

$$A_1=\{\mathrm{pA},\mathrm{pB},\mathrm{pC}\},m_1=1\left(\mathit{basket}\mathit{as}a\mathit{subset}\mathit{of}\mathit{products}\right);$$

$$A_2=\left\{\mathrm{tagSummer}\right\},m_2=0\left(a\mathit{sin}\mathit{gle}\mathit{tag}\mathit{element}\right).$$

Then the domain is

$$D={\mathcal{P}}_{m_1}\left(A_1\right)\times {\mathcal{P}}_{m_2}\left(A_2\right)=\mathcal{P}\left(A_1\right)\times A_2.$$

Take $k=2$ outputs with

$$B_1=A_1,n_1=1\left(a\mathit{recommended}\mathit{product}\mathit{set}\right);B_2=\left\{\mathrm{coupX}\right\},n_2=0\left(a\mathit{coupon}\mathit{token}\right).$$

Thus the codomain is $C=\mathcal{P}\left(B_1\right)\times B_2$. Define the single superhyperoperation $\circ :D\to C$ by

$$\circ (X,\mathrm{tagSummer}):=\left(Y_1,Y_2\right),Y_1:=A_1\setminus X,Y_2:=\mathrm{coupX}.$$

Meaning. Given a basket X and the summer context, recommend the missing products and issue coupon $\mathrm{coupX}$.

Concrete calculation. For $X=\{\mathrm{pA},\mathrm{pC}\}$,

$$Y_1=A_1\setminus X=\left\{\mathrm{pB}\right\}\in {\mathcal{P}}_1\left(B_1\right)=\mathcal{P}\left(A_1\right),Y_2=\mathrm{coupX}\in {\mathcal{P}}_0\left(B_2\right)=B_2,$$

so

$$\circ \left(\{\mathrm{pA},\mathrm{pC}\},\mathrm{tagSummer}\right)=\left(\left\{\mathrm{pB}\right\},\mathrm{coupX}\right)\in C.$$

Therefore $\mathcal{SH}=(D,C,\{\circ \left\}\right)$ is an $(h,k)$-ary SuperHyperstructure with $(h,k)=(2,2)$.

2. Main Results

This section presents the results of this paper.

2.1. Multiary Structure and Multiary Hyperstructure

A multiary hyperstructure extends algebraic systems, allowing operations with multiple inputs producing set-valued outputs, modeling uncertainty and multi-participant interactions.

Definition 11

(Multiary signature). Let I be a nonempty index set and let $\mathrm{ar}:I\to {\mathbb{N}}_{\ge 1}$ assign to each $i\in I$ a positive integer $\mathrm{ar}\left(i\right)$ (thearityof the symbol i). We call the pair $(I,\mathrm{ar})$ amultiary signature. Informally, one may think of the arity profile as an “alphabet” $(a,b,\dots ,z)$ of arities.

Example 8

(Multiary signature — smartphone imaging pipeline (real-life)). A mobile camera app exposes three tools that naturally come with different input counts:

$$I=\{\mathit{merge},\mathit{hdr},\mathit{pano}\},\mathrm{ar}\left(\mathit{merge}\right)=2,\mathrm{ar}\left(\mathit{hdr}\right)=3,\mathrm{ar}\left(\mathit{pano}\right)=5.$$

Interpretation. (i) merge blends two exposures of the same scene; (ii) hdr combines three bracketed shots to increase dynamic range; (iii) pano stitches five partially overlapping tiles into one panorama. Thus $(I,\mathrm{ar})$ is a concrete multiary signature whose “arity alphabet’’ is $(2,3,5)$.

Definition 12

(Multiary structure (algebra of type $(I,\mathrm{ar})$)). Given a multiary signature $(I,\mathrm{ar})$ and a nonempty set H, a multiary structure on H is a family of operations

$$\mathbf{H}:=\left(H;{\left(f_i\right)}_{i\in I}\right),f_i:H^{\mathrm{ar}\left(i\right)}⟶H(i\in I).$$

When $I=\left\{i\right\}$ is a singleton and $\mathrm{ar}\left(i\right)=n$, this reduces to the usual notion of an n-ary algebra.

Example 9

(Multiary structure — home-alarm decision fusion (real-life)). Let $H=\{0,1\}$ encodeno-alarm $=0$ andalarm $=1$. Take the signature $I=\{\wedge ,{\mathrm{Maj}}_3\}$ with $\mathrm{ar}(\wedge )=2$ and $\mathrm{ar}\left({\mathrm{Maj}}_3\right)=3$. Define 

$$f_{\wedge }(x,y):=x\wedge y,f_{{\mathrm{Maj}}_3}(x,y,z):=\left\{\begin{array}{cc}1,\hfill & x+y+z\ge 2,\hfill \\ 0,\hfill & x+y+z\le 1.\hfill \end{array}\right.$$

Interpretation. $f_{\wedge }$ triggers only if two independent gates both fire (e.g., door & motion). $f_{{\mathrm{Maj}}_3}$ triggers if at least two among {door, motion, glass} vote “alarm”. Numerical instance (door $=1$, motion $=0$, glass $=1$):

$$f_{\wedge }(1,0)=0,f_{{\mathrm{Maj}}_3}(1,0,1)=1(\sin \mathrm{ce}1+0+1=2\ge 2).$$

Hence $\mathbf{H}=(H;(f_{\wedge },f_{{\mathrm{Maj}}_3}))$ is a concrete multiary structure of type $(I,\mathrm{ar})$.

Example 10 ((a,b,c)-ary multiary structure on a finite set; here $(a,b,c)=(2,3,4)$). Let $H=\{0,1,\dots ,9\}$ with the usual order and addition modulo 10. Consider the signature $I=\{f,g,h\}$ with arities

$$\mathrm{ar}\left(f\right)=2,\mathrm{ar}\left(g\right)=3,\mathrm{ar}\left(h\right)=4.$$

Define the operations $f,g,h:H^{\mathrm{ar}(·)}\to H$ by

$$f(x,y):=(x+y)mod10,g(x,y,z):=\mathrm{med}\{x,y,z\},h(w,x,y,z):=max\{w,x,y,z\}.$$

Then

$$\mathbf{H}=\left(H;(f,g,h)\right)$$

is a concrete $(2,3,4)$-ary multiary structure (an algebra of type $(I,\mathrm{ar})$).Sample computations.

$$f(7,8)=(7+8)mod10=15mod10=5,g(2,9,3)=\mathrm{med}\{2,9,3\}=3,h(1,6,4,6)=6.$$

Example 11

(Multiary structure on strings: concatenation and round-robin interleaving). Let $\Sigma =\{\mathtt{A},\mathtt{B},\mathtt{C}\}$ and $H={\mathrm{\Sigma }}^{*}$ (all finite strings over Σ). Take $I=\{\mathrm{cat},\mathrm{mix}\}$ with arities $\mathrm{ar}\left(\mathrm{cat}\right)=2$ and $\mathrm{ar}\left(\mathrm{mix}\right)=3$. Define

$$\mathrm{cat}(u,v):=uv\left(\mathrm{concatenation}\right),$$

$$\mathrm{mix}(u,v,w):=\mathrm{the}\mathrm{string}\mathrm{obtained}\mathrm{by}\mathrm{round}-\mathrm{robin}\mathrm{taking}\mathrm{one}\mathrm{character}\mathrm{from}u,v,w,$$

skipping any exhausted word until all three are exhausted. Formally, if $u=u_1\dots u_m$, $v=v_1\dots v_n$, $w=w_1\dots w_p$, then

$$\mathrm{mix}(u,v,w)=u_1{v}_1{w}_1{u}_2{v}_2{w}_2\dots$$

with the convention that missing symbols are omitted once an index exceeds the word’s length. Then

$$\mathbf{H}=\left(H;(\mathrm{cat},\mathrm{mix})\right)$$

is a multiary structure of type $(I,\mathrm{ar})$. Sample computations. With $u=\mathtt{AB}$, $v=\mathtt{BAA}$, $w=\mathtt{B}$,

$$\mathrm{cat}(u,v)=\mathtt{ABBAA},\mathrm{mix}(u,v,w)=\mathtt{ABBABA}=\mathtt{ABBABA}.$$

(Explanation: round-robin picks $\mathtt{A}$ from u, $\mathtt{B}$ from v, $\mathtt{B}$ from w, then $\mathtt{B}$ from u, $\mathtt{A}$ from v; w is exhausted; finally $\mathtt{A}$ from v.)

Definition 13

(Homomorphism of multiary structures). Let $\mathbf{H}=(H;{\left(f_i\right)}_{i\in I})$ and ${\mathbf{H}}^{\prime }=(H^{\prime };{\left(f_i^{\prime }\right)}_{i\in I})$ be multiary structures of the same signature. A map $\phi :H\to H^{\prime }$ is a homomorphism if

$$\phi \left(f_i(x_1,\dots ,x_{\mathrm{ar}\left(i\right)})\right)=f_i^{\prime }\left(\phi \left(x_1\right),\dots ,\phi \left(x_{\mathrm{ar}\left(i\right)}\right)\right)\mathrm{for}\mathrm{all}i\in I\mathrm{and}(x_1,\dots ,x_{\mathrm{ar}\left(i\right)})\in H^{\mathrm{ar}\left(i\right)}.$$

(real-life)).Example 12 (Homomorphism of multiary structures — two encodings of the same policy Keep $\mathbf{H}=(H;(f_{\wedge },f_{{\mathrm{Maj}}_3}))$ from the previous example with $H=\{0,1\}$. Define a second system ${\mathbf{H}}^{\prime }=(H^{\prime };(f_{\wedge }^{\prime },f_{{\mathrm{Maj}}_3}^{\prime }))$ on $H^{\prime }=\{\mathrm{L},\mathrm{H}\}$ (low/high alert) with the total order $\mathrm{L}<\mathrm{H}$ and

$$f_{\wedge }^{\prime }(u,v):=min\{u,v\},f_{{\mathrm{Maj}}_3}^{\prime }(u,v,w):=\left\{\begin{array}{cc}\mathrm{H},\hfill & \#\{u,v,w=\mathrm{H}\}\ge 2,\hfill \\ \mathrm{L},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Let $\phi :H\to H^{\prime }$ be the code map $\phi \left(0\right)=\mathrm{L}$, $\phi \left(1\right)=\mathrm{H}$. We verify the homomorphism equalities for both operations.

(i) Binary gate:

$$\phi \left(f_{\wedge }(x,y)\right)=\phi (x\wedge y)=\left\{\begin{array}{cc}\mathrm{H},\hfill & x=y=1,\hfill \\ \mathrm{L},\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

while

$$f_{\wedge }^{\prime }\left(\phi \left(x\right),\phi \left(y\right)\right)=min\{\phi \left(x\right),\phi \left(y\right)\}=\left\{\begin{array}{cc}\mathrm{H},\hfill & \phi \left(x\right)=\phi \left(y\right)=\mathrm{H}\iff x=y=1,\hfill \\ \mathrm{L},\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Thus $\phi \left(f_{\wedge }(x,y)\right)=f_{\wedge }^{\prime }(\phi \left(x\right),\phi \left(y\right))$ for all $x,y\in H$.

(ii) Ternary majority:

$$\phi \left(f_{{\mathrm{Maj}}_3}(x,y,z)\right)=\left\{\begin{array}{cc}\mathrm{H},\hfill & x+y+z\ge 2,\hfill \\ \mathrm{L},\hfill & x+y+z\le 1,\hfill \end{array}\right.$$

$$f_{{\mathrm{Maj}}_3}^{\prime }\left(\phi \left(x\right),\phi \left(y\right),\phi \left(z\right)\right)=\left\{\begin{array}{cc}\mathrm{H},\hfill & \#\{x,y,z=1\}\ge 2,\hfill \\ \mathrm{L},\hfill & \mathit{otherwise},\hfill \end{array}\right.$$

which coincide since $\#\{x,y,z=1\}=x+y+z$ for $x,y,z\in \{0,1\}$.

Concrete checks. For $(x,y,z)=(1,0,1)$:

$$\phi \left(f_{{\mathrm{Maj}}_3}(1,0,1)\right)=\phi \left(1\right)=\mathrm{H}=f_{{\mathrm{Maj}}_3}^{\prime }(\mathrm{H},\mathrm{L},\mathrm{H}).$$

For $(x,y,z)=(0,0,1)$:

$$\phi \left(f_{{\mathrm{Maj}}_3}(0,0,1)\right)=\phi \left(0\right)=\mathrm{L}=f_{{\mathrm{Maj}}_3}^{\prime }(\mathrm{L},\mathrm{L},\mathrm{H}).$$

Hence φ is a homomorphism $\mathbf{H}\to {\mathbf{H}}^{\prime }$.

Definition 14

(Multiary hyperstructure). Given a multiary signature $(I,\mathrm{ar})$ and a nonempty set H, a multiary hyperstructure on H is a family

$$\mathbf{H}:=\left(H;{\left(h_i\right)}_{i\in I}\right),h_i:H^{\mathrm{ar}\left(i\right)}⟶{\mathcal{P}}^{*}\left(H\right)(i\in I).$$

For each fixed i with $\mathrm{ar}\left(i\right)=n$, one may impose the usual n-ary hyperalgebraic axioms on $h_i$ (e.g., commutativity, identities, associativity) in the sense of n-ary semihypergroups/hypergroups.

Example 13

(Multiary hyperstructure — noisy sensor fusion (real-life)). Let $H=\{0,1,2,\dots ,20\}$ denote discretized temperature in °C. Uncertain sensors produce set-valued outcomes with $\pm 1^{\circ }$ tolerance.

Define a binary hyperaddition (two sensors combined)

$$h_{+}(x,y):=\left\{t\in H\left|\right|t-(x+y)|\le 1\right\},$$

and a ternary hyperaverage (three sensors)

$$h_{\mathrm{avg}3}(x,y,z):=\left\{t\in H|\left|t-⌊\frac{x+y+z}{3}⌋\right|\le 1\right\}.$$

Then $\mathbf{H}=\left(H;(h_{+},h_{\mathrm{avg}3})\right)$ is a multiary hyperstructure of type $I=\{+,\mathrm{avg}3\}$ with arities $(2,3)$.

Concrete numerical outputs. (i) Two sensors read 7 and 8:

$$h_{+}(7,8)=\{14,15,16\}.$$

(ii) Three sensors read $10,12,15$:

$$⌊{\displaystyle \frac{10+12+15}{3}}⌋=⌊{\displaystyle \frac{37}{3}}⌋=12,h_{\mathrm{avg}3}(10,12,15)=\{11,12,13\}.$$

Both outputs are nonempty subsets of H, as required for a hyperoperation.

Notation 1

(Extension to subsets). If $h:H^n\to {\mathcal{P}}^{*}\left(H\right)$ is an n-ary hyperoperation and $A_1,\dots ,A_n\subseteq H$ are nonempty, set

$$h(A_1,\dots ,A_n):=\bigcup \{h(x_1,\dots ,x_n)\mid x_i\in A_i(1\le i\le n)\}.$$

This is standard in n-ary hyperstructure theory.

Definition 15

(Associativity for a fixed n-ary hyperoperation). Let $h:H^n\to {\mathcal{P}}^{*}\left(H\right)$. We say h is associative (i.e., $(H,h)$ is an n-ary semihypergroup) if, for all $x_1,\dots ,x_{2n-1}\in H$ and all $i,j\in \{1,\dots ,n\}$,

$$h\left(x_1,\dots ,x_{i-1},h(x_i,\dots ,x_{n+i-1}),x_{n+i},\dots ,x_{2n-1}\right)=h\left(x_1,\dots ,x_{j-1},h(x_j,\dots ,x_{n+j-1}),x_{n+j},\dots ,x_{2n-1}\right).$$

This is the usual n-ary associativity axiom in the literature.

Example 14

(Associativity of an n-ary hyperoperation — travel-time fusion with jitter (real-life)). Fix $n=2$ (binary case) and let $H=\{0,1,\dots ,180\}$ denote minutes. Define a jittered sum hyperoperation $h:H^2\to {\mathcal{P}}^{*}\left(H\right)$ by

$$h(x,y):=\{t\in H\mid |t-(x+y)|\le 1\}=\{x+y-1,x+y,x+y+1\}\cap H.$$

Interpretation. Combining two legs of a trip (e.g., bus and subway) yields the nominal total $x+y$ minutes, but real-world variability introduces a $\pm 1$ minute uncertainty. We adopt the standard extension to subsets $h(A,B):=\bigcup \left\{h\right(a,b)\mid a\in A,b\in B\}$.

We verify associativity on concrete data $x_1=17$, $x_2=23$, $x_3=12$:

$$\begin{array}{cc}\hfill h\left(h(17,23),12\right)& =h\left(\{39,40,41\},12\right)\hfill \\ & =\{38,39,40\}+12\cup \{39,40,41\}+12\cup \{40,41,42\}+12\hfill \\ & =\{50,51,52\}\cup \{51,52,53\}\cup \{52,53,54\}\hfill \\ & =\{50,51,52,53,54\},\hfill \\ \hfill h\left(17,h(23,12)\right)& =h\left(17,\{34,35,36\}\right)\hfill \\ & =17+\{33,34,35\}\cup 17+\{34,35,36\}\cup 17+\{35,36,37\}\hfill \\ & =\{50,51,52\}\cup \{51,52,53\}\cup \{52,53,54\}\hfill \\ & =\{50,51,52,53,54\}.\hfill \end{array}$$

Thus $h\left(h(x_1,x_2),x_3\right)=h\left(x_1,h(x_2,x_3)\right)$ for this instance. In fact, the jitter offsets $\{-1,0,1\}$ combine by (discrete) Minkowski sum, yielding

$$h\left(h(x_1,x_2),x_3\right)=h\left(x_1,h(x_2,x_3)\right)=\{x_1+x_2+x_3+u\mid u\in \{-2,-1,0,1,2\}\}\cap H,$$

so h is associative (i.e., $(H,h)$ is a semihypergroup) in this real-life model.

Definition 16

(Homomorphism of multiary hyperstructures). Let $\mathbf{H}=(H;{\left(h_i\right)}_{i\in I})$ and ${\mathbf{H}}^{\prime }=(H^{\prime };{\left(h_i^{\prime }\right)}_{i\in I})$ be multiary hyperstructures of the same signature. A map $\phi :H\to H^{\prime }$ is a (weak) homomorphism if for every $i\in I$ and every $(x_1,\dots ,x_{\mathrm{ar}\left(i\right)})\in H^{\mathrm{ar}\left(i\right)}$,

$$\phi \left(h_i(x_1,\dots ,x_{\mathrm{ar}\left(i\right)})\right)\subseteq h_i^{\prime }\left(\phi \left(x_1\right),\dots ,\phi \left(x_{\mathrm{ar}\left(i\right)}\right)\right),$$

where φ acts on subsets elementwise. This inclusion-style preservation is the standard hyperalgebraic homomorphism condition (cf. the $(m,n)$-ary hypermodule case).

Example 15

(Homomorphism of multiary hyperstructures — item counts → carton counts (real-life)). Let $H=\{0,1,\dots ,200\}$ be individual item counts in a warehouse. On H define the binary hyperaddition (counting uncertainty $\pm 1$ item)

$$h_{+}(x,y):=\{x+y-1,x+y,x+y+1\}\cap H.$$

Let $H^{\prime }=\{0,12,24,\dots ,2400\}$ be counts measured in cartons of 12 . On $H^{\prime }$ define the corresponding hyperaddition with carton-level tolerance $\pm 1$ carton:

$$h_{+}^{\prime }(u,v):=\{u+v-12,u+v,u+v+12\}\cap H^{\prime }.$$

Consider the map $\phi :H\to H^{\prime }$ given by packing into cartons,

$$\phi \left(x\right):=12x.$$

We claim φ is a (weak) homomorphism:

$$\phi \left(h_{+}(x,y)\right)\subseteq h_{+}^{\prime }\left(\phi \left(x\right),\phi \left(y\right)\right)(\forall x,y\in H).$$

Indeed,

$$\phi \left(h_{+}(x,y)\right)=\{12(x+y-1),12(x+y),12(x+y+1)\}=\{12(x+y)-12,12(x+y),12(x+y)+12\},$$

while

$$h_{+}^{\prime }\left(\phi \left(x\right),\phi \left(y\right)\right)=\{12x+12y-12,12x+12y,12x+12y+12\}.$$

Hence $\phi \left(h_{+}(x,y)\right)=h_{+}^{\prime }\left(\phi \left(x\right),\phi \left(y\right)\right)$, which in particular gives the inclusion.

Concrete check (items $x=18$, $y=7$):

$$h_{+}(18,7)=\{24,25,26\}\stackrel{\phi }{\to }\{288,300,312\},$$

$$h_{+}^{\prime }\left(\phi \left(18\right),\phi \left(7\right)\right)=h_{+}^{\prime }(216,84)=\{288,300,312\}.$$

Thus φ preserves the hyperoperation (equality here), so it is a homomorphism of the two multiary hyperstructures of type $I=\{+\}$.

Remark 1

(Important special cases).

• If $I=\left\{i\right\}$ and $\mathrm{ar}\left(i\right)=n$, a multiary hyperstructure is precisely an n-ary hypergroupoid; with associativity it becomes an n-ary semihypergroup, and with additional solvability it becomes an n-ary hypergroup.
• $(m,n)$-hyperrings have two basic operations: an m-ary hyperaddition f and an n-ary (single-valued) multiplication g, subject to distributivity of g over f; $(m,n)$-hypermodules add an $n-1$–ary scalar hyperaction k. These are instances of multiary hyperstructures with $I=\{+,·,\mathrm{act}\}$ and arities $(m,n,n-1)$, respectively. See the axioms in the cited sources.

2.2. Multiary Superhyperstructure

A multiary superhyperstructure lifts multiary hyperstructures onto higher powerset levels, encoding hierarchical, layered relationships with generalized superhyperoperations across domains.

Definition 17

(Multiary $(m,n)$-superhyperoperation). Fix $m,n\in {\mathbb{N}}_0$ and a nonempty base set S. Put $U:={\mathcal{P}}_m\left(S\right)$ (the m-level universe). Given a signature $(I,\mathrm{ar})$, a multiary $(m,n)$-superhyperoperation on S is a family

$${\left(h_i:U^{\mathrm{ar}\left(i\right)}⟶{\mathcal{P}}_n^{*}\left(S\right)\right)}_{i\in I}.$$

Thus each $h_i$ takes $\mathrm{ar}\left(i\right)$ many m-level arguments and returns a nonempty n-level object (a hyperimage). When $m=0$ and $n=1$ this specializes to the classical notion of (multiary) hyperoperations $H^r\to {\mathcal{P}}^{*}\left(H\right)$.

Example 16

(Multiary $(m,n)$-superhyperoperation — bundle suggestions from baskets (real-life)). Let the product universe be $S=\{\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}\}$. Fix $m=1$, $n=2$, so $U={\mathcal{P}}_1\left(S\right)=\mathcal{P}\left(S\right)$ and outputs lie in ${\mathcal{P}}_2\left(S\right)=\mathcal{P}\left(\mathcal{P}\left(S\right)\right)$ (i.e., families of bundles). Take the singleton signature $I=\left\{\mathrm{bund}\right\}$ with $\mathrm{ar}\left(\mathrm{bund}\right)=2$. Define the binary $(1,2)$-superhyperoperation

$$h_{\mathrm{bund}}:U\times U⟶{\mathcal{P}}_2^{*}\left(S\right),h_{\mathrm{bund}}(X,Y):=\left\{B\subseteq X\cup Y|2\le |B|\le 3\right\}.$$

Concrete input. Let $X=\{\mathrm{A},\mathrm{C}\}$ (basket#1) and $Y=\{\mathrm{B},\mathrm{C}\}$ (basket#2). Then

$$X\cup Y=\{\mathrm{A},\mathrm{B},\mathrm{C}\},$$

and the output is the nonempty element of ${\mathcal{P}}_2\left(S\right)$

$$h_{\mathrm{bund}}(X,Y)=\left\{\{\mathrm{A},\mathrm{B}\},\{\mathrm{A},\mathrm{C}\},\{\mathrm{B},\mathrm{C}\},\{\mathrm{A},\mathrm{B},\mathrm{C}\}\right\},$$

a family of candidate bundles proposed from the two baskets.

Definition 18

(Multiary $(m,n)$-superhyperstructure). With $S,m,n,U$ as above and a signature $(I,\mathrm{ar})$, amultiary $(m,n)$-superhyperstructure is an algebra

$${\mathbf{SH}}^{(m,n)}:=\left(U;{\left(h_i\right)}_{i\in I}\right),$$

where each $h_i:U^{\mathrm{ar}\left(i\right)}\to {\mathcal{P}}_n^{*}\left(S\right)$ is as in Definition 17. If one allows ${\mathcal{P}}_n\left(S\right)$ (possibly producing ⌀) as codomain, we speak of atotalmultiary $(m,n)$-superhyperstructure.

Example 17

(Multiary $(m,n)$-superhyperstructure — two real operators on shopping sets). Keep $S=\{\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}\}$ and fix $m=n=1$, hence $U=\mathcal{P}\left(S\right)$ and outputs are in ${\mathcal{P}}_1^{*}\left(S\right)={\mathcal{P}}^{*}\left(S\right)$. Let $I=\{\mathrm{merge},\mathrm{s}2\}$ with arities $\mathrm{ar}\left(\mathrm{merge}\right)=2$ and $\mathrm{ar}\left(\mathrm{s}2\right)=1$. Define

$$h_{\mathrm{merge}}(X,Y):=\{X\cup Y\},h_{\mathrm{s}2}\left(X\right):=\left\{\begin{array}{cc}\{B\subseteq X\mid |B|=2\},\hfill & \left|X\right|\ge 2,\hfill \\ \left\{X\right\},\hfill & \left|X\right|\le 1.\hfill \end{array}\right.$$

Then

$${\mathbf{SH}}^{(1,1)}=\left(U;h_{\mathrm{merge}},h_{\mathrm{s}2}\right)$$

is a multiary $(1,1)$-superhyperstructure.

Concrete computation. Let $X=\{\mathrm{A},\mathrm{C},\mathrm{D}\}$ and $Y=\left\{\mathrm{B}\right\}$.

$$h_{\mathrm{merge}}(X,Y)=\left\{\{\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}\}\right\},h_{\mathrm{s}2}\left(X\right)=\left\{\{\mathrm{A},\mathrm{C}\},\{\mathrm{A},\mathrm{D}\},\{\mathrm{C},\mathrm{D}\}\right\}.$$

Both outputs are nonempty families of item sets, as required.

Definition 19

(Associativity for a fixed operation). Let $r\ge 2$ and $h:U^r\to {\mathcal{P}}_n^{*}\left(S\right)$. Writing $x_i^j:=(x_i,\dots ,x_j)$ for blocks, h is (strongly) associative if for all $x_1,\dots ,x_{2r-1}\in U$ and all $1\le i<j\le r$,

$$h\left(x_1^{i-1},h\left(x_i^{i+r-1}\right),x_{i+r}^{2r-1}\right)=h\left(x_1^{j-1},h\left(x_j^{j+r-1}\right),x_{j+r}^{2r-1}\right).$$

It is weakly associative if the two sets above merely have nonempty intersection. This is the standard r-ary (hyper)associativity scheme, lifted here to the superhyper setting.

Example 18 (Associativity for a fixed operation — union-as-a-singleton family(real-life tags)). Let $m=n=1$, $U=\mathcal{P}\left(S\right)$ for $S=\{\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}\}$, and define the binary superhyperoperation

$$h(X,Y):=\{X\cup Y\}\in {\mathcal{P}}^{*}\left(S\right).$$

We use the standard extension $h(A,Z):={\bigcup }_{a\in A}h(a,Z)$ when an argument is a family. For the concrete tag-sets

$$X=\left\{\mathrm{A}\right\},Y=\{\mathrm{B},\mathrm{C}\},Z=\{\mathrm{C},\mathrm{D}\},$$

compute both nestings:

$$\begin{array}{cc}\hfill h\left(h(X,Y),Z\right)& =\bigcup _{W\in \{X\cup Y\}}h(W,Z)=h(X\cup Y,Z)=\{(X\cup Y)\cup Z\}=\left\{\{\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}\}\right\},\hfill \\ \hfill h\left(X,h(Y,Z)\right)& =\bigcup _{W\in \{Y\cup Z\}}h(X,W)=h\left(X,Y\cup Z\right)=\{X\cup (Y\cup Z)\}=\left\{\{\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}\}\right\}.\hfill \end{array}$$

Hence $h\left(h(X,Y),Z\right)=h\left(X,h(Y,Z)\right)$, and the binary case ($r=2$) satisfies strong associativity. Because h is induced by set-union, the same equality holds for all $X,Y,Z\in U$.

Definition 20

(Homomorphisms). Let ${\mathbf{SH}}^{(m,n)}=(U;{\left(h_i\right)}_{i\in I})$ over S and ${\mathbf{SH}}^{\prime (m,n)}=(U^{\prime };{\left(h_i^{\prime }\right)}_{i\in I})$ over $S^{\prime }$ share the same signature. A map $\phi :U\to U^{\prime }$ is ahomomorphismif for every $i\in I$ and every $(x_1,\dots ,x_{\mathrm{ar}\left(i\right)})\in U^{\mathrm{ar}\left(i\right)}$,

$$\phi \left(h_i(x_1,\dots ,x_{\mathrm{ar}\left(i\right)})\right)\subseteq h_i^{\prime }\left(\phi \left(x_1\right),\dots ,\phi \left(x_{\mathrm{ar}\left(i\right)}\right)\right),$$

where φ acts on subsets elementwise. (Equality gives a strong homomorphism.)

Example 19

(Homomorphism between multiary $(m,n)$-superhyperstructures — renaming items). Let $m=n=1$ and consider two bases

$$S=\{\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}\},S^{\prime }=\{\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\}.$$

Put $U=\mathcal{P}\left(S\right)$, $U^{\prime }=\mathcal{P}\left(S^{\prime }\right)$. On both sides use the same signature $I=\{\mathrm{merge},\mathrm{s}2\}$ and define

$$h_{\mathrm{merge}}(X,Y)=\{X\cup Y\},h_{\mathrm{s}2}\left(X\right)=\left\{\begin{array}{cc}\{B\subseteq X\mid |B|=2\},\hfill & \left|X\right|\ge 2,\hfill \\ \left\{X\right\},\hfill & \left|X\right|\le 1,\hfill \end{array}\right.$$

and analogously $h_{\mathrm{merge}}^{\prime },h_{\mathrm{s}2}^{\prime }$ on $U^{\prime }$.

Define $\phi :U\to U^{\prime }$ by the elementwise renaming $f:S\to S^{\prime }$ with $f\left(\mathrm{A}\right)=\mathrm{a}$, $f\left(\mathrm{B}\right)=\mathrm{b}$, $f\left(\mathrm{C}\right)=\mathrm{c}$, $f\left(\mathrm{D}\right)=\mathrm{d}$, and $\phi \left(X\right):=f\left[X\right]$ (image map). We verify the homomorphism inclusions (which in fact hold with equality).

1) Binary operator:

$$\phi \left(h_{\mathrm{merge}}(X,Y)\right)=\phi \left(\{X\cup Y\}\right)=\left\{f[X\cup Y]\right\}=\{f\left[X\right]\cup f\left[Y\right]\}=h_{\mathrm{merge}}^{\prime }\left(\phi \left(X\right),\phi \left(Y\right)\right).$$

2) Unary operator (case $\left|X\right|\ge 2$; the small cases are identical):

$$\phi \left(h_{\mathrm{s}2}\left(X\right)\right)=\left\{f\left[B\right]\mid B\subseteq X,\left|B\right|=2\right\}=\left\{B^{\prime }\subseteq f\left[X\right]\mid \left|B^{\prime }\right|=2\right\}=h_{\mathrm{s}2}^{\prime }\left(\phi \left(X\right)\right).$$

Concrete check. Let $X=\{\mathrm{A},\mathrm{C},\mathrm{D}\}$, $Y=\left\{\mathrm{B}\right\}$. Then

$$\phi \left(h_{\mathrm{merge}}(X,Y)\right)=\left\{\{\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\}\right\}=h_{\mathrm{merge}}^{\prime }\left(\{\mathrm{a},\mathrm{c},\mathrm{d}\},\left\{\mathrm{b}\right\}\right),$$

and

$$\phi \left(h_{\mathrm{s}2}\left(X\right)\right)=\left\{\{\mathrm{a},\mathrm{c}\},\{\mathrm{a},\mathrm{d}\},\{\mathrm{c},\mathrm{d}\}\right\}=h_{\mathrm{s}2}^{\prime }\left(\{\mathrm{a},\mathrm{c},\mathrm{d}\}\right).$$

Therefore φ is a (strong) homomorphism ${\mathbf{SH}}^{(1,1)}\to {\mathbf{SH}}^{\prime (1,1)}$.

Remark 2

(Reductions and special cases).

(a) 

If $I=\left\{i\right\}$ is a singleton and $r:=\mathrm{ar}\left(i\right)$, then ${\mathbf{SH}}^{(m,n)}$ reduces to an r-ary $(m,n)$-superhypergroupoid; imposing (weak) associativity on $h_i$ yields an r-ary $(m,n)$-superhypersemigroup.

(b) 

If $m=0$ and $n=1$, then $U=S$ and $h_i:S^{\mathrm{ar}\left(i\right)}\to {\mathcal{P}}^{*}\left(S\right)$, recovering classical multiary hyperstructures.

(c) 

If $n=0$, each $h_i$ is single-valued $U^{\mathrm{ar}\left(i\right)}\to U$, giving an ordinary multiary algebra on U.

(d) 

The choice of $U={\mathcal{P}}_m\left(S\right)$ realizes the superhyper viewpoint via the iterated powerset levels in the Definition.

Theorem 2

(Multiary $(m,n)$-superhyperstructure unifies three frameworks). Let $S\ne ⌀$.

(i) Every multiary hyperstructure $\left(S;{\left(g_i\right)}_{i\in I}\right)$ with $g_i:S^{\mathrm{ar}\left(i\right)}\to {\mathcal{P}}^{*}\left(S\right)$ is exactly a multiary $(0,1)$-superhyperstructure on S (take $m=0$, $n=1$ and $h_i:=g_i$).

(ii) Every SuperHyperStructure of order $(m,n)$ given by a single s-ary map ${\odot }^{(m,n)}:{\mathcal{P}}^m{\left(S\right)}^s\to {\mathcal{P}}^n{\left(S\right)}^{*}$ is a multiary $(m,n)$-superhyperstructure with singleton signature $I=\{*\}$ and $h_{*}:={\odot }^{(m,n)}$.

(iii) Every $(h,k)$-ary SuperHyperstructure

$${\circ }_{\alpha }:\underset{=:D}{\underbrace{{\mathcal{P}}^{m_1}\left(A_1\right)\times \dots \times {\mathcal{P}}^{m_h}\left(A_h\right)}}⟶\underset{=:C}{\underbrace{{\mathcal{P}}^{n_1}\left(B_1\right)\times \dots \times {\mathcal{P}}^{n_k}\left(B_k\right)}}(\alpha \in I)$$

canonically embeds into a multiary $(m,n)$-superhyperstructure on a tagged base $S^{†}:=S\times (\left\{0\right\}\cup K)$, where $K=\{1,\dots ,k\}$, $m:={max}_i{m}_i$ and $n:={max}_j{n}_j$.

Proof. (i) and (ii) are immediate by inspection of types: for (i) put $U:={\mathcal{P}}^0\left(S\right)=S$ and ${\mathcal{P}}^1{\left(S\right)}^{*}={\mathcal{P}}^{*}\left(S\right)$, then set $h_i:=g_i$; for (ii) use $I=\{*\}$ and $h_{*}:={\odot }^{(m,n)}$.

(iii) Fix $m:={max}_i{m}_i$, $n:={max}_j{n}_j$, $K:=\{1,\dots ,k\}$, and $S^{†}:=S\times (\left\{0\right\}\cup K)$. Define injective encoders, for each $i=1,\dots ,h$ and $j=1,\dots ,k$:

$$\begin{array}{cccc}\hfill {\iota }_i& :{\mathcal{P}}^{m_i}\left(A_i\right)⟶{\mathcal{P}}^m\left(S^{†}\right),\hfill & \hfill {\iota }_i& :={\iota }_{A_i\times \left\{0\right\}\hookrightarrow S^{†}}^{\left(m\right)}\circ {\mathrm{Tag}}_0^{\left(m\right)}\circ {\iota }_{A_i\hookrightarrow S}^{\left(m\right)}\circ {\mathrm{Up}}_{m_i}^m,\hfill \\ \hfill {\kappa }_j& :{\mathcal{P}}^{n_j}\left(B_j\right)⟶{\mathcal{P}}^n\left(S^{†}\right),\hfill & \hfill {\kappa }_j& :={\iota }_{B_j\times \left\{j\right\}\hookrightarrow S^{†}}^{\left(n\right)}\circ {\mathrm{Tag}}_j^{\left(n\right)}\circ {\iota }_{B_j\hookrightarrow S}^{\left(n\right)}\circ {\mathrm{Up}}_{n_j}^n.\hfill \end{array}$$

All constituents are injective, hence so are ${\iota }_i$ and ${\kappa }_j$.

Let $U:={\mathcal{P}}^m\left(S^{†}\right)$. For each $\alpha \in I$ define an h-ary $(m,n)$-superhyperoperation $h_{\alpha }:U^h\to {\mathcal{P}}^n{\left(S^{†}\right)}^{*}$ as follows. First, on the encoded domain

$$\widehat{D}:={\iota }_1\left({\mathcal{P}}^{m_1}\left(A_1\right)\right)\times \dots \times {\iota }_h\left({\mathcal{P}}^{m_h}\left(A_h\right)\right)\subseteq U^h,$$

set, for ${\widehat{X}}_i={\iota }_i\left(X_i\right)$,

$$h_{\alpha }({\widehat{X}}_1,\dots ,{\widehat{X}}_h):=\bigcup _{j=1}^k{\kappa }_j\left({\pi }_j\circ {\circ }_{\alpha }(X_1,\dots ,X_h)\right)\in {\mathcal{P}}^n{\left(S^{†}\right)}^{*},$$

(1)

where ${\pi }_j:C\to {\mathcal{P}}^{n_j}\left(B_j\right)$ is the j-th projection. (Outside $\widehat{D}$ one may extend arbitrarily, or work with the partial algebra on $\widehat{D}$.)

Decoding and faithfulness. For each j define the left-inverse decoder on the image of ${\kappa }_j$:

$${Dec}_j:={\left({\mathrm{Up}}_{n_j}^n\right)}^{-1}\circ {\left({\iota }_{B_j\hookrightarrow S}^{\left(n\right)}\right)}^{-1}\circ {\left({\mathrm{Tag}}_j^{\left(n\right)}\right)}^{-1}\circ {\left({\iota }_{B_j\times \left\{j\right\}\hookrightarrow S^{†}}^{\left(n\right)}\right)}^{-1},$$

which exists because each factor is injective. Then for all encoded inputs $({\widehat{X}}_1,\dots ,{\widehat{X}}_h)\in \widehat{D}$,

$$\begin{array}{cc}\hfill {Dec}_j\left(h_{\alpha }({\widehat{X}}_1,\dots ,{\widehat{X}}_h)\cap {\kappa }_j\left({\mathcal{P}}^{n_j}\left(B_j\right)\right)\right)& ={Dec}_j\left({\kappa }_j\left({\pi }_j\circ {\circ }_{\alpha }(X_1,\dots ,X_h)\right)\right)\hfill \\ \hfill & ={\pi }_j\circ {\circ }_{\alpha }(X_1,\dots ,X_h).\hfill \end{array}$$

Thus each coordinate of the original $(h,k)$-ary output is recovered exactly from the encoded output. Consequently, the assignment ${\circ }_{\alpha }\mapsto h_{\alpha }$ is faithful (injective on operations), and the $(h,k)$-ary SuperHyperstructure embeds into the multiary $(m,n)$-superhyperstructure $\left(U;{\left(h_{\alpha }\right)}_{\alpha \in I}\right)$. □

2.3. Some Examples for Multiary Structure

In this subsection, we examine how several concepts can be defined using the framework of a Multiary Structure. Concrete examples are provided below.

Example 20

(Multiary Topological Operator — closing a gap between coverage zones). Let $X=\mathbb{R}$ with its usual topology and take two open intervals

$$U_1=(0,1),U_2=(1,2).$$

Define the 2-ary operator $\mathrm{ClInt}(U_1,U_2):=int\left(\overline{U_1\cup U_2}\right)$. Meaning. Two Wi-Fi coverage zones just touch at 1; the “interior of the closure’’ fills the seam. Computation. $U_1\cup U_2=(0,1)\cup (1,2)$, so $\overline{U_1\cup U_2}=[0,2]$ and

$$\mathrm{ClInt}(U_1,U_2)=int\left([0,2]\right)=(0,2),$$

which adds the missing junction point 1 to the open coverage.

Example 21

(Multiary Metric Space — a 3-point “diameter’’ for carpool pickup). (cf.[33]) Define $D:{\mathbb{R}}^3\to [0,\infty )$ by

$$D(x,y,z):=max\{x,y,z\}-min\{x,y,z\}.$$

Meaning. The pickup spread of three riders is the farthest distance between any two of their locations on a line.Computation.For $(x,y,z)=(10,13,18)$ (kilometers along a corridor),

$$D(10,13,18)=18-10=8.$$

Smaller D means tighter clustering (easier pickup routing).

Example 22

(Multiary Convolution Algebra — total pipeline latency from three stages). (cf.[34,35]) Consider discrete signals on $\mathbb{Z}$ with convolution *. Let

$$f_1={\mathbf{1}}_{\{0,1\}},f_2={\mathbf{1}}_{\{0,2\}},f_3={\mathbf{1}}_{\left\{1\right\}},$$

where ${\mathbf{1}}_A\left(t\right)=1$ if $t\in A$ and 0 otherwise. Meaning. Stage 1 takes 0 or 1 time units, Stage 2 takes 0 or 2, Stage 3 takes 1. Computation. The 3-ary convolution

$$(f_1*f_2*f_3)\left(t\right)=\sum _{u+v+w=t}{f}_1\left(u\right)f_2\left(v\right)f_3\left(w\right)$$

has support $\{0,1\}+\{0,2\}+\left\{1\right\}=\{1,2,3,4\}$. In particular,

$$(f_1*f_2*f_3)\left(1\right)=1,\left(2\right)=2,\left(3\right)=1,\left(4\right)=1,$$

describing the distribution of total pipeline latency.

Example 23

(Multiary Capacity on Sets — joint risk coverage from three teams). Let the universe of controls be $X=\{a,b,c,d,e\}$. Define the 3-ary capacity

$${\mu }^{\left(3\right)}(A_1,A_2,A_3):={\displaystyle \frac{\left|A_1\cup A_2\cup A_3\right|}{\left|X\right|}}\in [0,1].$$

Meaning. Three audit teams flag subsets $A_i$; the score is the fraction of unique risks covered. Computation. With $A_1=\{a,b\}$, $A_2=\{b,c\}$, $A_3=\left\{d\right\}$,

$$A_1\cup A_2\cup A_3=\{a,b,c,d\},{\mu }^{\left(3\right)}=4/5=0.8.$$

Example 24

(Multiary Probability Coupler — barycenter of point forecasts). On $\mathbb{R}$, let the three distributions be point masses

$${\nu }_1={\delta }_0,{\nu }_2={\delta }_2,{\nu }_3={\delta }_4.$$

Define the fused forecast as the 2-Wasserstein barycenter

$${\mathrm{Fuse}}_3({\nu }_1,{\nu }_2,{\nu }_3):=\mathrm{ar}g\underset{\nu }{min}\sum _{i=1}^3{W}_2{(\nu ,{\nu }_i)}^2.$$

Meaning. Combine three scenario forecasts into one central plan. Computation (1D, equal weights). The barycenter is the point mass at the mean:

$${\mathrm{Fuse}}_3={\delta }_{(0+2+4)/3}={\delta }_2.$$

Example 25

(Multiary Graph Product Operator — multilayer “AND’’ connectivity). (cf.[36]) Let $G_1=(\{0,1\},\left\{\{0,1\}\right\})$ and $G_2=(\{0,1\},\left\{\{0,1\}\right\})$ be undirected edges. Define the product graph on $V=\{0,1\}\times \{0,1\}$ with adjacency

$$\{(u_1,u_2),(v_1,v_2)\}\in E⟺\{u_1,v_1\}\in E\left(G_1\right)\mathrm{AND}\{u_2,v_2\}\in E\left(G_2\right).$$

Meaning. Two cities are connected if both road layer and rail layer connect their components. Computation. The edges are

$$\left\{\right(0,0),(1,1\left)\right\},\left\{\right(0,1),(1,0\left)\right\},$$

i.e., the two diagonals on the $2\times 2$ grid.

Example 26

(Multiary Access-Control Hyperalgebra — consolidating roles). A hyperalgebra generalizes algebraic systems by replacing operations with hyperoperations, producing sets of possible results instead of single outcomes [37,38,39]. Let the permission universe be $U=\{\mathrm{read},\mathrm{write},\mathrm{deploy}\}$. Given two departmental roles $R_1=\left\{\mathrm{read}\right\}$ and $R_2=\{\mathrm{read},\mathrm{write}\}$, define the hyperoperation

$$h(R_1,R_2):=\{R\subseteq U\mid R_1\subseteq R\subseteq R_1\cup R_2\}.$$

Meaning. All admissible enterprise policies between intersection and union. Computation. Since $R_1\cup R_2=\{\mathrm{read},\mathrm{write}\}$,

$$h(R_1,R_2)=\left\{\left\{\mathrm{read}\right\},\{\mathrm{read},\mathrm{write}\}\right\}.$$

Example 27

(Multiary Gradient Aggregator — robust trimming in federated learning). A gradient aggregator combines multiple gradient updates from distributed learners into a unified estimate, ensuring robustness, stability, and convergence efficiency (cf.[40,41]). Suppose $m=5$ one-dimensional gradients:

$$g_1=-1.0,g_2=-0.9,g_3=0.0,g_4=0.1,g_5=5.0.$$

Define ${\mathrm{Agg}}_5$ as the convex hull of the $\alpha =0.2$ trimmed set (drop top/bottom 20%). Meaning. Remove one extreme low and one extreme high client; average within survivors. Computation. Trim to $\{-0.9,0.0,0.1\}$; the convex hull is the interval

$${\mathrm{Agg}}_5(g_1,\dots ,g_5)=[-0.9,0.1],$$

a set of robust aggregate directions.

Example 28

(Multiary Temporal Modality — “since all prerequisites occurred earlier’’). Temporal modalities extend logic with time-based operators, expressing statements about necessity, possibility, or sequence of events across temporal dimensions (cf.[42,43,44]). Let discrete time $T=\{1,2,3,4,5\}$ and for $A\subseteq T$ set ${\mathrm{Past}}_t\left(A\right)=A\cap \{1,\dots ,t-1\}$. Let ${\phi }_1$ hold at $\{2,4\}$ and ${\phi }_2$ at $\{1,3,4\}$. Define the 2-ary modality

$${\mathrm{Since}}_2({\phi }_1,{\phi }_2):=\bigcup _{t\in T}\bigcap _{i=1}^2{\mathrm{Past}}_t\left(〚{\phi }_i〛\right).$$

Meaning.Times when both prerequisites have already happened sometime in the past. Computation. At $t=5$,

$${\mathrm{Past}}_5\left({\phi }_1\right)=\{2,4\},{\mathrm{Past}}_5\left({\phi }_2\right)=\{1,3,4\},$$

so the intersection is $\left\{4\right\}\ne ⌀$; thus $5\in {\mathrm{Since}}_2({\phi }_1,{\phi }_2)$.

Example 29

(Multiary Data-Imputation Hyperoperation — reconciling three partial records). Data-imputation is the statistical process of filling missing values with plausible estimates, preserving dataset integrity for analysis and prediction (cf.[45,46,47]). Fields: City $\in \{\mathrm{Tokyo},\mathrm{Osaka}\}$, Age $\in \{30,31,32\}$, Gender $\in \{\mathrm{F},\mathrm{M}\}$. Three partial records:

$$D_1=(\mathrm{City}=\mathrm{Tokyo},\mathrm{Age}=?,\mathrm{Gender}=\mathrm{F}),$$

$$D_2=(\mathrm{City}=?,\mathrm{Age}=31,\mathrm{Gender}=\mathrm{F}),$$

$$D_3=(\mathrm{City}=\mathrm{Tokyo},\mathrm{Age}=?,\mathrm{Gender}=?).$$

Define the 3-ary hyperoperation ${Impute}_3(D_1,D_2,D_3)$ as the set of fully filled records consistent with all evidence. Computation. Consistency forces

$$\mathrm{City}=\mathrm{Tokyo},\mathrm{Age}=31,\mathrm{Gender}=\mathrm{F}.$$

Hence

$${Impute}_3(D_1,D_2,D_3)=\left\{(\mathrm{Tokyo},31,\mathrm{F})\right\},$$

the unique jointly consistent completion.

3. Conclusion

In this paper, we investigated Multiary Hyperstructures and Multiary Superhyperstructures, which generalize these constructions. In future work, we expect further research to explore extensions incorporating Fuzzy Sets [48,49], Intuitionistic Fuzzy Sets [50,51], Hesitant Fuzzy Set [52,53], HyperFuzzy Set [54,55], Neutrosophic Sets [56,57,58], Quadripartitioned Neutrosophic Sets [59,60], and Plithogenic Sets [61,62].