Review of Structured-Degree Uncertain Sets: Vector-, Matrix-, and Tensor-Valued Membership

1. Preliminaries

This section fixes notation and reviews the background material needed in the sequel. Unless stated otherwise, all sets considered in this paper are finite.

1.1. Fuzzy Sets and Neutrosophic Sets

In classical set theory, membership is a two-valued notion: an element either belongs to a set or it does not. In many real situations, however, assessments are gradual rather than strictly yes/no. Fuzzy set theory formalizes graded membership by assigning to each element a degree in the unit interval $[0,1]$ [2]. As related concepts, intuitionistic fuzzy sets [3], bipolar fuzzy sets [4], and hesitant fuzzy sets [5,6] are well known. We recall their standard definitions.

Definition 1

(Fuzzy Set). [2,7] Let Y be a nonempty universe. Afuzzy seton Y is a mapping

$$\tau :Y⟶[0,1].$$

Afuzzy relationon Y is a mapping

$$\delta :Y\times Y⟶[0,1].$$

We say that δ is afuzzy relation onτ if, for all $y,z\in Y$,

$$\delta (y,z)\le min\left\{\tau \right(y),\tau (z\left)\right\}.$$

Neutrosophic sets further refine graded membership by recording three (typically independent) components that quantify support, hesitation, and rejection. These components are commonly interpreted as the degrees of truth, indeterminacy, and falsity, respectively [8,9,10,11]. Related notions are also well known, including Double-valued Neutrosophic Sets [12,13], Bipolar Neutrosophic Sets [14,15], and Plithogenic Sets [16,17]. We adopt the widely used single-valued formulation.

Definition 2

(Neutrosophic Set). [8,18] Let X be a nonempty set. ANeutrosophic Set (NS)A on X is specified by three functions

$$T_A:X\to [0,1],I_A:X\to [0,1],F_A:X\to [0,1],$$

where, for each $x\in X$, the values $T_A\left(x\right)$, $I_A\left(x\right)$, and $F_A\left(x\right)$ represent, respectively, the degrees of truth, indeterminacy, and falsity of x with respect to A. These degrees satisfy

$$0\le T_A\left(x\right)+I_A\left(x\right)+F_A\left(x\right)\le 3(\forall x\in X).$$

1.2. Uncertain Set and Functorial Set

An Uncertain Set provides a uniform way to attach “uncertainty values” to elements, where the values take their meaning from a chosen degree-domain. By selecting the degree-domain appropriately, one can recover fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, and many other degree-based models as special cases [1,19].

Definition 3 Uncertain Set (U-Set)). [1] Let X be a nonempty universe and let M be an uncertainty model with degree-domain

$$Dom\left(M\right)\subseteq {[0,1]}^k$$

for some integer $k\ge 1$. AnUncertain Set of type M(briefly, aU-Set of type M) on X is a mapping

$${\mu }_M:X⟶Dom\left(M\right).$$

For each $x\in X$, the value ${\mu }_M\left(x\right)\in Dom\left(M\right)$ is called theM-membership degree(orM-uncertainty value) of x. Different choices of M yield the usual fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, and other degree-based set models.

2. Main Results

In this section, we define the vector-, matrix-, and tensor-valued Uncertain Sets summarized in Table 1 and examine their fundamental properties.

Note that vectors[20,21], matrices[22,23,24], and tensors [25,26,27,28] have been studied extensively and have a wide range of applications in fields such as machine learning and artificial intelligence. Moreover, for reference, we also examine in the Appendix A the notion of a MetaTensor (a tensor whose entries are tensors), as well as its iterated extension, the Iterated MetaTensor (a tensor of … of tensors). In addition, hierarchical tensors referred to as HyperTensors and SuperHyperTensors are also examined in the Appendix.

2.1. Vector-valued Fuzzy, Neutrosophic, and Uncertain Sets

In many applications, a single scalar degree in $[0,1]$ is too coarse: one may wish to record several aspects of uncertainty simultaneously (e.g., confidence, reliability, relevance, or scores coming from multiple sensors). A simple way to model such situations is to replace the unit interval by a vector-valued degree domain.

Definition 4

(Vector-valued unit interval and componentwise order). Fix an integer $d\ge 1$ and define thed-dimensional unit cube

$${[0,1]}^d:=\{(x_1,\dots ,x_d)\in {\mathbb{R}}^d\mid 0\le x_i\le 1(1\le i\le d)\}.$$

For $u=(u_1,\dots ,u_d)$ and $v=(v_1,\dots ,v_d)$ in ${[0,1]}^d$, define thecomponentwise order

$$u⪯v:⟺u_i\le v_i(1\le i\le d).$$

We also use componentwise operations: for instance,

$${(u+v)}_i:=u_i+v_i,{(min\{u,v\})}_i:=min\{u_i,v_i\},$$

and we write $\mathbf{0}:=(0,\dots ,0)$ and $\mathbf{1}:=(1,\dots ,1)$.

Definition 5

(Vector-valued fuzzy set). Let X be a nonempty universe and fix $d\ge 1$. Avector-valued fuzzy set of dimension don X is a mapping

$$\mu :X⟶{[0,1]}^d.$$

For each $x\in X$, the value $\mu \left(x\right)=({\mu }_1\left(x\right),\dots ,{\mu }_d\left(x\right))$ is interpreted as a d-component membership/uncertainty vector associated with x. When $d=1$, this definition reduces to the classical (scalar) fuzzy set.

Optionally, avector-valued fuzzy relationon X is a mapping

$$\delta :X\times X⟶{[0,1]}^d.$$

We say that δ iscompatiblewith μ if, for all $x,y\in X$,

$$\delta (x,y)⪯min\left\{\mu \right(x),\mu (y\left)\right\},$$

where the minimum is taken componentwise.

Definition 6

(Vector-valued neutrosophic set). Let X be a nonempty universe and fix $d\ge 1$. Avector-valued neutrosophic set of dimension don X consists of three mappings

$$T:X\to {[0,1]}^d,I:X\to {[0,1]}^d,F:X\to {[0,1]}^d,$$

where, for each $x\in X$, the vectors $T\left(x\right)$, $I\left(x\right)$, and $F\left(x\right)$ represent, respectively, the truth, indeterminacy, and falsity degrees of x.

These degrees satisfy the componentwise constraint

$$\mathbf{0}⪯T\left(x\right)+I\left(x\right)+F\left(x\right)⪯3\mathbf{1}(\forall x\in X),$$

where addition and inequalities are interpreted componentwise. When $d=1$, this is exactly the usual single-valued neutrosophic set constraint $0\le T\left(x\right)+I\left(x\right)+F\left(x\right)\le 3$.

Example 1

(Real-life example of a vector-valued neutrosophic set). Consider a hospital triage unit that must assess whether a patient should be treated as ahigh-risk respiratory case. Let X be the set of patients arriving in a given day. Fix $d=3$ and interpret the three dimensions as independent evidence channels:

$$\left(1\right)\mathrm{symptoms},\left(2\right)\mathrm{rapid}\mathrm{test},\left(3\right)\mathrm{imaging}(\mathrm{e}.\mathrm{g}.,\mathrm{X}-\mathrm{ray}/\mathrm{CT}).$$

A vector-valued neutrosophic set on X assigns to each patient $x\in X$ three vectors

$$T\left(x\right)=(T_1\left(x\right),T_2\left(x\right),T_3\left(x\right))\in {[0,1]}^3,$$

$$I\left(x\right)=(I_1\left(x\right),I_2\left(x\right),I_3\left(x\right))\in {[0,1]}^3,$$

$$F\left(x\right)=(F_1\left(x\right),F_2\left(x\right),F_3\left(x\right))\in {[0,1]}^3,$$

where, for each channel $j\in \{1,2,3\}$:

• $T_j\left(x\right)$ measures how strongly channel j supports the statement “patient x is a high-risk respiratory case”;
• $F_j\left(x\right)$ measures how strongly channel j supports the negation “patient x is not a high-risk respiratory case”;
• $I_j\left(x\right)$ measures the indeterminacy in channel j (e.g., missing data, borderline values, poor image quality, or conflicting findings).

For instance, suppose a patient x presents clear symptoms but has an inconclusive rapid test and a moderately suggestive image. One may encode this as

$$T\left(x\right)=(0.85,0.40,0.65),I\left(x\right)=(0.10,0.45,0.20),F\left(x\right)=(0.20,0.55,0.30).$$

The interpretation is: symptoms provide strong support ($T_1=0.85$) with low uncertainty ($I_1=0.10$); the rapid test is ambiguous ($I_2=0.45$) and leans against high-risk ($F_2=0.55$); imaging provides moderate support ($T_3=0.65$) with moderate uncertainty ($I_3=0.20$).

The componentwise constraint

$$\mathbf{0}⪯T\left(x\right)+I\left(x\right)+F\left(x\right)⪯3\mathbf{1}$$

holds automatically here because each coordinate lies in $[0,1]$, hence each coordinate-sum lies in $[0,3]$. Such a representation is useful when decisions must be justified by multiple evidence channels: it keeps track of support, opposition, and uncertaintyper channel, rather than collapsing everything into a single score.

Definition 7

(Vector-valued uncertainty model). Fix integers $d\ge 1$ and $k\ge 1$. Avector-valued uncertainty model of shape $(k,d)$is specified by a degree-domain

$$Dom\left(M\right)\subseteq {\left({[0,1]}^d\right)}^k\cong {[0,1]}^{dk}.$$

Elements of $Dom\left(M\right)$ are viewed as k-tuples of d-dimensional degree vectors.

Definition 8 (Vector-valued Uncertain Set (vector-valued U-Set)). Let X be a nonempty universe and let M be a vector-valued uncertainty model with

$$Dom\left(M\right)\subseteq {\left({[0,1]}^d\right)}^k.$$

Avector-valued Uncertain Set of type M(briefly, avector-valued U-Set of type M) on X is a mapping

$${\mu }_M:X⟶Dom\left(M\right).$$

For each $x\in X$, the value ${\mu }_M\left(x\right)\in Dom\left(M\right)$ is called theM-membership degree(orM-uncertainty value) of x.

Remark 1

(Special cases and interpretation). The vector-valued U-Set template subsumes the preceding notions by suitable choices of $Dom\left(M\right)$.

(i)

If $k=1$ and $Dom\left(M\right)={[0,1]}^d$, then a vector-valued U-Set of type M is exactly a vector-valued fuzzy set $\mu :X\to {[0,1]}^d$.

(ii)

If $k=3$ and $Dom\left(M\right)={\left({[0,1]}^d\right)}^3$, then a vector-valued U-Set of type M can be written as ${\mu }_M\left(x\right)=(T\left(x\right),I\left(x\right),F\left(x\right))$ and thus corresponds to a vector-valued neutrosophic description.

(iii)

The parameter d represents the number of simultaneously tracked components (criteria, sensors, or features). The set $Dom\left(M\right)$ can impose additional structural constraints beyond simple box constraints, for example admissible correlations, normalization conditions, or application-specific feasibility regions.

2.2. Matrix-valued Fuzzy, Neutrosophic, and Uncertain Sets

In some settings, uncertainty is naturally structured as a matrix rather than a scalar or a vector. Typical examples include uncertainty indexed simultaneously by (criterion × scenario), (role × time), (sensor × feature), or (agent × attribute). To capture such two-dimensional structure, we replace the unit interval $[0,1]$ by a matrix-valued degree domain.

Definition 9

(Matrix-valued unit cube and componentwise order). Fix integers $p,q\ge 1$ and define

$${[0,1]}^{p\times q}:=\left\{X=\left(x_{ij}\right)\in {\mathbb{R}}^{p\times q}\mid 0\le x_{ij}\le 1(1\le i\le p,1\le j\le q)\right\}.$$

For $X=\left(x_{ij}\right)$ and $Y=\left(y_{ij}\right)$ in ${[0,1]}^{p\times q}$, define thecomponentwise order

$$X⪯Y:⟺x_{ij}\le y_{ij}(1\le i\le p,1\le j\le q).$$

We also use componentwise operations:

$${(X+Y)}_{ij}:=x_{ij}+y_{ij},{(min\{X,Y\})}_{ij}:=min\{x_{ij},y_{ij}\}.$$

Let ${\mathbf{0}}_{p\times q}$ and ${\mathbf{1}}_{p\times q}$ denote the $p\times q$ zero and all-ones matrices, respectively.

Definition 10

(Matrix-valued fuzzy set). Let X be a nonempty universe and fix $p,q\ge 1$. Amatrix-valued fuzzy set of shape $(p,q)$on X is a mapping

$$\mu :X⟶{[0,1]}^{p\times q}.$$

For each $x\in X$, the matrix $\mu \left(x\right)=\left(\mu {\left(x\right)}_{ij}\right)$ is interpreted as a two-dimensional membership/uncertainty profile of x.

Optionally, amatrix-valued fuzzy relationon X is a mapping

$$\delta :X\times X⟶{[0,1]}^{p\times q}.$$

We say that δ iscompatiblewith μ if, for all $x,y\in X$,

$$\delta (x,y)⪯min\left\{\mu \right(x),\mu (y\left)\right\},$$

where the minimum is taken componentwise.

Definition 11

(Matrix-valued neutrosophic set). Let X be a nonempty universe and fix $p,q\ge 1$. Amatrix-valued neutrosophic set of shape $(p,q)$on X consists of three mappings

$$T:X\to {[0,1]}^{p\times q},I:X\to {[0,1]}^{p\times q},F:X\to {[0,1]}^{p\times q},$$

where for each $x\in X$, the matrices $T\left(x\right)$, $I\left(x\right)$, and $F\left(x\right)$ represent, respectively, the truth, indeterminacy, and falsity degrees of x in a two-dimensional index system.

These degrees satisfy the componentwise constraint

$${\mathbf{0}}_{p\times q}⪯T\left(x\right)+I\left(x\right)+F\left(x\right)$$

$$⪯3{\mathbf{1}}_{p\times q}(\forall x\in X),$$

where addition and inequalities are interpreted componentwise. When $(p,q)=(1,1)$, this reduces to the usual single-valued neutrosophic set constraint $0\le T\left(x\right)+I\left(x\right)+F\left(x\right)\le 3$.

Example 2

(Real-life example of a matrix-valued neutrosophic set). Consider an enterprise security team that must assess whether each software service should be classified ashigh cyber-riskfor prioritizing remediation. Let X be the set of services (or applications) deployed in the organization.

Fix two indexing dimensions:

$$p=3\mathrm{evidence}\mathrm{sources}\left(\mathrm{static}\mathrm{scan},\mathrm{runtime}\mathrm{telemetry},\mathrm{external}\mathrm{intelligence}\right),$$

$$q=4\mathrm{risk}\mathrm{aspects}\left(\mathrm{vulnerability},\mathrm{exposure},\mathrm{impact},\mathrm{exploitability}\right).$$

For each service $x\in X$, a matrix-valued neutrosophic set assigns three matrices

$$T\left(x\right),I\left(x\right),F\left(x\right)\in {[0,1]}^{3\times 4},$$

where entry $(s,a)$ (source s, aspect a) has the following meaning:

• $T_{s,a}\left(x\right)$: how strongly source s supports the statement “service x is high-risk” with respect to aspect a;
• $F_{s,a}\left(x\right)$: how strongly source s supports the opposite statement “service x is not high-risk” with respect to aspect a;
• $I_{s,a}\left(x\right)$: how indeterminate the assessment from source s is for aspect a (e.g., missing logs, noisy signals, stale intelligence, or conflicting findings).

For example, for a particular service x, suppose a static scanner finds multiple severe CVEs, telemetry shows intermittent suspicious behavior but incomplete logging, and external intelligence suggests active exploitation of similar stacks. One may encode:

$$T\left(x\right)=\left(\begin{array}{cccc}0.90& 0.70& 0.60& 0.80\\ 0.55& 0.65& 0.50& 0.60\\ 0.75& 0.60& 0.55& 0.85\end{array}\right),$$

$$I\left(x\right)=\left(\begin{array}{cccc}0.05& 0.10& 0.15& 0.05\\ 0.30& 0.20& 0.35& 0.25\\ 0.15& 0.20& 0.20& 0.10\end{array}\right),$$

$$F\left(x\right)=\left(\begin{array}{cccc}0.20& 0.30& 0.40& 0.15\\ 0.35& 0.25& 0.45& 0.30\\ 0.25& 0.30& 0.35& 0.20\end{array}\right).$$

Here the first row corresponds tostatic scan, the second toruntime telemetry, and the third toexternal intelligence; the four columns correspond, respectively, tovulnerability,exposure,impact, andexploitability.

This representation keeps a two-dimensional audit trail: it separateswhich sourceproduced the evidence andwhich risk aspectit concerns, while still recording support (T), uncertainty (I), and opposition (F). The componentwise constraint

$${\mathbf{0}}_{3\times 4}⪯T\left(x\right)+I\left(x\right)+F\left(x\right)⪯3{\mathbf{1}}_{3\times 4}$$

is satisfied because each entry lies in $[0,1]$ and thus each entrywise sum lies in $[0,3]$. Such matrix-valued neutrosophic modeling is useful in practice because remediation decisions can be justified granularly (per source and per aspect) rather than by a single aggregated score.

Definition 12

(Matrix-valued uncertainty model). Fix integers $p,q\ge 1$ and $k\ge 1$. Amatrix-valued uncertainty model of shape $(k;p,q)$is specified by a degree-domain

$$Dom\left(M\right)\subseteq {\left({[0,1]}^{p\times q}\right)}^k.$$

Elements of $Dom\left(M\right)$ are viewed as k-tuples of $(p\times q)$-matrices, and $Dom\left(M\right)$ may encode additional admissibility constraints (e.g., normalization, sparsity, monotonicity, or correlations) beyond entrywise bounds.

Definition 13 (Matrix-valued Uncertain Set (matrix-valued U-Set)). Let X be a nonempty universe and let M be a matrix-valued uncertainty model with

$$Dom\left(M\right)\subseteq {\left({[0,1]}^{p\times q}\right)}^k.$$

Amatrix-valued Uncertain Set of type M(briefly, amatrix-valued U-Set of type M) on X is a mapping

$${\mu }_M:X⟶Dom\left(M\right).$$

For each $x\in X$, the value ${\mu }_M\left(x\right)\in Dom\left(M\right)$ is called theM-membership degree(orM-uncertainty value) of x.

Remark 2

(Special cases and interpretation). The matrix-valued U-Set framework subsumes the preceding notions via suitable choices of $Dom\left(M\right)$:

(i)

If $k=1$ and $Dom\left(M\right)={[0,1]}^{p\times q}$, then a matrix-valued U-Set is exactly a matrix-valued fuzzy set $\mu :X\to {[0,1]}^{p\times q}$.

(ii)

If $k=3$ and $Dom\left(M\right)={\left({[0,1]}^{p\times q}\right)}^3$, then ${\mu }_M\left(x\right)=(T\left(x\right),I\left(x\right),F\left(x\right))$ yields a matrix-valued neutrosophic description.

(iii)

The indices $(i,j)$ can represent, for example, (criterion × scenario) or (time × role), so the matrix encodes structured uncertainty that is not naturally captured by a single scalar degree. The admissible domain $Dom\left(M\right)$ can enforce application-specific constraints on such matrices.

2.3. Tensor-valued Fuzzy, Neutrosophic, and Uncertain Sets

In complex applications, uncertainty may be indexed by more than two dimensions, for example (agent × criterion × time), (sensor × feature × location), or (scenario × stage × resource × constraint). Such structures are naturally represented by multi-order tensors. Accordingly, we replace the unit interval $[0,1]$ by a tensor-valued degree domain.

Definition 14

(Tensor-valued unit cube). Fix an integer $n\ge 1$ (the order) and a size vector

$$\mathbf{d}=(d_1,\dots ,d_n)\in {\mathbb{N}}_{\ge 1}^n.$$

Let

$${[0,1]}^{d_1\times \dots \times d_n}:=\left\{X=\left(x_{i_1\dots i_n}\right)\in {\mathbb{R}}^{d_1\times \dots \times d_n}|0\le x_{i_1\dots i_n}\le 1\forall (i_1,\dots ,i_n)\right\}.$$

Elements of ${[0,1]}^{d_1\times \dots \times d_n}$ are callednth-order tensors(with mode sizes $d_1,\dots ,d_n$) whose entries lie in $[0,1]$. We write ${\mathbf{0}}_{\mathbf{d}}$ and ${\mathbf{1}}_{\mathbf{d}}$ for the all-zeros and all-ones tensors of this shape.

Definition 15

(Componentwise order and operations on tensors). For $X=\left(x_{i_1\dots i_n}\right)$ and $Y=\left(y_{i_1\dots i_n}\right)$ in ${[0,1]}^{d_1\times \dots \times d_n}$, define thecomponentwise order

$$X⪯Y:⟺x_{i_1\dots i_n}\le y_{i_1\dots i_n}\mathrm{for}\mathrm{all}\mathrm{indices}(i_1,\dots ,i_n).$$

We also use componentwise operations:

$${(X+Y)}_{i_1\dots i_n}:=x_{i_1\dots i_n}+y_{i_1\dots i_n},{(min\{X,Y\})}_{i_1\dots i_n}:=min\{x_{i_1\dots i_n},y_{i_1\dots i_n}\}.$$

Scalar multiplication is defined entrywise, so that ${\left(\lambda X\right)}_{i_1\dots i_n}:=\lambda x_{i_1\dots i_n}$.

Definition 16

(Tensor-valued fuzzy set). Let X be a nonempty universe and fix an order $n\ge 1$ and shape $\mathbf{d}=(d_1,\dots ,d_n)$. Atensor-valued fuzzy set of shape $\mathbf{d}$on X is a mapping

$$\mu :X⟶{[0,1]}^{d_1\times \dots \times d_n}.$$

For each $x\in X$, the tensor $\mu \left(x\right)$ is interpreted as an n-way membership/uncertainty profile of x.

Optionally, atensor-valued fuzzy relationon X is a mapping

$$\delta :X\times X⟶{[0,1]}^{d_1\times \dots \times d_n}.$$

We say that δ iscompatiblewith μ if, for all $x,y\in X$,

$$\delta (x,y)⪯min\left\{\mu \right(x),\mu (y\left)\right\},$$

where the minimum is taken componentwise.

Definition 17

(Tensor-valued neutrosophic set). Let X be a nonempty universe and fix an order $n\ge 1$ and shape $\mathbf{d}=(d_1,\dots ,d_n)$. Atensor-valued neutrosophic set of shape $\mathbf{d}$on X consists of three mappings

$$T:X\to {[0,1]}^{d_1\times \dots \times d_n},I:X\to {[0,1]}^{d_1\times \dots \times d_n},F:X\to {[0,1]}^{d_1\times \dots \times d_n},$$

where, for each $x\in X$, the tensors $T\left(x\right)$, $I\left(x\right)$, and $F\left(x\right)$ represent, respectively, the truth, indeterminacy, and falsity degrees of x in an n-dimensional index system.

These degrees satisfy the componentwise constraint

$${\mathbf{0}}_{\mathbf{d}}⪯T\left(x\right)+I\left(x\right)+F\left(x\right)⪯3{\mathbf{1}}_{\mathbf{d}}(\forall x\in X),$$

where addition and inequalities are interpreted componentwise. When $n=1$ and $d_1=1$, this reduces to the usual single-valued neutrosophic set constraint $0\le T\left(x\right)+I\left(x\right)+F\left(x\right)\le 3$.

Example 3

(Real-life example of a tensor-valued neutrosophic set). Consider a global manufacturer that must assess whether eachsuppliershould be classified ashigh disruption-riskfor procurement planning. Let X be the set of candidate suppliers.

Risk assessment is inherently multi-dimensional: it depends onwhich criterionis considered,which regionthe supplier operates in,which time windowis relevant, andwhich scenario(e.g., baseline vs. stress) is assumed. We therefore fix an order $n=4$ tensor shape

$$\mathbf{d}=(d_1,d_2,d_3,d_4)=(5,3,4,2),$$

with the following index semantics:

• $d_1=5$ criteria: (1) financial stability, (2) delivery performance, (3) quality history, (4) geopolitical exposure, (5) capacity flexibility;
• $d_2=3$ regions: Asia, Europe, Americas;
• $d_3=4$ time horizons: next month, next quarter, next half-year, next year;
• $d_4=2$ scenarios: baseline, stress (e.g., port congestion or regional conflict escalation).

A tensor-valued neutrosophic set on X assigns to each supplier $x\in X$ three tensors

$$T\left(x\right),I\left(x\right),F\left(x\right)\in {[0,1]}^{5\times 3\times 4\times 2},$$

where, for each multi-index $(i,r,t,s)$,

• $T_{i,r,t,s}\left(x\right)$ quantifies the degree to which evidence supports the statement “supplier x is high disruption-risk” under criterion i, region r, horizon t, scenario s;
• $F_{i,r,t,s}\left(x\right)$ quantifies the degree to which evidence supports the opposite statement “supplier x is not high disruption-risk” in the same context;
• $I_{i,r,t,s}\left(x\right)$ captures indeterminacy in that context (e.g., incomplete audit data, unreliable forecasts, missing geopolitical intelligence, or conflicting reports).

For a concrete supplier x, the procurement team might encode the following qualitative situation:

• In Asia under the stress scenario, geopolitical exposure is judged strongly risky in the short term, but forecasts are uncertain;
• In Europe under baseline conditions, delivery performance is consistently reliable with low uncertainty;
• For the Americas, capacity flexibility is unclear because the latest capacity audit is outdated.

This can be represented, for example, by entries such as

$$\begin{array}{cc}& T_{\mathrm{geo},\mathrm{Asia},\mathrm{quarter},\mathrm{stress}}\left(x\right)=0.80,\hfill \\ & I_{\mathrm{geo},\mathrm{Asia},\mathrm{quarter},\mathrm{stress}}\left(x\right)=0.35,\hfill \\ & F_{\mathrm{geo},\mathrm{Asia},\mathrm{quarter},\mathrm{stress}}\left(x\right)=0.20,\hfill \\ & T_{\mathrm{deliv},\mathrm{Europe},\mathrm{half}-\mathrm{year},\mathrm{baseline}}\left(x\right)=0.15,\hfill \\ & I_{\mathrm{deliv},\mathrm{Europe},\mathrm{half}-\mathrm{year},\mathrm{baseline}}\left(x\right)=0.05,\hfill \\ & F_{\mathrm{deliv},\mathrm{Europe},\mathrm{half}-\mathrm{year},\mathrm{baseline}}\left(x\right)=0.85,\hfill \\ & T_{\mathrm{cap},\mathrm{Americas},\mathrm{year},\mathrm{baseline}}\left(x\right)=0.45,\hfill \\ & I_{\mathrm{cap},\mathrm{Americas},\mathrm{year},\mathrm{baseline}}\left(x\right)=0.40,\hfill \\ & F_{\mathrm{cap},\mathrm{Americas},\mathrm{year},\mathrm{baseline}}\left(x\right)=0.35.\hfill \end{array}$$

The first line indicates strong support for high risk (large T) with substantial uncertainty (large I), the second indicates strong evidence against high risk (large F) with low uncertainty, and the third reflects an ambiguous capacity assessment (high indeterminacy).

Finally, the componentwise neutrosophic constraint

$${\mathbf{0}}_{\mathbf{d}}⪯T\left(x\right)+I\left(x\right)+F\left(x\right)⪯3{\mathbf{1}}_{\mathbf{d}}$$

is satisfied because each tensor entry lies in $[0,1]$, hence each entrywise sum lies in $[0,3]$. Tensor-valued neutrosophic modeling is practically useful here because it preserves the full context (criterion × region × horizon × scenario) needed for transparent risk governance and targeted mitigation actions.

Definition 18

(Tensor-valued uncertainty model). Fix an order $n\ge 1$, a shape $\mathbf{d}=(d_1,\dots ,d_n)$, and an integer $k\ge 1$. Atensor-valued uncertainty model of shape $(k;\mathbf{d})$is specified by a degree-domain

$$Dom\left(M\right)\subseteq {\left({[0,1]}^{d_1\times \dots \times d_n}\right)}^k.$$

Elements of $Dom\left(M\right)$ are viewed as k-tuples of tensors of shape $\mathbf{d}$. The admissible set $Dom\left(M\right)$ may enforce additional structural constraints (e.g., normalization, low-rank structure, sparsity, monotonicity across modes, or correlations), beyond the entrywise bounds $[0,1]$.

Definition 19 (Tensor-valued Uncertain Set (tensor-valued U-Set)). Let X be a nonempty universe and let M be a tensor-valued uncertainty model with

$$Dom\left(M\right)\subseteq {\left({[0,1]}^{d_1\times \dots \times d_n}\right)}^k.$$

Atensor-valued Uncertain Set of type M(briefly, atensor-valued U-Set of type M) on X is a mapping

$${\mu }_M:X⟶Dom\left(M\right).$$

For each $x\in X$, the value ${\mu }_M\left(x\right)\in Dom\left(M\right)$ is called theM-membership degree(orM-uncertainty value) of x.

Remark 3

(Special cases and interpretation). The tensor-valued U-Set framework recovers vector- and matrix-valued notions as lower-order cases.

(i)

If $n=1$ and $\mathbf{d}=\left(d\right)$, then ${[0,1]}^{d_1\times \dots \times d_n}={[0,1]}^d$, and tensor-valued fuzzy/neutrosophic/U-sets reduce to the vector-valued definitions.

(ii)

If $n=2$ and $\mathbf{d}=(p,q)$, then ${[0,1]}^{d_1\times \dots \times d_n}={[0,1]}^{p\times q}$, and tensor-valued fuzzy/neutrosophic/U-sets reduce to the matrix-valued definitions.

(iii)

Taking $k=1$ and $Dom\left(M\right)={[0,1]}^{d_1\times \dots \times d_n}$ yields a tensor-valued fuzzy set. Taking $k=3$ and $Dom\left(M\right)={\left({[0,1]}^{d_1\times \dots \times d_n}\right)}^3$ yields a tensor-valued neutrosophic description.

3. Additional Results: Tensor-valued Functorial Set

A Functorial Set packages a family of structured sets by specifying a category together with a functor to $\mathbf{Set}$. Objects are interpreted as carriers of structure, while morphisms describe how structure is transported[1]. In particular, the functor assigns to each object a set of admissible structures, and to each morphism a structure-preserving map between such sets.

In the present work we focus on tensor-valued structures. Informally, a tensor-valued functorial set associates to each set X the collection of all tensor-valued assignments on X, and transports those assignments along functions by pullback (precomposition). This perspective is convenient because it makes change-of-variables and restriction maps automatic consequences of functoriality.

Definition 20

(Functorial Set). [1] Let $\mathcal{C}$ be a category and

$$F:\mathcal{C}⟶\mathbf{Set}$$

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

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

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

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

Definition 21

(Tensor-valued structure functor). Fix a tensor degree-domain ${\mathsf{Ten}}_{k,\mathbf{d}}$. Define a functor

$${\mathcal{F}}_{k,\mathbf{d}}:{\mathbf{Set}}^{\mathrm{op}}⟶\mathbf{Set}$$

by:

• for each set X, set

  $${\mathcal{F}}_{k,\mathbf{d}}\left(X\right):=\{\mu :X\to {\mathsf{Ten}}_{k,\mathbf{d}}\}={\left({\mathsf{Ten}}_{k,\mathbf{d}}\right)}^X;$$
• for each function $f:X\to Y$ in $\mathbf{Set}$ (hence a morphism $f^{\mathrm{op}}:Y\to X$ in ${\mathbf{Set}}^{\mathrm{op}}$), define

  $${\mathcal{F}}_{k,\mathbf{d}}\left(f^{\mathrm{op}}\right):{\mathcal{F}}_{k,\mathbf{d}}\left(Y\right)⟶{\mathcal{F}}_{k,\mathbf{d}}\left(X\right),\mu ⟼\mu \circ f.$$

Proposition 1.

${\mathcal{F}}_{k,\mathbf{d}}$ is a well-defined covariant functor ${\mathbf{Set}}^{\mathrm{op}}\to \mathbf{Set}$.

Proof. 

Let X be a set. For the identity ${id}_X:X\to X$, we have for any $\mu \in {\mathcal{F}}_{k,\mathbf{d}}\left(X\right)$:

$${\mathcal{F}}_{k,\mathbf{d}}\left({\left({id}_X\right)}^{\mathrm{op}}\right)\left(\mu \right)=\mu \circ {id}_X=\mu ,$$

so ${\mathcal{F}}_{k,\mathbf{d}}\left({\left({id}_X\right)}^{\mathrm{op}}\right)={id}_{{\mathcal{F}}_{k,\mathbf{d}}\left(X\right)}$. Next, for composable functions $X\stackrel{f}{\to }Y\stackrel{g}{\to }Z$ and any $\mu \in {\mathcal{F}}_{k,\mathbf{d}}\left(Z\right)$,

$${\mathcal{F}}_{k,\mathbf{d}}\left({(g\circ f)}^{\mathrm{op}}\right)\left(\mu \right)=\mu \circ (g\circ f)=(\mu \circ g)\circ f={\mathcal{F}}_{k,\mathbf{d}}\left(f^{\mathrm{op}}\right)\left({\mathcal{F}}_{k,\mathbf{d}}\left(g^{\mathrm{op}}\right)\left(\mu \right)\right).$$

Hence ${\mathcal{F}}_{k,\mathbf{d}}\left({(g\circ f)}^{\mathrm{op}}\right)={\mathcal{F}}_{k,\mathbf{d}}\left(f^{\mathrm{op}}\right)\circ {\mathcal{F}}_{k,\mathbf{d}}\left(g^{\mathrm{op}}\right)$. Therefore ${\mathcal{F}}_{k,\mathbf{d}}$ is a functor. □

Definition 22

(Tensor-valued Functorial Set). Fix $(k;\mathbf{d})$. The pair $\left({\mathbf{Set}}^{\mathrm{op}},{\mathcal{F}}_{k,\mathbf{d}}\right)$ is called thetensor-valued functorial set(of shape $(k;\mathbf{d})$). For a set X, an element $\mu \in {\mathcal{F}}_{k,\mathbf{d}}\left(X\right)$ is called atensor-valued ${\mathcal{F}}_{k,\mathbf{d}}$-set over X, i.e., a tensor-valued assignment $\mu :X\to {\mathsf{Ten}}_{k,\mathbf{d}}$.

Example 4

(A concrete example of a tensor-valued functorial set). Fix an order $n\ge 1$ and a shape $\mathbf{d}=(d_1,\dots ,d_n)$, and take $k=1$ for simplicity. Let

$${\mathsf{Ten}}_{\mathbf{d}}:={[0,1]}^{d_1\times \dots \times d_n}.$$

Consider the category $\mathbf{FinSet}$ of finite sets and functions, and work with its opposite category ${\mathbf{FinSet}}^{\mathrm{op}}$.

Define a functor

$${\mathcal{F}}_{\mathbf{d}}:{\mathbf{FinSet}}^{\mathrm{op}}⟶\mathbf{Set}$$

by:

$${\mathcal{F}}_{\mathbf{d}}\left(X\right):={\mathsf{Ten}}_{\mathbf{d}}^X=\{\mu :X\to {\mathsf{Ten}}_{\mathbf{d}}\},$$

and for any function $f:X\to Y$ in $\mathbf{FinSet}$ (hence $f^{\mathrm{op}}:Y\to X$ in ${\mathbf{FinSet}}^{\mathrm{op}}$),

$${\mathcal{F}}_{\mathbf{d}}\left(f^{\mathrm{op}}\right):{\mathcal{F}}_{\mathbf{d}}\left(Y\right)\to {\mathcal{F}}_{\mathbf{d}}\left(X\right),\mu ⟼\mu \circ f.$$

Then $({\mathbf{FinSet}}^{\mathrm{op}},{\mathcal{F}}_{\mathbf{d}})$ is a tensor-valued functorial set in the sense of Definition 22 (restricted from ${\mathbf{Set}}^{\mathrm{op}}$ to ${\mathbf{FinSet}}^{\mathrm{op}}$).

Interpretation (data refinement by reindexing).Think of an object $X\in \mathbf{FinSet}$ as a finite collection ofitems(products, patients, services, etc.). An element $\mu \in {\mathcal{F}}_{\mathbf{d}}\left(X\right)$ assigns to each item $x\in X$ a tensor

$$\mu \left(x\right)\in {\mathsf{Ten}}_{\mathbf{d}},$$

which may encode a structured, multi-index score profile (e.g., criterion × time × scenario).

Now let $f:X\to Y$ be acoarse-grainingmap that sends each fine item $x\in X$ to its group label $f\left(x\right)\in Y$ (e.g., product ↦ product-category, patient ↦ ward, service ↦ business-unit). Given a tensor assignment $\mu :Y\to {\mathsf{Ten}}_{\mathbf{d}}$ at the coarse level, the functorial pullback

$${\mathcal{F}}_{\mathbf{d}}\left(f^{\mathrm{op}}\right)\left(\mu \right)=\mu \circ f:X\to {\mathsf{Ten}}_{\mathbf{d}}$$

produces the induced fine-level assignment byreplicating the group tensor to every memberof that group. Concretely, for each $x\in X$,

$$\left(\mu \circ f\right)\left(x\right)=\mu \left(f\left(x\right)\right).$$

A small explicit instance.Let $Y=\{\mathrm{A},\mathrm{B}\}$ be two categories and $X=\{1,2,3,4\}$ four items with

$$f\left(1\right)=f\left(2\right)=\mathrm{A},f\left(3\right)=f\left(4\right)=\mathrm{B}.$$

Take $n=2$ and $\mathbf{d}=(2,2)$, so ${\mathsf{Ten}}_{\mathbf{d}}={[0,1]}^{2\times 2}$. Define $\mu \in {\mathcal{F}}_{\mathbf{d}}\left(Y\right)$ by

$$\mu \left(\mathrm{A}\right)=\left(\begin{array}{cc}0.9& 0.4\\ 0.2& 0.7\end{array}\right),\mu \left(\mathrm{B}\right)=\left(\begin{array}{cc}0.3& 0.8\\ 0.6& 0.1\end{array}\right).$$

Then the pulled-back assignment $\mu \circ f\in {\mathcal{F}}_{\mathbf{d}}\left(X\right)$ is

$$(\mu \circ f)\left(1\right)=(\mu \circ f)\left(2\right)=\mu \left(\mathrm{A}\right),(\mu \circ f)\left(3\right)=(\mu \circ f)\left(4\right)=\mu \left(\mathrm{B}\right).$$

Why this is a tensor-valued functorial example.This example illustrates the essential functorial mechanism: tensor-valued structure is assigned to each object (as ${\mathsf{Ten}}_{\mathbf{d}}$-valued functions), and transported along morphisms by pullback (precomposition), so that identities and compositions are respected automatically.

Definition 23

(Subfunctor determined by an uncertainty model). Let M be a tensor-valued uncertainty model with $Dom\left(M\right)\subseteq {\mathsf{Ten}}_{k,\mathbf{d}}$. Define a functor

$${\mathcal{F}}_M:{\mathbf{Set}}^{\mathrm{op}}⟶\mathbf{Set}$$

by restricting the codomain:

$${\mathcal{F}}_M\left(X\right):=\{\mu :X\to Dom\left(M\right)\}=Dom{\left(M\right)}^X,{\mathcal{F}}_M\left(f^{\mathrm{op}}\right)\left(\mu \right):=\mu \circ f.$$

Proposition 2.

${\mathcal{F}}_M$ is a well-defined functor and a subfunctor of ${\mathcal{F}}_{k,\mathbf{d}}$ (i.e., ${\mathcal{F}}_M\left(X\right)\subseteq {\mathcal{F}}_{k,\mathbf{d}}\left(X\right)$ for all X, and the action on morphisms agrees).

Proof. 

Functoriality follows by the same identity/composition calculation as Proposition 1. Moreover, since $Dom\left(M\right)\subseteq {\mathsf{Ten}}_{k,\mathbf{d}}$, any map $\mu :X\to Dom\left(M\right)$ is also a map $\mu :X\to {\mathsf{Ten}}_{k,\mathbf{d}}$, hence ${\mathcal{F}}_M\left(X\right)\subseteq {\mathcal{F}}_{k,\mathbf{d}}\left(X\right)$. The morphism action is identical (precomposition), so ${\mathcal{F}}_M$ is a subfunctor. □

Theorem 1

(Tensor-valued Functorial Sets subsume Tensor-valued Uncertain Sets). Let M be a tensor-valued uncertainty model with $Dom\left(M\right)\subseteq {\mathsf{Ten}}_{k,\mathbf{d}}$. For every set X, there is a canonical bijection between:

• tensor-valued Uncertain Sets of type M on X (i.e., maps ${\mu }_M:X\to Dom\left(M\right)$), and
• ${\mathcal{F}}_M$-sets over X (i.e., elements of ${\mathcal{F}}_M\left(X\right)$).

Moreover, this identification is functorial: for any function $f:X\to Y$, the transport of tensor-valued uncertain sets along f is exactly given by the functor action ${\mathcal{F}}_M\left(f^{\mathrm{op}}\right)\left(\mu \right)=\mu \circ f$.

Proof. 

Fix X. By definition, a tensor-valued Uncertain Set of type M on X is precisely a map ${\mu }_M:X\to Dom\left(M\right)$. On the other hand,

$${\mathcal{F}}_M\left(X\right)=Dom{\left(M\right)}^X=\{\mu :X\to Dom\left(M\right)\}.$$

Hence the correspondence “take the same function” gives a bijection between the two collections.

For functoriality, let $f:X\to Y$ and let $\mu :Y\to Dom\left(M\right)$ represent a tensor-valued uncertain set on Y. The induced assignment on X obtained by pullback along f is $\mu \circ f:X\to Dom\left(M\right)$. But by Definition 23, this is exactly ${\mathcal{F}}_M\left(f^{\mathrm{op}}\right)\left(\mu \right)$. Therefore the identification respects the morphism action, and the tensor-valued functorial set framework indeed generalizes tensor-valued uncertain sets. □

4. Conclusions

We introduced vector-valued, matrix-valued, and tensor-valued Uncertain Sets (including the corresponding fuzzy, neutrosophic, and related special cases) and investigate their fundamental properties. We expect that future work will further advance extensions based on hypergraphs [29,30] and SuperHyperGraphs [31,32].