HyperInterval-Valued and SuoerHyperInterval-Valued Fuzzy/Neutrosophic Set

1. Preliminaries

We collect the basic terminology and notation used in what follows. The definitions in this paper are assumed to be finite.

1.1. Interval Set, HyperInterval Set, and SuperHyperInterval Set

An interval set collects subsets A with lower bound $A_{\ell }$ and upper bound $A_u$, requiring $A_{\ell }\subseteq A\subseteq A_u$ [1,2,3,4,5]. A HyperInterval set assigns each base interval $[A_{\ell },A_u]$ a nonempty family of admissible interval sets, satisfying $A_{\ell }\subseteq B_{\ell }\subseteq B_u\subseteq A_u$ [6]. A SuperHyperInterval set of order n maps elements of ${\mathcal{P}}^n\left(U\right)$ to $(n-1)$-nested families of interval sets, supporting hierarchical interval evidence [6].

Definition 1  

(Universe). Let U be a nonempty finite set, called theuniverseorbase set. All subsequent powerset constructions are formed relative to U.

Definition 2  

(Powerset [7]). Thepowersetof a set S, denoted $\mathcal{P}\left(S\right)$, is the family of all subsets of S, including both the empty set and S itself:

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

Definition 3  

(n-th Powerset [8,9,10,11]). For a nonempty set H and integer $n\ge 1$, then-th powersetis defined recursively by

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

Analogously, then-th nonempty powerset, denoted ${\mathcal{P}}_n^{*}\left(H\right)$, is constructed 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 for Menu Planning Across Horizons). Let the base set (“atomic choices”) be $H=\{\mathrm{Salad},\mathrm{Soup},\mathrm{Pasta}\}$.First powerset${\mathcal{P}}_1\left(H\right)=\mathcal{P}\left(H\right)$ lists all meal options as sets of dishes (e.g., ⌀, $\left\{\mathrm{Salad}\right\}$, $\{\mathrm{Soup},\mathrm{Pasta}\}$, $\{\mathrm{Salad},\mathrm{Soup},\mathrm{Pasta}\}$). ASecond powersetelement is a collection of such options; interpret each as adaily plan with admissible alternatives. For instance,

$$D=\left\{\left\{\mathrm{Salad}\right\},\{\mathrm{Soup},\mathrm{Pasta}\}\right\}\in {\mathcal{P}}_2\left(H\right)$$

means “one acceptable meal is just Salad, another acceptable choice is Soup together with Pasta.” AThird powersetelement bundles multiple daily plans into aweekly (or multi-day) template; for example

$$W=\left\{D,\left\{\right\{\mathrm{Soup}\},\{\mathrm{Pasta}\left\}\right\}\right\}\in {\mathcal{P}}_3\left(H\right)$$

encodes a set of admissible daily-plan choices (e.g., day one uses D, day two allows either only Soup or only Pasta).

Thus, moving from ${\mathcal{P}}_1\left(H\right)$ to ${\mathcal{P}}_2\left(H\right)$ to ${\mathcal{P}}_3\left(H\right)$ adds planning layers: fromsets of dishes(meal options), tosets of meal options(daily plans with alternatives), tosets of daily plans(weekly templates). In everyday terms, the n-th powerset models a hierarchy of “choices of choices’’ across n decision horizons (dish → day → week → ...).

Definition 4  

(Interval set). (cf. [1]) Let U be a nonempty universe and let $\mathcal{P}\left(U\right)$ be its powerset, ordered by ⊆. For any two subsets $A_{\ell },A_u\subseteq U$ with $A_{\ell }\subseteq A_u$, theinterval setdetermined by $(A_{\ell },A_u)$ is

$$[A_{\ell },A_u]:=\{A\subseteq U\mid A_{\ell }\subseteq A\subseteq A_u\}.$$

We call $A_{\ell }$ thelower boundand $A_u$ theupper bound. The class of all (closed) interval sets on U is

$$\mathsf{I}\left(\mathcal{P}\left(U\right)\right):=\{[A_{\ell },A_u]:A_{\ell },A_u\subseteq U,A_{\ell }\subseteq A_u\}.$$

Remark 1.  

Each $[A_{\ell },A_u]$ is a complete sublattice of $\left(\mathcal{P}\right(U),\subseteq )$ with bottom $A_{\ell }$ and top $A_u$. Degenerate cases: $[A,A]=\left\{A\right\}$ (a crisp set) and $[⌀,U]=\mathcal{P}\left(U\right)$ (the largest interval set).

Example 2  

(Interval set for a Shopping List Envelope). Let the universe of items be

$$U=\{\mathrm{milk},\mathrm{bread},\mathrm{eggs},\mathrm{apples},\mathrm{cheese}\}$$

. Suppose themust–buycore is $A_{\ell }=\{\mathrm{milk},\mathrm{eggs}\}$ and themay–buysuperset is $A_u=\{\mathrm{milk},\mathrm{bread},\mathrm{eggs},\mathrm{apples}\}$ (cheese is excluded today). The interval set

$$[A_{\ell },A_u]=\{A\subseteq U\mid A_{\ell }\subseteq A\subseteq A_u\}$$

collects all acceptable shopping lists between these bounds. Since $|A_u\setminus A_{\ell }|=2$ (bread, apples), the members are exactly the $2^2=4$ sets

$$\{\mathrm{milk},\mathrm{eggs}\},\{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\},\{\mathrm{milk},\mathrm{eggs},\mathrm{apples}\},\{\mathrm{milk},\mathrm{eggs},\mathrm{bread},\mathrm{apples}\}.$$

For instance, $\{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\}\in [A_{\ell },A_u]$ while $\{\mathrm{milk},\mathrm{eggs},\mathrm{cheese}\}\notin [A_{\ell },A_u]$ (cheese $\notin A_u$).

Definition 5  

(HyperInterval set). Let $\mathsf{I}\left(\mathcal{P}\right(U\left)\right)$ be as in Definition 4. AHyperInterval seton U is a map

$$\mathrm{HI}:\mathsf{I}\left(\mathcal{P}\left(U\right)\right)⟶{\mathcal{P}}^{*}\left(\mathsf{I}\left(\mathcal{P}\right(U\left)\right)\right),$$

assigning to each base interval set $I=[A_{\ell },A_u]$ a nonempty family $\mathrm{HI}\left(I\right)\subseteq \mathsf{I}\left(\mathcal{P}\right(U\left)\right)$ of admissible interval sets.(Refinement semantics, optional.)If, moreover, every $[B_{\ell },B_u]\in \mathrm{HI}\left([A_{\ell },A_u]\right)$ satisfies $A_{\ell }\subseteq B_{\ell }\subseteq B_u\subseteq A_u$, we call $\mathrm{HI}$ arefinement HyperInterval set.

Remark 2.  

Via pointwise union and intersection, $({\mathrm{HI}}_1\bigsqcup {\mathrm{HI}}_2)\left(I\right):={\mathrm{HI}}_1\left(I\right)\cup {\mathrm{HI}}_2\left(I\right)$ and $({\mathrm{HI}}_1\sqcap {\mathrm{HI}}_2)\left(I\right):={\mathrm{HI}}_1\left(I\right)\cap {\mathrm{HI}}_2\left(I\right),$ the set of all HyperInterval sets on U inherits a natural (hyper)structure.

Example 3  

(HyperInterval set refining an Envelope by Store Policies). Let $I_0=[A_{\ell },A_u]$ be as in Example 2. Define a HyperInterval set $\mathrm{HI}$ by assigning to $I_0$ tworefinementscoming from different stores’ policies:

$$\mathrm{HI}\left(I_0\right)=\left\{I_1=[\{\mathrm{milk},\mathrm{eggs}\},\{\mathrm{milk},\mathrm{eggs},\mathrm{bread},\mathrm{apples}\}],I_2=[\{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\},\{\mathrm{milk},\mathrm{eggs},\mathrm{bread},\mathrm{apples}\}]\right\}.$$

Both satisfy $A_{\ell }\subseteq B_{\ell }\subseteq B_u\subseteq A_u$. Theconjunctive core(intersection in the lattice $\mathcal{P}\left(U\right)$) is

$${\bigcap }_{k=1}^2{I}_k$$

$$=\left[\underset{=\{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\}}{\underbrace{\{\mathrm{milk},\mathrm{eggs}\}\cup \{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\}}},\underset{=\{\mathrm{milk},\mathrm{eggs},\mathrm{bread},\mathrm{apples}\}}{\underbrace{\{\mathrm{milk},\mathrm{eggs},\mathrm{bread},\mathrm{apples}\}\cap \{\mathrm{milk},\mathrm{eggs},\mathrm{bread},\mathrm{apples}\}}}\right]$$

$$=\left[\right\{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\},\{\mathrm{milk},\mathrm{eggs},\mathrm{bread},\mathrm{apples}\left\}\right],$$

which is feasible since the lower bound is contained in the upper bound. Thedisjunctive hull(least interval containing $I_1\cup I_2$) equals

$$\left[\{\mathrm{milk},\mathrm{eggs}\}\cap \{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\},\{\mathrm{milk},\mathrm{eggs},\mathrm{bread},\mathrm{apples}\}\cup \{\mathrm{milk},\mathrm{eggs},\mathrm{bread},\mathrm{apples}\}\right]=[A_{\ell },A_u]=I_0.$$

Thus the hyper-interval captures store–specific refinements without changing the global hull.

Definition 6  

(SuperHyperInterval set of order n). For $n\ge 0$ define the iterated powersets by ${\mathcal{P}}^0\left(U\right)=U$ and ${\mathcal{P}}^{n+1}\left(U\right)=\mathcal{P}\left({\mathcal{P}}^n\left(U\right)\right)$. Also write ${\mathcal{P}}^r\left(\mathsf{I}\left(\mathcal{P}\right(U\left)\right)\right)$ for iterated powersets of the interval–set universe. ASuperHyperInterval set of order $n\ge 1$on U is a map

$${\mathrm{SHI}}^{\left(n\right)}:{\mathcal{P}}^n\left(U\right)⟶{\mathcal{P}}^{n-1}\left(\mathsf{I}\left(\mathcal{P}\right(U\left)\right)\right).$$

Thus, to each n–nested subset $A\in {\mathcal{P}}^n\left(U\right)$ the map assigns an $(n-1)$–nested family of interval sets. (When a refinement discipline is desired, one can require that the interval bounds appearing at the leaves are chosen compatibly with the subsets occurring in A.)

(order $n=2$) for Two Scenarios of a Dinner Plan).

Example 4 (SuperHyperInterval set (order n=2) for Two Scenarios of a Dinner Plan. Keep

$$U=\{\mathrm{milk},\mathrm{bread},\mathrm{eggs},\mathrm{apples},\mathrm{cheese}\}$$

. Consider two scenario sets (elements of $\mathcal{P}\left(U\right)$):

$$S_{\mathrm{family}}=\{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\},$$

$$S_{\mathrm{guests}}=\{\mathrm{bread},\mathrm{cheese}\}.$$

Form the nested input $A=\{S_{\mathrm{family}},S_{\mathrm{guests}}\}\in {\mathcal{P}}^2\left(U\right)$. Define the SuperHyperInterval map of order 2 by

$${\mathrm{SHI}}^{\left(2\right)}\left(A\right)=\{I_{\mathrm{family}}=[\{\mathrm{milk},\mathrm{eggs}\},\{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\}],$$

$$I_{\mathrm{guests}}=[\left\{\mathrm{bread}\right\},\{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\}]\}\in \mathcal{P}\left(\mathsf{I}\left(\mathcal{P}\right(U\left)\right)\right).$$

Here the family scenario insists on milk and eggs; the guest scenario insists on bread but allows the same upper bound to keep cooking logistics unified.

If one demands a list acceptable tobothscenarios, the core is

$$I_{\mathrm{family}}\cap I_{\mathrm{guests}}=\left[\{\mathrm{milk},\mathrm{eggs}\}\cup \left\{\mathrm{bread}\right\},\{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\}\cap \{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\}\right]$$

$$=\left[\right\{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\},\{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\left\}\right],$$

a crisp recommendation $\{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\}$. If instead one allows either scenario, the hull is

$$\left[\{\mathrm{milk},\mathrm{eggs}\}\cap \left\{\mathrm{bread}\right\},\{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\}\cup \{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\}\right]$$

$$=[⌀,\{\mathrm{milk},\mathrm{eggs},\mathrm{bread}\left\}\right],$$

recording every intermediate list up to the common upper bound. This illustrates how order-2 nesting organizesscenario familiesbefore interval selection.

1.2. Interval-Valued Fuzzy Set

A fuzzy set assigns each element a membership degree between zero and one, modeling belonging and vagueness beyond crisp classification [12,13]. An interval-valued fuzzy set assigns to each element u an interval $[\alpha ,\beta ]\subseteq [0,1]$ of admissible membership degrees, modeling imprecision about exact values [14,15,16,17,18,19]. As related concepts, interval-valued intuitionistic fuzzy sets [20,21,22], interval-valued picture fuzzy sets [23,24], and interval-valued hesitant fuzzy sets [25,26,27] have also been studied in the literature.

Definition 7  

(Interval-valued fuzzy set). [14,15,16] Let $U\ne ⌀$ be a universe. Write

$$L\left(\right[0,1\left]\right)=\left\{\right[\alpha ,\beta ]\mid 0\le \alpha \le \beta \le 1\},$$

the set of allclosedsubintervals of $[0,1]$. Aninterval-valued fuzzy set (IVFS)on U is a mapping

$$A:U⟶L\left(\right[0,1\left]\right),$$

so that each $u\in U$ is assigned an interval $A\left(u\right)=[\underline{A}\left(u\right),\overline{A}\left(u\right)]\in L\left([0,1]\right)$ of admissible membership degrees. We denote the class of all IVFSs on U by $\mathrm{IVFS}\left(U\right)$.

Example 5  

(IVFS for Fruit Ripeness in a Grocery Store). Let $U=\{{\mathrm{A}}_1,{\mathrm{A}}_2,{\mathrm{A}}_3\}$ be three avocados on display. Consider the interval-valued fuzzy set $A:U\to L\left(\right[0,1\left]\right)$ where $A\left(u\right)$ is the degree to which u is “ripe enough to eat tonight.” Sensor readings (color, firmness) and staff judgment are summarized as

$$A\left({\mathrm{A}}_1\right)=[0.65,0.82],A\left({\mathrm{A}}_2\right)=[0.30,0.50],A\left({\mathrm{A}}_3\right)=[0.75,0.90].$$

A conservative customer requires membership at least $\alpha =0.70$ under thenecessaryview (use lower bounds), yielding the α-cut

$$N_{0.70}=\{u\in U\mid \underline{A}\left(u\right)\ge 0.70\}=\left\{{\mathrm{A}}_3\right\}.$$

An optimistic customer accepts if it ispossibleto meet $\alpha =0.70$ (use upper bounds), giving

$$P_{0.70}=\{u\in U\mid \overline{A}\left(u\right)\ge 0.70\}=\{{\mathrm{A}}_1,{\mathrm{A}}_3\}.$$

(Optionally) the IVFS complement $A^{\complement }\left(u\right)=[1-\overline{A}\left(u\right),1-\underline{A}\left(u\right)]$ quantifies “not ripe tonight”: for ${\mathrm{A}}_1$, $A^{\complement }\left({\mathrm{A}}_1\right)=[0.18,0.35]$.

1.3. Interval-Valued Neutrosophic Set

A neutrosophic set assigns independent truth, indeterminacy, and falsity degrees to each element, capturing inconsistency and uncertainty beyond fuzzy membership [28,29,30]. An interval-valued neutrosophic set assigns each element intervals for truth, indeterminacy, and falsity, allowing independent bounded ranges for all three [31,32,33,34,35].

Definition 8 (Interval-valued neutrosophic set (IVNS)). Let U be a nonempty universe. Aninterval-valued neutrosophic set A on U is specified by three maps

$$T_A,I_A,F_A:U⟶\mathrm{Int}\left([0,1]\right),$$

assigning to each $u\in U$ closed intervals

$$T_A\left(u\right)=[T_A^{-}\left(u\right),T_A^{+}\left(u\right)],I_A\left(u\right)=[I_A^{-}\left(u\right),I_A^{+}\left(u\right)],F_A\left(u\right)=[F_A^{-}\left(u\right),F_A^{+}\left(u\right)]$$

interpreted respectively as thetruth,indeterminacy, andfalsitymembership degrees of u. These components are independent; the only numeric bound required is

$$0\le T_A^{+}\left(u\right)+I_A^{+}\left(u\right)+F_A^{+}\left(u\right)\le 3,\forall u\in U.$$

Equivalently, one writes

$$A=\left\{\langle T_A\left(u\right),I_A\left(u\right),F_A\left(u\right)\rangle /u\in U\right\}.$$

Example 6  

(IVNS for Spam Detection of a Single Email). Let $U=\left\{e\right\}$ where e is an incoming email. Define an IVNS $S=(T,I,F)$ for the statement “e is spam.” From sender reputation, content filters, and user history we obtain

$$T\left(e\right)=[0.72,0.86],I\left(e\right)=[0.10,0.20],F\left(e\right)=[0.04,0.12].$$

The numeric constraint holds:

$$T^{+}\left(e\right)+I^{+}\left(e\right)+F^{+}\left(e\right)=0.86+0.20+0.12=1.18\le 3.$$

A conservative rule declares spam when $T^{-}\left(e\right)\ge 0.70$ and $F^{+}\left(e\right)\le 0.20$. Here $T^{-}\left(e\right)=0.72(\ge 0.70)$ and $F^{+}\left(e\right)=0.12(\le 0.20)$, so e is classified as spam. The interval $T\left(e\right)=[0.72,0.86]$ captures admissible truth, $I\left(e\right)=[0.10,0.20]$ the uncertainty due to weak indicators, and $F\left(e\right)=[0.04,0.12]$ the bounded counter-evidence (e.g., some benign content).

1.4. Multistructure

MultiStructure is a carrier set equipped with indexed multi-operations mapping tuples to sets of outcomes, enabling nondeterministic, multi-arity algebraic computation [36,37].

Definition 9  

(MultiOperation). Fix an integer $m\ge 1$ and let H be a nonempty set. Anm-ary multi-operationon H is a map

$${\#}^{\left(m\right)}:H^m⟶\mathcal{M}\left(H\right),(x_1,\dots ,x_m)\mapsto {\#}^{\left(m\right)}(x_1,\dots ,x_m),$$

assigning to each m-tuple $(x_1,\dots ,x_m)$ a finite multiset of elements of H rather than a single element.

Definition 10  

(MultiStructure). AMultiStructureis a pair

$$\mathcal{MS}=\left(H,{\{{\#}^{\left(m\right)}:H^m\to \mathcal{M}\left(H\right)\}}_{m\in \mathcal{I}}\right),$$

where H is a nonempty carrier set and $\mathcal{I}\subseteq {\mathbb{Z}}_{>0}$ indexes a family of multi-operations of various arities. No further axioms are imposed unless explicitly stated.

Example 7  

(MultiStructure on Interval Sets for Aggregating Requirements). Let $H=\mathsf{I}\left(\mathcal{P}\right(U\left)\right)$ with U as above, and take $\mathcal{M}\left(H\right)=\mathcal{P}\left(H\right)$. Define two multi-operations for any $m\ge 1$ and intervals $I_r=[L_r,U_r]\in H$:

$${\#}_{\wedge }^{\left(m\right)}(I_1,\dots ,I_m)=\left\{\begin{array}{cc}\left\{[\bigcup _{r=1}^m{L}_r,\bigcap _{r=1}^m{U}_r]\right\},\hfill & \mathrm{if}\bigcup _r{L}_r\subseteq \bigcap _r{U}_r,\hfill \\ ⌀,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

$${\#}_{\vee \mathrm{hull}}^{\left(m\right)}(I_1,\dots ,I_m)=\left\{[\bigcap _{r=1}^m{L}_r,\bigcup _{r=1}^m{U}_r]\right\}.$$

Then $\mathcal{MS}=(H,{\{{\#}_{\wedge }^{\left(m\right)},{\#}_{\vee \mathrm{hull}}^{\left(m\right)}\}}_{m\ge 1})$ is a MultiStructure.

Concrete computation (two requirements).Let

$$I_1=[\left\{\mathrm{milk}\right\},\{\mathrm{milk},\mathrm{bread},\mathrm{eggs}\}],$$

$$I_2=[\left\{\mathrm{bread}\right\},\{\mathrm{milk},\mathrm{bread},\mathrm{eggs}\}].$$

Then

$${\#}_{\wedge }^{\left(2\right)}(I_1,I_2)=\left\{\left[\left\{\mathrm{milk}\right\}\cup \left\{\mathrm{bread}\right\},\{\mathrm{milk},\mathrm{bread},\mathrm{eggs}\}\cap \{\mathrm{milk},\mathrm{bread},\mathrm{eggs}\}\right]\right\}$$

$$=\left\{\left[\right\{\mathrm{milk},\mathrm{bread}\},\{\mathrm{milk},\mathrm{bread},\mathrm{eggs}\left\}\right]\right\},$$

while

$${\#}_{\vee \mathrm{hull}}^{\left(2\right)}(I_1,I_2)=\left\{\left[\left\{\mathrm{milk}\right\}\cap \left\{\mathrm{bread}\right\},\{\mathrm{milk},\mathrm{bread},\mathrm{eggs}\}\cup \{\mathrm{milk},\mathrm{bread},\mathrm{eggs}\}\right]\right\}$$

$$=\left\{[⌀,\{\mathrm{milk},\mathrm{bread},\mathrm{eggs}\left\}\right]\right\}.$$

Thus, the ∧–operation returns thejointlyacceptable envelope, and the ∨–hull returns theleastenvelope containing both requirements—illustrating multi-arity, set-valued outputs in a real aggregation workflow.

2. Main Results

In this section, we present the main results of this paper.

2.1. MultiInterval-Valued Fuzzy Set

A MultiInterval-valued Fuzzy Set assigns each element a finite family of membership intervals, aggregating evidence using intersection cores and hulls.

Definition 11  

(MultiInterval). Let $U\ne ⌀$ and let J be a finite, nonempty index set. AMultiIntervalon U is a family

$$\mathbf{I}=\left\{[A_{\ell }^{\left(j\right)},A_u^{\left(j\right)}]|j\in J,A_{\ell }^{\left(j\right)}\subseteq A_u^{\left(j\right)}\subseteq U\right\}\in {\mathcal{P}}^{*}\left(\mathsf{I}\left(\mathcal{P}\right(U\left)\right)\right).$$

Itsconjunctive semantics(feasible core) is the set

$$〚\mathbf{I}{〛}_{\wedge }:=\bigcap _{j\in J}[A_{\ell }^{\left(j\right)},A_u^{\left(j\right)}]=\left\{A\subseteq U|\forall j\in J:A_{\ell }^{\left(j\right)}\subseteq A\subseteq A_u^{\left(j\right)}\right\}.$$

Itsdisjunctive hullis the smallest interval set containing ${\bigcup }_{j\in J}[A_{\ell }^{\left(j\right)},A_u^{\left(j\right)}]$:

$$〚\mathbf{I}{〛}_{\vee }^{\mathrm{hull}}:=\left[\bigcap _{j\in J}{A}_{\ell }^{\left(j\right)},\bigcup _{j\in J}{A}_u^{\left(j\right)}\right].$$

Lemma 1  

(Exact formula for the conjunctive core). Let $\mathbf{I}$ be as in Definition 11 and set

$$L^{\wedge }:=\bigcup _{j\in J}{A}_{\ell }^{\left(j\right)},U^{\wedge }:=\bigcap _{j\in J}{A}_u^{\left(j\right)}.$$

Then

$$\bigcap _{j\in J}[A_{\ell }^{\left(j\right)},A_u^{\left(j\right)}]=\left\{\begin{array}{cc}[L^{\wedge },U^{\wedge }],\hfill & \mathrm{if}{L}^{\wedge }\subseteq U^{\wedge },\hfill \\ ⌀,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Proof. (⊆) Let $A\in {\bigcap }_j[A_{\ell }^{\left(j\right)},A_u^{\left(j\right)}]$. Then $A_{\ell }^{\left(j\right)}\subseteq A\subseteq A_u^{\left(j\right)}$ for all j. Taking unions of the left bounds gives $L^{\wedge }={\bigcup }_j{A}_{\ell }^{\left(j\right)}\subseteq A$, and taking intersections of the right bounds gives $A\subseteq {\bigcap }_j{A}_u^{\left(j\right)}=U^{\wedge }$. Hence $A\in [L^{\wedge },U^{\wedge }]$, which already forces $L^{\wedge }\subseteq U^{\wedge }$ whenever the intersection is nonempty.

(⊇) Conversely, if $L^{\wedge }\subseteq U^{\wedge }$ and $A\in [L^{\wedge },U^{\wedge }]$, then $A_{\ell }^{\left(j\right)}\subseteq L^{\wedge }\subseteq A$ and $A\subseteq U^{\wedge }\subseteq A_u^{\left(j\right)}$ for each j, so $A\in [A_{\ell }^{\left(j\right)},A_u^{\left(j\right)}]$. Therefore A lies in the intersection. □

Remark 3  

(Order and feasibility). Define a preorder ⪯ on MultiIntervals by ${\mathbf{I}}_1⪯{\mathbf{I}}_2$ iff $〚{\mathbf{I}}_1{〛}_{\wedge }\subseteq 〚{\mathbf{I}}_2{〛}_{\wedge }$. By Lemma 1, feasibility of $\mathbf{I}$ is equivalent to the numeric inequality ${\bigcup }_j{A}_{\ell }^{\left(j\right)}\subseteq {\bigcap }_j{A}_u^{\left(j\right)}$.

Let $H:=\mathsf{I}\left(\mathcal{P}\right(U\left)\right)$. We define two canonical multi-operations on H.

Definition 12  

(Meet and hull multi-operations on H). For $m\ge 1$ and $(I_1,\dots ,I_m)\in H^m$ with $I_r=[L_r,U_r]$, set

$${\#}_{\wedge }^{\left(m\right)}(I_1,\dots ,I_m):=\left\{\begin{array}{cc}\left\{[\bigcup _{r=1}^m{L}_r,\bigcap _{r=1}^m{U}_r]\right\},\hfill & \mathrm{if}\bigcup L_r\subseteq \bigcap U_r,\hfill \\ ⌀,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

and

$${\#}_{\vee \mathrm{hull}}^{\left(m\right)}(I_1,\dots ,I_m):=\left\{[\bigcap _{r=1}^m{L}_r,\bigcup _{r=1}^m{U}_r]\right\}.$$

Then ${\mathcal{MS}}_U:=(H,{\{{\#}_{\wedge }^{\left(m\right)},{\#}_{\vee \mathrm{hull}}^{\left(m\right)}\}}_{m\ge 1})$ is a MultiStructure in the sense of Definition 10.

Proposition 1  

(MultiInterval as a MultiStructure computation). Let $\mathbf{I}={\left\{I_j\right\}}_{j\in J}$ be a MultiInterval on U with $I_j=[A_{\ell }^{\left(j\right)},A_u^{\left(j\right)}]$. Then

$$〚\mathbf{I}{〛}_{\wedge }=\bigcap _{j\in J}{I}_j=\left\{\begin{array}{cc}{I}^{★},\hfill & \mathrm{if}{\#}_{\wedge }^{\left(\right|J\left|\right)}\left({\left(I_j\right)}_{j\in J}\right)=\left\{I^{★}\right\},\hfill \\ ⌀,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

with $I^{★}=\left[{\bigcup }_j{A}_{\ell }^{\left(j\right)},{\bigcap }_j{A}_u^{\left(j\right)}\right]$. Moreover, the hull semantics satisfies

$$〚\mathbf{I}{〛}_{\vee }^{\mathrm{hull}}={\#}_{\vee \mathrm{hull}}^{\left(\right|J\left|\right)}\left({\left(I_j\right)}_{j\in J}\right).$$

Proof. 

Immediate from Definition 12 and Lemma 1, by expanding unions and intersections of bounds elementwise. □

Definition 13  

(Singleton embedding). Define $\iota :\mathsf{I}\left(\mathcal{P}\left(U\right)\right)\to {\mathcal{P}}^{*}\left(\mathsf{I}\left(\mathcal{P}\left(U\right)\right)\right)$ by

$$\iota \left([A_{\ell },A_u]\right):=\left\{[A_{\ell },A_u]\right\}.$$

We identify $\iota \left([A_{\ell },A_u]\right)$ with a MultiInterval having index set $J=\left\{1\right\}$.

Theorem 1  

(MultiInterval generalizes Interval). For every interval set $I=[A_{\ell },A_u]\in \mathsf{I}\left(\mathcal{P}\left(U\right)\right)$,

$$〚\iota \left(I\right){〛}_{\wedge }=I\mathrm{and}〚\iota \left(I\right){〛}_{\vee }^{\mathrm{hull}}=I.$$

Hence the map ι is an order-embedding from $\left(\mathsf{I}\right(\mathcal{P}\left(U\right)),\subseteq )$ into the preorder of MultiIntervals under ⪯, and MultiIntervals strictly generalize intervals.

Proof. 

Let $J=\left\{1\right\}$ and $I=[A_{\ell },A_u]$. By Lemma 1,

$$〚\iota \left(I\right){〛}_{\wedge }=\bigcap _{j\in J}[A_{\ell }^{\left(j\right)},A_u^{\left(j\right)}]=\left[\bigcup _{j\in J}{A}_{\ell }^{\left(j\right)},\bigcap _{j\in J}{A}_u^{\left(j\right)}\right]=[A_{\ell },A_u]=I,$$

since the union and intersection over a singleton index set return $A_{\ell }$ and $A_u$ respectively. The hull equality is analogous:

$$〚\iota \left(I\right){〛}_{\vee }^{\mathrm{hull}}=\left[\bigcap _{j\in J}{A}_{\ell }^{\left(j\right)},\bigcup _{j\in J}{A}_u^{\left(j\right)}\right]=[A_{\ell },A_u]=I.$$

Monotonicity of $\iota$ with respect to ⊆ and ⪯ follows directly from these equalities. □

Definition 14 (MultiInterval-valued fuzzy set (MIVFS)) Let $U\ne ⌀$ and $L\left(\right[0,1\left]\right)$ as above. AMIVFSon U assigns to each $u\in U$ a finite nonempty family of numeric intervals

$$\mathcal{A}\left(u\right)=\left\{[{\alpha }_j\left(u\right),{\beta }_j\left(u\right)]\in L\left([0,1]\right)|j\in J\left(u\right)\mathrm{finite},\mathrm{nonempty}\right\}.$$

Its conjunctive (intersection) semantics and disjunctive (hull) semantics are

$$〚\mathcal{A}\left(u\right){〛}_{\wedge }=\left\{\begin{array}{cc}[\underset{j}{max}{\alpha }_j\left(u\right),\underset{j}{min}{\beta }_j\left(u\right)],\hfill & \mathrm{if}\underset{j}{max}{\alpha }_j\left(u\right)\le \underset{j}{min}{\beta }_j\left(u\right),\hfill \\ ⌀,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

$$〚\mathcal{A}\left(u\right){〛}_{\vee }^{\mathrm{hull}}=[\underset{j}{min}{\alpha }_j\left(u\right),\underset{j}{max}{\beta }_j\left(u\right)].$$

Example 8  

(MIVFS for Software Release Readiness Today). Let $U=\left\{\nu \right\}$, where ν denotes the proposition

$$``\mathrm{Version}\mathrm{v}1.2\mathrm{is}\mathrm{ready}\mathrm{for}\mathrm{production}\mathrm{today}.’’$$

As a MultiInterval-valued fuzzy set, we assign to ν multiple membership intervals from independent engineering sources:

$$\mathcal{A}\left(\nu \right)=\{I_1,I_2,I_3,I_4\}\subseteq L\left([0,1]\right),$$

with

$$\begin{array}{cc}\hfill I_1& =[0.78,0.88](\mathrm{unit}+\mathrm{integration}\mathrm{test}\mathrm{trends}),\hfill \\ \hfill I_2& =[0.72,0.85](\mathrm{QA}\mathrm{exploratory}\mathrm{testing}\mathrm{outcomes}),\hfill \\ \hfill I_3& =[0.80,0.90](\mathrm{SRE}\mathrm{load}/\mathrm{performance}\mathrm{confidence}),\hfill \\ \hfill I_4& =[0.75,0.83](\sec \mathrm{urity}/\mathrm{compliance}\mathrm{quick}\mathrm{audit}).\hfill \end{array}$$

By Definition (MIVFS), the conjunctive (intersection) semantics and the disjunctive (hull) semantics are

$$〚\mathcal{A}\left(\nu \right){〛}_{\wedge }=\left\{\begin{array}{cc}[\underset{j}{max}{\alpha }_j,\underset{j}{min}{\beta }_j],\hfill & \mathrm{if}\underset{j}{max}{\alpha }_j\le \underset{j}{min}{\beta }_j,\hfill \\ ⌀,\hfill & \mathrm{otherwise},\hfill \end{array}\right.〚\mathcal{A}\left(\nu \right){〛}_{\vee }^{\mathrm{hull}}=[\underset{j}{min}{\alpha }_j,\underset{j}{max}{\beta }_j],$$

where $I_j=[{\alpha }_j,{\beta }_j]$.

We compute the bounds explicitly.

Lower endpoints:

$${\alpha }_1=0.78,{\alpha }_2=0.72,{\alpha }_3=0.80,{\alpha }_4=0.75⟹\underset{j}{max}{\alpha }_j=max\{0.78,0.72,0.80,0.75\}=0.80.$$

Upper endpoints:

$${\beta }_1=0.88,{\beta }_2=0.85,{\beta }_3=0.90,{\beta }_4=0.83⟹\underset{j}{min}{\beta }_j=min\{0.88,0.85,0.90,0.83\}=0.83.$$

Since $0.80\le 0.83$, the intersection is feasible and hence

$$〚\mathcal{A}\left(\nu \right){〛}_{\wedge }=[0.80,0.83].$$

Hull:

$$\underset{j}{min}{\alpha }_j=min\{0.78,0.72,0.80,0.75\}=0.72,\underset{j}{max}{\beta }_j=max\{0.88,0.85,0.90,0.83\}=0.90,$$

so

$$〚\mathcal{A}\left(\nu \right){〛}_{\vee }^{\mathrm{hull}}=[0.72,0.90].$$

The interval $[0.80,0.83]$ is the conservative consensus range endorsed byallengineering evidence streams, suited to a strict release gate. The hull $[0.72,0.90]$ captures the full plausible readiness reported byat least onestream, useful for exploratory planning and risk negotiation.

Proposition 2  

(Reduction to IVFS). If each $J\left(u\right)$ is a singleton, say $J\left(u\right)=\left\{1\right\}$ with $\mathcal{A}\left(u\right)=\left\{[{\alpha }_1\left(u\right),{\beta }_1\left(u\right)]\right\}$, then

$$〚\mathcal{A}\left(u\right){〛}_{\wedge }=〚\mathcal{A}\left(u\right){〛}_{\vee }^{\mathrm{hull}}=[{\alpha }_1\left(u\right),{\beta }_1\left(u\right)],$$

so every IVFS is a special case of a MIVFS.

Proof. 

Compute ${max}_j{\alpha }_j\left(u\right)={min}_j{\beta }_j\left(u\right)$ over a singleton index set. □

2.2. MultiInterval-Valued Neutrosophic Set

A MultiInterval-valued Neutrosophic Set assigns each element interval families for truth, indeterminacy, and falsity, aggregated componentwise by cores and hulls.

Definition 15  

(MIVNS). AMultiInterval-valued neutrosophic set (MIVNS)on U is a triple

$$\mathcal{A}=(\mathcal{T},\mathcal{I},\mathcal{F}),$$

where for each $u\in U$, $\mathcal{T}\left(u\right),\mathcal{I}\left(u\right),\mathcal{F}\left(u\right)$ are finite nonempty families of numeric intervals in $L\left(\right[0,1\left]\right)$. Conjunctive semantics are computed componentwise by interval intersection:

$$〚\mathcal{T}\left(u\right){〛}_{\wedge }=[\underset{j}{max}{T}_j^{-}\left(u\right),\underset{j}{min}{T}_j^{+}\left(u\right)],〚\mathcal{I}\left(u\right){〛}_{\wedge }=[\underset{j}{max}{I}_j^{-}\left(u\right),\underset{j}{min}{I}_j^{+}\left(u\right)],〚\mathcal{F}\left(u\right){〛}_{\wedge }=[\underset{j}{max}{F}_j^{-}\left(u\right),\underset{j}{min}{F}_j^{+}\left(u\right)],$$

with the usual feasibility conditions $max\le min$ in each component; hull semantics take $[min,max]$ componentwise.

Example 9  

(MIVNS for a Clinical Decision: “Pneumonia Present Today?”). Let $U=\left\{p\right\}$, where p denotes a particular patient. We assess the statement

$$``p\mathrm{has}\mathrm{community}–\mathrm{acquired}\mathrm{pneumonia}\left(\mathrm{CAP}\right)\mathrm{today}.’’$$

as a MultiInterval–valued neutrosophic set $\mathcal{A}=(\mathcal{T},\mathcal{I},\mathcal{F})$. Here $\mathcal{T}\left(p\right)$ collectssupportingevidence intervals (truth), $\mathcal{I}\left(p\right)$ collectsuncertaintyintervals (indeterminacy), and $\mathcal{F}\left(p\right)$ collectscounter–evidenceintervals (falsity). Each interval $[\alpha ,\beta ]\subseteq [0,1]$ encodes an admissible range for the corresponding degree.

Concrete (illustrative) sources and intervals:

$$\begin{array}{cc}\hfill \mathcal{T}\left(p\right)& =\left\{[0.72,0.88](\mathrm{chest}\mathrm{X}–\mathrm{ray}\mathrm{report}),[0.65,0.80](\mathrm{clinical}\mathrm{score}),[0.70,0.90](\mathrm{C}–\mathrm{reactive}\mathrm{protein}\mathrm{model})\right\},\hfill \\ \hfill \mathcal{I}\left(p\right)& =\left\{[0.10,0.25](\mathrm{equivocal}\mathrm{imaging}),[0.15,0.30](\mathrm{atypical}\mathrm{symptom}\mathrm{onset}),[0.05,0.20](\mathrm{comorbidity}\mathrm{confounding})\right\},\hfill \\ \hfill \mathcal{F}\left(p\right)& =\left\{[0.08,0.22](\mathrm{normal}\mathrm{WBC}),[0.12,0.18](\mathrm{viral}/\mathrm{bacterial} \mathrm{panel} \mathrm{result}),[0.05,0.15](\mathrm{stable}\mathrm{oxygenation})\right\}.\hfill \end{array}$$

By the MIVNS semantics, theconjunctive coreanddisjunctive hullare computedcomponentwiseusing interval intersection and least–containing–interval, respectively.

Truth component.

$$\begin{array}{cc}\hfill max\underline{\mathcal{T}}\left(p\right)& =max\{0.72,0.65,0.70\}=0.72,\hfill \\ \hfill min\overline{\mathcal{T}}\left(p\right)& =min\{0.88,0.80,0.90\}=0.80⟹〚\mathcal{T}\left(p\right){〛}_{\wedge }=[0.72,0.80](\mathrm{feasible}\sin \mathrm{ce}0.72\le 0.80),\hfill \\ \hfill min\underline{\mathcal{T}}\left(p\right)& =min\{0.72,0.65,0.70\}=0.65,max\overline{\mathcal{T}}\left(p\right)=max\{0.88,0.80,0.90\}=0.90\hfill \\ & \Rightarrow 〚\mathcal{T}\left(p\right){〛}_{\vee }^{\mathrm{hull}}=[0.65,0.90].\hfill \end{array}$$

Indeterminacy component.

$$\begin{array}{cc}\hfill max\underline{\mathcal{I}}\left(p\right)& =max\{0.10,0.15,0.05\}=0.15,\hfill \\ \hfill min\overline{\mathcal{I}}\left(p\right)& =min\{0.25,0.30,0.20\}=0.20⟹〚\mathcal{I}\left(p\right){〛}_{\wedge }=[0.15,0.20],\hfill \\ \hfill min\underline{\mathcal{I}}\left(p\right)& =min\{0.10,0.15,0.05\}=0.05,max\overline{\mathcal{I}}\left(p\right)=max\{0.25,0.30,0.20\}=0.30\hfill \\ & \Rightarrow 〚\mathcal{I}\left(p\right){〛}_{\vee }^{\mathrm{hull}}=[0.05,0.30].\hfill \end{array}$$

Falsity component.

$$\begin{array}{cc}\hfill max\underline{\mathcal{F}}\left(p\right)& =max\{0.08,0.12,0.05\}=0.12,\hfill \\ \hfill min\overline{\mathcal{F}}\left(p\right)& =min\{0.22,0.18,0.15\}=0.15⟹〚\mathcal{F}\left(p\right){〛}_{\wedge }=[0.12,0.15],\hfill \\ \hfill min\underline{\mathcal{F}}\left(p\right)& =min\{0.08,0.12,0.05\}=0.05,max\overline{\mathcal{F}}\left(p\right)=max\{0.22,0.18,0.15\}=0.22\hfill \\ & \Rightarrow 〚\mathcal{F}\left(p\right){〛}_{\vee }^{\mathrm{hull}}=[0.05,0.22].\hfill \end{array}$$

Vector–valued result.

$${\mathrm{core}}_{\wedge }\left(\mathcal{A}\right)\left(p\right)=\left([0.72,0.80],[0.15,0.20],[0.12,0.15]\right),$$

$${\mathrm{hull}}_{\vee }\left(\mathcal{A}\right)\left(p\right)=\left([0.65,0.90],[0.05,0.30],[0.05,0.22]\right).$$

The conjunctive core gives the consensus ranges simultaneously supported by all sources in each component (truth, indeterminacy, falsity). The hulls summarize the full plausible spans reported by at least one source. A conservative decision maker would consult the core; exploratory or safety–margin assessments can reference the hulls.

Theorem 2  

(MIVNS generalizes IVNS). Define the embedding

$$\iota :{\left(L\left(\right[0,1\left]\right)\right)}^3⟶{\left({\mathcal{P}}^{*}\left(L\left(\right[0,1\left]\right)\right)\right)}^3,\iota \left([a,b],[c,d],[e,f]\right):=\left(\left\{\right[a,b\left]\right\},\left\{\right[c,d\left]\right\},\left\{\right[e,f\left]\right\}\right).$$

If $A=(T,I,F)$ is an IVNS on U, then $\mathcal{A}:=\iota \circ A$ is a MIVNS and, for every $u\in U$,

$${\mathrm{core}}_{\wedge }\left(\mathcal{A}\right)\left(u\right)=(T\left(u\right),I\left(u\right),F\left(u\right))\mathrm{and}{\mathrm{hull}}_{\vee }\left(\mathcal{A}\right)\left(u\right)=(T\left(u\right),I\left(u\right),F\left(u\right)).$$

Hence every IVNS is a (canonically embedded) special case of a MIVNS.

Proof. 

Fix $u\in U$ and write $T\left(u\right)=[t^{-}\left(u\right),t^{+}\left(u\right)]$, $I\left(u\right)=[i^{-}\left(u\right),i^{+}\left(u\right)]$, $F\left(u\right)=[f^{-}\left(u\right),f^{+}\left(u\right)]$. By definition of $\iota$,

$$\mathcal{T}\left(u\right)=\left\{[t^{-}\left(u\right),t^{+}\left(u\right)]\right\},\mathcal{I}\left(u\right)=\left\{[i^{-}\left(u\right),i^{+}\left(u\right)]\right\},\mathcal{F}\left(u\right)=\left\{[f^{-}\left(u\right),f^{+}\left(u\right)]\right\}.$$

Evaluating maxima/minima over a singleton gives, componentwise,

$$\underset{[\alpha ,\beta ]\in \mathcal{T}\left(u\right)}{max}\alpha =t^{-}\left(u\right),\underset{[\alpha ,\beta ]\in \mathcal{T}\left(u\right)}{min}\beta =t^{+}\left(u\right)⟹{\mathrm{core}}_{\wedge }\left(\mathcal{T}\left(u\right)\right)=[t^{-}\left(u\right),t^{+}\left(u\right)]=T\left(u\right),$$

and similarly ${\mathrm{core}}_{\wedge }\left(\mathcal{I}\left(u\right)\right)=I\left(u\right)$, ${\mathrm{core}}_{\wedge }\left(\mathcal{F}\left(u\right)\right)=F\left(u\right)$. The hull equalities follow analogously: ${\mathrm{hull}}_{\vee }\left(\mathcal{T}\left(u\right)\right)=[t^{-}\left(u\right),t^{+}\left(u\right)]=T\left(u\right)$, etc. Thus ${\mathrm{core}}_{\wedge }\left(\mathcal{A}\right)\left(u\right)={\mathrm{hull}}_{\vee }\left(\mathcal{A}\right)\left(u\right)=(T\left(u\right),I\left(u\right),F\left(u\right))$. □

Theorem 3  

(MIVNS generalizes MIVFS). Let ${\mathcal{A}}_{\mathrm{F}}$ be a MIVFS on U. Define $\Phi \left({\mathcal{A}}_{\mathrm{F}}\right)$ to be the MIVNS $\left(\mathcal{T},\mathcal{I},\mathcal{F}\right)$ given by

$$\mathcal{T}\left(u\right):={\mathcal{A}}_{\mathrm{F}}\left(u\right),\mathcal{I}\left(u\right):=\left\{[0,0]\right\},\mathcal{F}\left(u\right):=\left\{[0,0]\right\},\mathrm{for}\mathrm{each}u\in U.$$

Then, for every $u\in U$,

$${\mathrm{core}}_{\wedge }\left(\mathcal{T}\left(u\right)\right)={\mathrm{core}}_{\wedge }\left({\mathcal{A}}_{\mathrm{F}}\left(u\right)\right),{\mathrm{hull}}_{\vee }\left(\mathcal{T}\left(u\right)\right)={\mathrm{hull}}_{\vee }\left({\mathcal{A}}_{\mathrm{F}}\left(u\right)\right),$$

and

$${\mathrm{core}}_{\wedge }\left(\mathcal{I}\left(u\right)\right)={\mathrm{hull}}_{\vee }\left(\mathcal{I}\left(u\right)\right)=[0,0],{\mathrm{core}}_{\wedge }\left(\mathcal{F}\left(u\right)\right)={\mathrm{hull}}_{\vee }\left(\mathcal{F}\left(u\right)\right)=[0,0].$$

Consequently, projecting the MIVNS $\Phi \left({\mathcal{A}}_{\mathrm{F}}\right)$ onto its truth component recovers exactly the original MIVFS semantics (both core and hull), so MIVNS strictly extends MIVFS.

Proof. 

Fix $u\in U$ and write ${\mathcal{A}}_{\mathrm{F}}\left(u\right)={\left\{[{\alpha }_j\left(u\right),{\beta }_j\left(u\right)]\right\}}_{j\in J\left(u\right)}$ with $J\left(u\right)$ finite nonempty. By definition of MIVNS core/hull, applied to $\mathcal{T}\left(u\right)={\mathcal{A}}_{\mathrm{F}}\left(u\right)$,

$${\mathrm{core}}_{\wedge }\left(\mathcal{T}\left(u\right)\right)=\left\{\begin{array}{cc}[\underset{j\in J\left(u\right)}{max}{\alpha }_j\left(u\right),\underset{j\in J\left(u\right)}{min}{\beta }_j\left(u\right)],\hfill & \mathrm{if}max{\alpha }_j\le min{\beta }_j,\hfill \\ ⌀,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

$${\mathrm{hull}}_{\vee }\left(\mathcal{T}\left(u\right)\right)=[\underset{j\in J\left(u\right)}{min}{\alpha }_j\left(u\right),\underset{j\in J\left(u\right)}{max}{\beta }_j\left(u\right)],$$

which are exactly the MIVFS core/hull for ${\mathcal{A}}_{\mathrm{F}}\left(u\right)$. For the neutrosophic indeterminacy and falsity components, each is the singleton family $\left\{\right[0,0\left]\right\}$; hence

$$\underset{[\alpha ,\beta ]\in \left\{\right[0,0\left]\right\}}{max}\alpha =0,\underset{[\alpha ,\beta ]\in \left\{\right[0,0\left]\right\}}{min}\beta =0⟹{\mathrm{core}}_{\wedge }\left(\left\{[0,0]\right\}\right)=[0,0],$$

and trivially ${\mathrm{hull}}_{\vee }\left(\left\{[0,0]\right\}\right)=[0,0]$. Therefore the truth-component of $\Phi \left({\mathcal{A}}_{\mathrm{F}}\right)$ reproduces the MIVFS, while the neutrosophic extras are neutralized at $[0,0]$. □

2.3. HyperInterval-Valued Fuzzy Set

A HyperInterval-Valued Fuzzy Set permits arbitrary sets of membership intervals per element, deriving intersection cores and minimal containing hulls afterward.

Definition 16 (HyperInterval-valued fuzzy set (HIVFS)). AHIVFSon U assigns to each $u\in U$ a nonempty family $\mathrm{H}\left(u\right)\subseteq L\left(\right[0,1\left]\right).$ Its conjunctive core and hull are, respectively,

$$〚\mathrm{H}\left(u\right){〛}_{\wedge }=\bigcap _{I\in \mathrm{H}\left(u\right)}I=\left\{\begin{array}{cc}[\underset{I\in \mathrm{H}\left(u\right)}{sup}infI,\underset{I\in \mathrm{H}\left(u\right)}{inf}supI],\hfill & \mathrm{if}\mathrm{feasible},\hfill \\ ⌀,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

$$〚\mathrm{H}\left(u\right){〛}_{\vee }^{\mathrm{hull}}=[\underset{I\in \mathrm{H}\left(u\right)}{inf}infI,\underset{I\in \mathrm{H}\left(u\right)}{sup}supI].$$

Example 10  

(HIVFS in a Morning Commute Decision). Let U be the set of candidate routes for today’s commute. For the proposition “Arrive on time via route $r_1$”, define the HyperInterval-valued fuzzy set

$$\mathrm{H}\left(r_1\right)=\left\{I_1,I_2,I_3\right\}\subseteq L\left([0,1]\right),$$

where each interval encodes an admissible membership range (probability-like confidence) from an independent source:

$$I_1=[0.62,0.78](\mathrm{live}\mathrm{traffic}\mathrm{app}),$$

$$I_2=[0.55,0.72](\mathrm{weather}-\mathrm{adjusted}\mathrm{model}),$$

$$I_3=[0.68,0.80](\mathrm{historical}\mathrm{punctuality}).$$

By Definition (HIVFS), theconjunctive coreanddisjunctive hullat $r_1$ are

$$〚\mathrm{H}\left(r_1\right){〛}_{\wedge }=\bigcap _{I\in \mathrm{H}\left(r_1\right)}I=\left\{\begin{array}{cc}\left[\underset{I\in \mathrm{H}\left(r_1\right)}{sup}infI,\underset{I\in \mathrm{H}\left(r_1\right)}{inf}supI\right],\hfill & \mathrm{if}\mathrm{feasible},\hfill \\ ⌀,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

$$〚\mathrm{H}\left(r_1\right){〛}_{\vee }^{\mathrm{hull}}=\left[\underset{I\in \mathrm{H}\left(r_1\right)}{inf}infI,\underset{I\in \mathrm{H}\left(r_1\right)}{sup}supI\right].$$

We compute these bounds explicitly. First the lower endpoints:

$$inf{I}_1=0.62,inf{I}_2=0.55,inf{I}_3=0.68⟹\underset{I\in \mathrm{H}\left(r_1\right)}{sup}infI=max\{0.62,0.55,0.68\}=0.68.$$

Then the upper endpoints:

$$sup{I}_1=0.78,sup{I}_2=0.72,sup{I}_3=0.80⟹\underset{I\in \mathrm{H}\left(r_1\right)}{inf}supI=min\{0.78,0.72,0.80\}=0.72.$$

Therefore the conjunctive core is feasible and equals

$$〚\mathrm{H}\left(r_1\right){〛}_{\wedge }=[0.68,0.72].$$

The disjunctive hull aggregates the most permissive range:

$$\underset{I\in \mathrm{H}\left(r_1\right)}{inf}infI=min\{0.62,0.55,0.68\}=0.55,\underset{I\in \mathrm{H}\left(r_1\right)}{sup}supI=max\{0.78,0.72,0.80\}=0.80,$$

so

$$〚\mathrm{H}\left(r_1\right){〛}_{\vee }^{\mathrm{hull}}=[0.55,0.80].$$

The interval $[0.68,0.72]$ is the consensus range supportedsimultaneouslyby all sources for “arrive on time via $r_1$”. The hull $[0.55,0.80]$ captures the full plausible spectrum reported byat least onesource. Decision makers can require the core (conservative) or consult the hull (exploratory) depending on risk tolerance.

2.4. SuperHyperInterval-Valued Fuzzy Set

A SuperHyperInterval-Valued Fuzzy Set assigns nested families of membership intervals via iterated powersets, flattened to compute intersection cores and hulls.

Definition 17 (SuperHyperInterval-valued fuzzy set (order n)). Fix $n\ge 1$. ASuperHyperInterval-valued fuzzy set of order non U is a map

$${\mathrm{SH}}^{\left(n\right)}:U⟶{\mathcal{P}}^{n-1}\left(L\left(\right[0,1\left]\right)\right),$$

assigning to each u an $(n-1)$–nested family of numeric intervals. Conjunctive cores and hulls are computed by iterated application of intersection and hull at the leaves.

(order $n=3$) for a Same-Day Delivery Promise).Example 11 (SHIVFS Let $U=\left\{\tau \right\}$, where τ denotes today’s task “Deliver the parcel by 6 pm”. Fix $n=3$. A SuperHyperInterval-valued fuzzy set of order 3 maps τ to a two-level nested family of numeric intervals (elements of $L\left(\right[0,1\left]\right)$), grouped by evidence sources:

$${\mathrm{SH}}^{\left(3\right)}\left(\tau \right)=\left\{S_{\mathrm{logistics}},S_{\mathrm{conditions}},S_{\mathrm{recipient}}\right\}\in {\mathcal{P}}^2\left(L\left(\right[0,1\left]\right)\right),$$

where each $S_{•}$ is a finite set of closed intervals $[\alpha ,\beta ]\subseteq [0,1]$.

We instantiate concretely (all numbers are unit-free confidence levels):

$$\begin{array}{cc}\hfill S_{\mathrm{logistics}}& =\left\{I_1=[0.78,0.90](\mathrm{Carrier}\mathrm{A}\mathrm{dispatch}+\mathrm{capacity}),I_2=[0.80,0.88](\mathrm{Carrier}\mathrm{B}\mathrm{historical}\mathrm{on}-\mathrm{time})\right\},\hfill \\ \hfill S_{\mathrm{conditions}}& =\left\{I_3=[0.75,0.92](\mathrm{traffic}\mathrm{nowcast}),I_4=[0.77,0.89](\mathrm{weather}-\mathrm{adjusted}\mathrm{travel}\mathrm{time})\right\},\hfill \\ \hfill S_{\mathrm{recipient}}& =\left\{I_5=[0.76,0.93](\mathrm{recipient}\mathrm{availability}\mathrm{window}),I_6=[0.82,0.87](\mathrm{building}\mathrm{access}\mathrm{constraints})\right\}.\hfill \end{array}$$

By definition, conjunctive cores and hulls are computed at the leaves. Flatten the nesting by taking the union of groups:

$$\mathrm{Leaves}\left(\tau \right):=S_{\mathrm{logistics}}\cup S_{\mathrm{conditions}}\cup S_{\mathrm{recipient}}=\{I_1,I_2,I_3,I_4,I_5,I_6\}\subseteq L\left([0,1]\right).$$

Conjunctive core (intersection of all leaf intervals) is

$$〚{\mathrm{SH}}^{\left(3\right)}\left(\tau \right){〛}_{\wedge }=\bigcap _{k=1}^6{I}_k$$

$$=\left[\underset{1\le k\le 6}{sup}inf{I}_k,\underset{1\le k\le 6}{inf}sup{I}_k\right],$$

provided feasibility $supinf\le infsup$ holds. We compute explicitly:

$$inf{I}_1=0.78,inf{I}_2=0.80,inf{I}_3=0.75,inf{I}_4=0.77,inf{I}_5=0.76,inf{I}_6=0.82$$

$$⟹\underset{k}{sup}inf{I}_k=max\{0.78,0.80,0.75,0.77,0.76,0.82\}=0.82,$$

$$sup{I}_1=0.90,sup{I}_2=0.88,sup{I}_3=0.92,sup{I}_4=0.89,sup{I}_5=0.93,sup{I}_6=0.87$$

$$⟹\underset{k}{inf}sup{I}_k=min\{0.90,0.88,0.92,0.89,0.93,0.87\}=0.87.$$

Since $0.82\le 0.87$, the intersection is feasible and

$$〚{\mathrm{SH}}^{\left(3\right)}\left(\tau \right){〛}_{\wedge }=[0.82,0.87].$$

Disjunctive hull (least interval containing all leaf intervals) is

$$〚{\mathrm{SH}}^{\left(3\right)}\left(\tau \right){〛}_{\vee }^{\mathrm{hull}}=\left[\underset{k}{inf}inf{I}_k,\underset{k}{sup}sup{I}_k\right]$$

$$=\left[min\{0.78,0.80,0.75,0.77,0.76,0.82\},max\{0.90,0.88,0.92,0.89,0.93,0.87\}\right]=[0.75,0.93].$$

The nested structure recordshierarchicalevidence: logistics providers, external conditions, and recipient constraints. Flattening aggregates every leaf interval; the core $[0.82,0.87]$ captures the consensus range simultaneously supported byallgroups and sources, while the hull $[0.75,0.93]$ captures the full plausible spectrum reported byat least oneleaf source. A planner requiring high reliability would use the core; exploratory planning or contingency analysis may reference the hull.

2.5. HyperInterval-Valued Neutrosophic Set

A HyperInterval-Valued Neutrosophic Set uses sets of intervals for truth, indeterminacy, and falsity, yielding componentwise intersection cores and hulls semantics.

Definition 18  (HyperInterval-Valued Neutrosophic Set (HIVNS)). Let $U\ne ⌀$ be a universe, and let

$$L\left([0,1]\right):=\left\{[\alpha ,\beta ]\subseteq [0,1]|0\le \alpha \le \beta \le 1\right\}$$

be the set of all closed numeric intervals in $[0,1]$. AHyperInterval-Valued Neutrosophic Set (HIVNS)on U is a triple of maps

$$\mathcal{A}=(\mathcal{T},\mathcal{I},\mathcal{F}),$$

$$\mathcal{T},\mathcal{I},\mathcal{F}:U⟶{\mathcal{P}}^{*}\left(L\left(\right[0,1\left]\right)\right),$$

such that for each $u\in U$ the sets $\mathcal{T}\left(u\right)$, $\mathcal{I}\left(u\right)$, and $\mathcal{F}\left(u\right)$ arefinite, nonemptyfamilies of intervals in $L\left(\right[0,1\left]\right)$.

For a finite family $\mathcal{S}={\left\{[{\alpha }_j,{\beta }_j]\right\}}_{j\in J}\subseteq L\left([0,1]\right)$, define theconjunctive coreand thedisjunctive hullby

$${\mathrm{core}}_{\wedge }\left(\mathcal{S}\right)=\left\{\begin{array}{cc}\left[\underset{j\in J}{max}{\alpha }_j,\underset{j\in J}{min}{\beta }_j\right],\hfill & \mathrm{if}\underset{j}{max}{\alpha }_j\le \underset{j}{min}{\beta }_j,\hfill \\ ⌀,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

$${\mathrm{hull}}_{\vee }\left(\mathcal{S}\right)=\left[\underset{j\in J}{min}{\alpha }_j,\underset{j\in J}{max}{\beta }_j\right].$$

We interpret ${\mathrm{core}}_{\wedge }\left(\mathcal{S}\right)$ as the precise admissible range underall hyper-interval declarations, and ${\mathrm{hull}}_{\vee }\left(\mathcal{S}\right)$ as the least interval containingsomedeclaration.

Theconjunctive semanticsandhull semanticsof $\mathcal{A}$ at $u\in U$ are then given componentwise by

$${\mathrm{core}}_{\wedge }\left(\mathcal{A}\right)\left(u\right):=\left({\mathrm{core}}_{\wedge }\left(\mathcal{T}\left(u\right)\right),{\mathrm{core}}_{\wedge }\left(\mathcal{I}\left(u\right)\right),{\mathrm{core}}_{\wedge }\left(\mathcal{F}\left(u\right)\right)\right),$$

$${\mathrm{hull}}_{\vee }\left(\mathcal{A}\right)\left(u\right):=\left({\mathrm{hull}}_{\vee }\left(\mathcal{T}\left(u\right)\right),{\mathrm{hull}}_{\vee }\left(\mathcal{I}\left(u\right)\right),{\mathrm{hull}}_{\vee }\left(\mathcal{F}\left(u\right)\right)\right),$$

with feasibility of the conjunctive semantics requiring, for each component,

$$\underset{[\alpha ,\beta ]\in \mathcal{T}\left(u\right)}{max}\alpha \le \underset{[\alpha ,\beta ]\in \mathcal{T}\left(u\right)}{min}\beta ,$$

$$\underset{[\alpha ,\beta ]\in \mathcal{I}\left(u\right)}{max}\alpha \le \underset{[\alpha ,\beta ]\in \mathcal{I}\left(u\right)}{min}\beta ,$$

$$\underset{[\alpha ,\beta ]\in \mathcal{F}\left(u\right)}{max}\alpha \le \underset{[\alpha ,\beta ]\in \mathcal{F}\left(u\right)}{min}\beta .$$

Theorem 4  

(HIVNS generalizes IVNS). Let anInterval-Valued Neutrosophic Set (IVNS)be given by three maps

$$T,I,F:U⟶L\left([0,1]\right),T\left(u\right)=[t^{-}\left(u\right),t^{+}\left(u\right)],I\left(u\right)=[i^{-}\left(u\right),i^{+}\left(u\right)],F\left(u\right)=[f^{-}\left(u\right),f^{+}\left(u\right)].$$

Define the embedding

$$\iota :{\left(L\left(\right[0,1\left]\right)\right)}^3⟶{\left({\mathcal{P}}^{*}\left(L\left(\right[0,1\left]\right)\right)\right)}^3,$$

$$\iota \left([a,b],[c,d],[e,f]\right):=\left(\left\{\right[a,b\left]\right\},\left\{\right[c,d\left]\right\},\left\{\right[e,f\left]\right\}\right).$$

Then $\mathcal{A}:=\iota (T,I,F)$ is a HIVNS and, for every $u\in U$,

$${\mathrm{core}}_{\wedge }\left(\mathcal{A}\right)\left(u\right)=\left(T\left(u\right),I\left(u\right),F\left(u\right)\right)={\mathrm{hull}}_{\vee }\left(\mathcal{A}\right)\left(u\right).$$

Consequently, every IVNS is (canonically) a special case of a HIVNS.

Proof. 

Fix $u\in U$. By construction,

$$\mathcal{T}\left(u\right)=\left\{[t^{-}\left(u\right),t^{+}\left(u\right)]\right\},$$

$$\mathcal{I}\left(u\right)=\left\{[i^{-}\left(u\right),i^{+}\left(u\right)]\right\},$$

$$\mathcal{F}\left(u\right)=\left\{[f^{-}\left(u\right),f^{+}\left(u\right)]\right\}.$$

Evaluating the maxima/minima over a singleton set gives

$$\underset{[\alpha ,\beta ]\in \mathcal{T}\left(u\right)}{max}\alpha =t^{-}\left(u\right),$$

$$\underset{[\alpha ,\beta ]\in \mathcal{T}\left(u\right)}{min}\beta =t^{+}\left(u\right),$$

and analogously for the I and F components. Hence

$${\mathrm{core}}_{\wedge }\left(\mathcal{T}\left(u\right)\right)=[t^{-}\left(u\right),t^{+}\left(u\right)],$$

$${\mathrm{hull}}_{\vee }\left(\mathcal{T}\left(u\right)\right)=[t^{-}\left(u\right),t^{+}\left(u\right)],$$

with identical equalities for $\mathcal{I}\left(u\right)$ and $\mathcal{F}\left(u\right)$. Therefore

$${\mathrm{core}}_{\wedge }\left(\mathcal{A}\right)\left(u\right)={\mathrm{hull}}_{\vee }\left(\mathcal{A}\right)\left(u\right)=(T\left(u\right),I\left(u\right),F\left(u\right)).$$

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Example 12  

(Concrete computation at a point). Let $\mathcal{T}\left(u\right)=\left\{\right[0.6,0.9],[0.7,0.8\left]\right\}$, $\mathcal{I}\left(u\right)=\left\{\right[0.1,0.3\left]\right\}$, $\mathcal{F}\left(u\right)=\left\{\right[0.05,0.2],[0.0,0.15\left]\right\}$. Then

$${\mathrm{core}}_{\wedge }\left(\mathcal{T}\left(u\right)\right)=[max\{0.6,0.7\},min\{0.9,0.8\}]=[0.7,0.8],$$

$${\mathrm{hull}}_{\vee }\left(\mathcal{T}\left(u\right)\right)=[min\{0.6,0.7\},max\{0.9,0.8\}]=[0.6,0.9],$$

$${\mathrm{core}}_{\wedge }\left(\mathcal{I}\left(u\right)\right)={\mathrm{hull}}_{\vee }\left(\mathcal{I}\left(u\right)\right)=[0.1,0.3],$$

$${\mathrm{core}}_{\wedge }\left(\mathcal{F}\left(u\right)\right)=[max\{0.05,0.0\},min\{0.2,0.15\}]=[0.05,0.15],$$

$${\mathrm{hull}}_{\vee }\left(\mathcal{F}\left(u\right)\right)=[min\{0.05,0.0\},max\{0.2,0.15\}]=[0.0,0.2].$$

Thus ${\mathrm{core}}_{\wedge }\left(\mathcal{A}\right)\left(u\right)=\left([0.7,0.8],[0.1,0.3],[0.05,0.15]\right)$ and ${\mathrm{hull}}_{\vee }\left(\mathcal{A}\right)\left(u\right)=\left([0.6,0.9],[0.1,0.3],[0.0,0.2]\right)$.

2.6. SuperHyperInterval-Valued Neutrosophic Set

A SuperHyperInterval-Valued Neutrosophic Set organizes nested interval families for truth, indeterminacy, and falsity, flattening to compute componentwise cores and hulls.

Definition 19  

(Flattening operator on iterated powersets). For a base set X and $k\ge 0$, define ${\mathrm{Flat}}_k:{\mathcal{P}}^k\left(X\right)\to \mathcal{P}\left(X\right)$ recursively by

$${\mathrm{Flat}}_0\left(S\right):=S\subseteq X,{\mathrm{Flat}}_{k+1}\left(\mathcal{S}\right):=\bigcup _{S\in \mathcal{S}}{\mathrm{Flat}}_k\left(S\right)(\mathcal{S}\subseteq {\mathcal{P}}^k\left(X\right)).$$

Definition 20  (SuperHyperInterval-Valued Neutrosophic Set (order n)). Fix $n\ge 1$. ASuperHyperInterval-Valued Neutrosophic Set (SHIVNS) of order non U is a triple

$${\mathcal{S}}^{\left(n\right)}=\left({\mathsf{T}}^{\left(n\right)},{\mathsf{I}}^{\left(n\right)},{\mathsf{F}}^{\left(n\right)}\right),{\mathsf{T}}^{\left(n\right)},{\mathsf{I}}^{\left(n\right)},{\mathsf{F}}^{\left(n\right)}:U⟶{\mathcal{P}}^{n-1}\left(L\left(\right[0,1\left]\right)\right),$$

with the requirement that for each $u\in U$ the sets ${\mathsf{T}}^{\left(n\right)}\left(u\right)$, ${\mathsf{I}}^{\left(n\right)}\left(u\right)$, ${\mathsf{F}}^{\left(n\right)}\left(u\right)$ are finite at every nesting level.

Write theleaf familiesat u as

$${\mathcal{T}}_{\mathrm{leaf}}\left(u\right):={\mathrm{Flat}}_{n-1}\left({\mathsf{T}}^{\left(n\right)}\left(u\right)\right)\subseteq L\left([0,1]\right),{\mathcal{I}}_{\mathrm{leaf}}\left(u\right):={\mathrm{Flat}}_{n-1}\left({\mathsf{I}}^{\left(n\right)}\left(u\right)\right),{\mathcal{F}}_{\mathrm{leaf}}\left(u\right):={\mathrm{Flat}}_{n-1}\left({\mathsf{F}}^{\left(n\right)}\left(u\right)\right).$$

Theconjunctive semanticsandhull semanticsof ${\mathcal{S}}^{\left(n\right)}$ are defined by applying Definition 18 to these leaf families:

$${\mathrm{core}}_{\wedge }\left({\mathcal{S}}^{\left(n\right)}\right)\left(u\right):=\left({\mathrm{core}}_{\wedge }\left({\mathcal{T}}_{\mathrm{leaf}}\left(u\right)\right),{\mathrm{core}}_{\wedge }\left({\mathcal{I}}_{\mathrm{leaf}}\left(u\right)\right),{\mathrm{core}}_{\wedge }\left({\mathcal{F}}_{\mathrm{leaf}}\left(u\right)\right)\right),$$

$${\mathrm{hull}}_{\vee }\left({\mathcal{S}}^{\left(n\right)}\right)\left(u\right):=\left({\mathrm{hull}}_{\vee }\left({\mathcal{T}}_{\mathrm{leaf}}\left(u\right)\right),{\mathrm{hull}}_{\vee }\left({\mathcal{I}}_{\mathrm{leaf}}\left(u\right)\right),{\mathrm{hull}}_{\vee }\left({\mathcal{F}}_{\mathrm{leaf}}\left(u\right)\right)\right).$$

Definition 21  

(Canonical nesting). For any set S and $k\ge 0$, define ${\mathrm{Nest}}_0\left(S\right):=S$ and ${\mathrm{Nest}}_{k+1}\left(S\right):=\left\{{\mathrm{Nest}}_k\left(S\right)\right\}$. Then ${\mathrm{Nest}}_k\left(S\right)\in {\mathcal{P}}^k\left(S\right)$ and, crucially,

$${\mathrm{Flat}}_k\left({\mathrm{Nest}}_k\left(S\right)\right)=S\mathrm{for}\mathrm{all}k\ge 0.$$

(1)

Lemma 2  

(Flattening a canonical nest). Equality (1) holds by induction on k.

Proof. 

For $k=0$ the claim is ${\mathrm{Flat}}_0\left(S\right)=S$, true by definition. If ${\mathrm{Flat}}_k\left({\mathrm{Nest}}_k\left(S\right)\right)=S$, then

$${\mathrm{Flat}}_{k+1}\left({\mathrm{Nest}}_{k+1}\left(S\right)\right)={\mathrm{Flat}}_{k+1}\left(\left\{{\mathrm{Nest}}_k\left(S\right)\right\}\right)={\mathrm{Flat}}_k\left({\mathrm{Nest}}_k\left(S\right)\right)=S.$$

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Theorem 5  

(SHIVNS generalizes HIVNS). Let $\mathcal{A}=(\mathcal{T},\mathcal{I},\mathcal{F})$ be a HIVNS on U (Definition 18). For any $n\ge 1$, define the embedding

$$j_n:{\left({\mathcal{P}}^{*}\left(L\left([0,1]\right)\right)\right)}^3⟶{\left({\mathcal{P}}^{n-1}\left(L\left([0,1]\right)\right)\right)}^3$$

by

$$\left(\mathcal{T}\left(u\right),\mathcal{I}\left(u\right),\mathcal{F}\left(u\right)\right)⟼\left({\mathrm{Nest}}_{n-1}\left(\mathcal{T}\left(u\right)\right),{\mathrm{Nest}}_{n-1}\left(\mathcal{I}\left(u\right)\right),{\mathrm{Nest}}_{n-1}\left(\mathcal{F}\left(u\right)\right)\right).$$

Then ${\mathcal{S}}^{\left(n\right)}:=j_n\left(\mathcal{A}\right)$ is a SHIVNS of order n and, for every $u\in U$,

$${\mathrm{core}}_{\wedge }\left({\mathcal{S}}^{\left(n\right)}\right)\left(u\right)={\mathrm{core}}_{\wedge }\left(\mathcal{A}\right)\left(u\right),{\mathrm{hull}}_{\vee }\left({\mathcal{S}}^{\left(n\right)}\right)\left(u\right)={\mathrm{hull}}_{\vee }\left(\mathcal{A}\right)\left(u\right).$$

Hence every HIVNS is (canonically) a special case of a SHIVNS (for any $n\ge 1$).

Proof. 

Fix $u\in U$. By Lemma 2 with $S=\mathcal{T}\left(u\right)$ (and similarly for $\mathcal{I},\mathcal{F}$),

$${\mathcal{T}}_{\mathrm{leaf}}\left(u\right)={\mathrm{Flat}}_{n-1}\left({\mathrm{Nest}}_{n-1}\left(\mathcal{T}\left(u\right)\right)\right)=\mathcal{T}\left(u\right).$$

Therefore the leaf families of ${\mathcal{S}}^{\left(n\right)}$ coincide with the original hyper-interval families of $\mathcal{A}$:

$${\mathcal{T}}_{\mathrm{leaf}}\left(u\right)=\mathcal{T}\left(u\right),{\mathcal{I}}_{\mathrm{leaf}}\left(u\right)=\mathcal{I}\left(u\right),{\mathcal{F}}_{\mathrm{leaf}}\left(u\right)=\mathcal{F}\left(u\right).$$

Since both ${\mathrm{core}}_{\wedge }$ and ${\mathrm{hull}}_{\vee }$ depend only on the sets of leaf intervals via $max/min$ and $min/max$, it follows immediately that

$${\mathrm{core}}_{\wedge }\left({\mathcal{S}}^{\left(n\right)}\right)\left(u\right)={\mathrm{core}}_{\wedge }\left(\mathcal{A}\right)\left(u\right),{\mathrm{hull}}_{\vee }\left({\mathcal{S}}^{\left(n\right)}\right)\left(u\right)={\mathrm{hull}}_{\vee }\left(\mathcal{A}\right)\left(u\right).$$

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3. Conclusion

In this paper, we introduced HyperInterval- and SuperHyperInterval-valued fuzzy/neutrosophic sets, defined conjunctive “core” (intersection) and disjunctive “hull” semantics, and proved embedding theorems showing that classical interval, fuzzy, and neutrosophic models appear as singleton or degenerate cases. For future work, we plan to investigate extended concepts using HyperFuzzy Sets[38,39,40,41,42], HyperNeutrosophic Sets[43,44], Plithogenic Sets [45,46,47], as well as structural frameworks such as Graphs[48], HyperGraphs[49,50,51], and SuperHyperGraphs[52,53,54,55].

Disclaimer: This work presents theoretical concepts that have not yet undergone practical testing or validation. Future researchers are encouraged to apply and assess these ideas in empirical contexts. While every effort has been made to ensure accuracy and appropriate referencing, unintentional errors or omissions may still exist. Readers are advised to verify referenced materials on their own. The views and conclusions expressed here are the authors’ own and do not necessarily reflect those of their affiliated organizations.