A New Class of Interval-Valued Discrete Sugeno-like Integrals

Abstract

Discrete Sugeno integrals form a significant family of aggregation functions. Several variants of these integrals have been proposed, most of which replace the minimum operation with alternative operations, such as product, overlap functions, and t-norms. Notably, the associativity of t-norms in discrete Sugeno integrals has no significant consequences, leading us to relinquish this property and introduce the concept of partial t-norms. Although interval-valued versions of the discrete Sugeno integral or its variants have been limited, this paper presents a novel approach based on interval-valued partial t-norms, admissible orders, and interval-valued fuzzy measures. We provide rigorous proofs of the properties of this operator class.

1. Introduction

In 1974, Michio Sugeno introduced the concept of the “fuzzy integral” in his doctoral thesis [1], now widely known as the Sugeno integral. Sugeno demonstrated the basic properties of these integrals and presented several important and interesting applications [2,3,4]. The Sugeno integral, along with the Choquet integral, has played a crucial role in establishing a new era of generalized measurement and integration theory [5]. Several variants of the Choquet and Sugeno integrals have been proposed [6,7,8,9,10,11]. Notably, references [11,12] employ arbitrary t-norms instead of the minimum t-norm in the definition of Sugeno integrals. However, since the associativity of the t-norm has no impact on this variant of the Sugeno integral, we introduce a new class of conjunctive aggregation functions, called partial triangular norms (pt-norms), which results from removing the associativity axiom from the definition of t-norms.

Independently, references [13,14,15] proposed the use of closed subintervals of $[0, 1]$ as membership degrees of fuzzy sets. According to [16], assigning an exact value as the degree of membership to elements of the universe in a fuzzy set can be contradictory. However, interval-valued fuzzy sets can mitigate this issue. Moreover, when an expert provides degrees of relevance, they may be uncertain about the exact value, making the use of an interval a more suitable approach [17,18,19].

It is noteworthy that in various fuzzy theory concepts, such as aggregation functions, fuzzy implications, overlap indices, fuzzy entropy, and similarity measures, the usual order in $[0, 1]$ is crucial [20,21,22]. These notions have been extended to the context of closed subintervals of $[0, 1]$, considering the product order, which is not a total order, as seen in [23,24,25,26]. However, in some fuzzy set theory concepts, such as ordered weighted aggregation functions, discrete Sugeno and Choquet integrals, and their variants, the totality of order plays a fundamental role. In these cases, it is preferable to consider admissible orders, i.e., total orders in the set of closed subintervals of $[0, 1]$, refining the product order [27], as in [28,29,30,31,32].

Despite the numerous variants of the discrete Sugeno integral, only a few interval-valued versions have been proposed, including those in [29,30,33]. This paper focuses on a subclass of interval-valued fuzzy measures, as introduced in [31], but uses the product order instead of an admissible order. Furthermore, we extend the notion of pt-norms to the domain of closed subintervals of $[0, 1]$ endowed with an admissible order. By combining these two concepts, we propose a novel version of the discrete Sugeno integral for interval-valued data and investigate its properties.

This paper is organized as follows. In Section 2, some preliminary concepts of fuzzy set theory necessary for the rest of the sections are reviewed, but also the concept of pt-norm is introduced. In Section 3, some concepts from the theory of interval-valued fuzzy sets are revisited and the pt-norms with interval values are introduced. Section 4 is dedicated to defining and investigating the notion of a fuzzy measure with an interval value considering the product order. The main contributions of the paper are given in Section 5, with the definition and study of properties of the interval-valued discrete Sugeno-like integral proposal based on interval-valued pt-norms and the interval fuzzy measures proposed in Section 4 and admissible orders. Finally, some final considerations are offered in Section 6.

2. Preliminary Concepts

This section presents some basic mathematical concepts necessary for the next sections.

2.1. n-Ary Aggregation Functions

Let $A,B:{[0,1]}^n\to [0,1]$. Then, $A\le B$ if $A(x_1,\dots ,x_n)\le B(x_1,\dots ,x_n)$ for each $x_1,\dots ,x_n\in [0,1]$.

Definition 1

([21]). Function $A:{[0,1]}^n\to [0,1]$ is an n-ary aggregation function whenever the following conditions hold:

(A1) 

A is non-decreasing in each argument: for each $i\in \{1,\dots ,n\}$, if $x_i\le y$, then $A(x_1,\dots ,x_n)\le A(x_1,\dots ,x_{i-1},y,x_{i+1},\dots ,x_n)$;

(A2) 

A satisfies boundary conditions (i) $A(0,\dots ,0)=0$ and (ii) $A(1,\dots ,1)=1$.

In addition, an aggregation function A is conjunctive if $A\le min$, A is an average if $min\le A\le max$ and disjunctive if $max\le A$.

Definition 2

([34]). Function $T:{[0,1]}^2\to [0,1]$, such that for all $x,y,z\in [0,1]$, that satisfies conditions

(T1) 

Symmetry: $T(x,y)=T(y,x)$;

(T2) 

Associativity: $T(x,T(y,z\left)\right)=T\left(T\right(x,y),z)$;

(T3) 

Non-decreasing: if $y\le z$, then $T(x,y)\le T(x,z)$;

(T4) 

Boundary condition: $T(1,x)=x$.

is called a triangular norm or just a t-norm.

Example 1.

The best known t-norms are the minimum ($T_{min}$), the weak ($T_W$), the product ($T_P$), and the ukasiewicz ($T_L$) defined for each $x,y\in [0,1]$ by $T_{min}(x,y)=min(x,y)$, $T_W(x,y)=0$ if $max(x,y)<1$, and elsewhere $T_W(x,y)=x·y$, $T_P(x,y)=x·y$ and $T_L(x,y)=max(0,x+y-1)$.

Several generalizations of the t-norms have been proposed by relaxing or dropping some of Conditions (T1)–(T4), leading to concepts such as pseudo-t-norm [35,36], weak t-norm [37], and t-subnorm [34]. Notably, a non-associative t-norm was introduced in [38], but this concept still relies on continuity. In contrast, we discard the associativity requirement for t-norms in this work, without imposing continuity.

Definition 3.

Function $PT:{[0,1]}^2\to [0,1]$ is a partial t-norm (pt-norm, for short) if it satisfies Conditions (T1), (T3), and (T4). Let $\mathbb{PT}$ be the set of all pt-norms.

A pt-norm $PT$ is strict if for each $x,y,z\in [0,1]$ such that $y<z$, it holds that $PT(x,y)<PT(x,z)$. It is worth noting that each t-norm T is a pt-norm and each pt-norm is a conjunctive aggregation function. Furthermore, as in the case of t-norms, the set of pt-norms is also bounded by $T_W$ and $T_{min}$, i.e., for each pt-norm $PT$ and $x,y\in [0,1]$, $T_W(x,y)\le PT(x,y)\le T_{min}(x,y)$. The next result presents a structure for the set of all pt-norms.

Proposition 1.

Structure $\langle \mathbb{PT},\bigsqcup ,\sqcap ,T_W,T_{min}\rangle$ where $P{T}_1\bigsqcup P{T}_2(x,y)=P{T}_1(x,y)\vee P{T}_2(x,y)$ and $P{T}_1\sqcap P{T}_2(x,y)=P{T}_1(x,y)\wedge P{T}_2(x,y)$ for all pt-norms $P{T}_1$ and $P{T}_2$ and for all $x,y\in [0,1]$ is a bounded lattice.

Proof. 

By the symmetry of both $P{T}_1$ and $P{T}_2$, the symmetry of $P{T}_1\bigsqcup P{T}_2$ and $P{T}_1\sqcap P{T}_2$ follows. It holds that $P{T}_1\bigsqcup P{T}_2(1,x)=P{T}_1(1,x)\vee P{T}_2(1,x)=x\vee x=x$ for the boundary condition of $P{T}_1$ and $P{T}_2$. Hence, $P{T}_1\bigsqcup P{T}_2$ satisfies the boundary condition and analogously $P{T}_1\sqcap P{T}_2$ also does. Suppose that $y\le z$. Thus, $P{T}_1\bigsqcup P{T}_2(x,z)=P{T}_1(x,z)\vee P{T}_2(x,z)\ge P{T}_1(x,z)\ge P{T}_1(x,y)$ and $P{T}_1(x,z)\vee P{T}_2(x,z)\ge P{T}_2(x,z)\ge P{T}_2(x,y)$ for the non-decreasing condition of $P{T}_1$ and $P{T}_2$. Hence, $P{T}_1\bigsqcup P{T}_2(x,z)\ge P{T}_1(x,y)\vee P{T}_2(x,y)=P{T}_1\bigsqcup P{T}_2(x,y)$. Therefore, $P{T}_1\bigsqcup P{T}_2$ is non-decreasing. Also, by the non-decreasing condition of $P{T}_1$ and $P{T}_2$, it holds that $P{T}_1\sqcap P{T}_2(x,y)=P{T}_1(x,y)\wedge P{T}_2(x,y)\le P{T}_1(x,y)\le P{T}_1(x,z)$ and $P{T}_1(x,y)\wedge P{T}_2(x,y)\le P{T}_2(x,y)\le P{T}_2(x,z)$. Hence, $P{T}_1\sqcap P{T}_2(x,y)\le P{T}_1(x,z)\wedge P{T}_2(x,z)=P{T}_1\sqcap P{T}_2(x,z)$. Therefore, $P{T}_1\sqcap P{T}_2$ is non-decreasing, either. It holds trivially that $P{T}_1\bigsqcup P{T}_2$ and $P{T}_1\sqcap P{T}_2$ are, respectively, the least upper bound and greatest lower bound of $\{P{T}_1,P{T}_2\}$. □

2.2. Fuzzy Measures

The concept of capabilities was introduced in [39] to express the relevance or weight of a coalition in a large and important class of aggregation functions known today as Choquet integrals [21]. Michio Sugeno in [1] introduced a new class of integrals, today called the Sugeno integrals, which are based on the notion of fuzzy measures. The main difference between the notion of fuzzy measures and capabilities is the range of the function: while in capabilities the range is the set of real numbers, in fuzzy measures the range is $[0,1]$ [40]. In fact, every capability can be normalized, generating a fuzzy measure, and every fuzzy measure is a capability. In this paper, we are only interested in fuzzy measures defined on finite discrete sets.

Some notations: Let $N=\{1,\dots ,n\}$ be for an arbitrary $n>0$. In what follows $\wp \left(N\right)$ is the power set of N, $\#X$ the cardinality of X, $X^c$ is the complement of a $X\subseteq N$, i.e., $X^c=N\setminus X$.

Definition 4

([21]). Function $\mathfrak{m}:\wp \left(N\right)\to [0,1]$ is said to be a fuzzy measure if for all $X,Y\subseteq N$, the following conditions hold:

($\mathfrak{m}$1) 

Increasingness: if $X\subseteq Y$ then $\mathfrak{m}\left(X\right)\le \mathfrak{m}\left(Y\right)$;

($\mathfrak{m}$2) 

Boundary conditions: $\mathfrak{m}(\varnothing )=0$ and $\mathfrak{m}\left(N\right)=1$.

There are several types of fuzzy measures, the following among them [21]:

Symmetric if $\mathfrak{m}\left(X\right)=\mathfrak{m}\left(Y\right)$ for all $X,Y\subseteq N$ such that $\#X=\#Y$;

Additive if $\mathfrak{m}(X\cup Y)=\mathfrak{m}\left(X\right)+\mathfrak{m}\left(Y\right)$ for all $X,Y\subseteq N$ such that $X\cap Y=\varnothing$;

Boolean if $\mathfrak{m}\left(X\right)\in \{0,1\}$ for each $X\subseteq N$;

Self dual if $\mathfrak{m}\left(X\right)+\mathfrak{m}\left(X^c\right)=1$ for each $X\subseteq N$.

Note that each additive measure is self-dual, but the converse is not valid as can be seen in the next example, in which it is also highlighted that the symmetric, additive, and Boolean properties are completely independent.

Example 2.

Some fuzzy measures and their respective types are listed below and shown in Table 1, respectively. Let $X\subseteq N$. Hence,

1. 

The power measure: ${\mathfrak{m}}_{PM}^q\left(X\right)={\left({\displaystyle \frac{\#X}{n}}\right)}^q$ for a fix $q>0$ with $q\ne 1$;

2. 

The relative measure [41]: ${\mathfrak{m}}_R\left(X\right)={\displaystyle \frac{{\displaystyle \sum _{i\in X}}i}{{\displaystyle \sum _{i=1}^n}i}}$;

3. 

Let $k\in N\setminus \{1,n\}$. The ${max}_k$ measure ${\mathfrak{m}}_{{max}_k}\left(X\right)=\left\{\begin{array}{cc}1\hfill & ifmaxX\ge k\hfill \\ 0\hfill & ifmaxX<k;\hfill \end{array}\right.$

4. 

The λ-Dirac measure for $k\in N\setminus \{1,n\}$ and $\lambda \in (0.5,1)$:

${\mathfrak{m}}_{\lambda D}\left(X\right)=\left\{\begin{array}{cc}1\hfill & if X=N\hfill \\ 0\hfill & if X=\varnothing \hfill \\ \lambda \hfill & if k\in X and X\ne N\hfill \\ 1-\lambda \hfill & if k\notin X and X\ne \varnothing ;\hfill \end{array}\right.$

5. 

The uniform measure: ${\mathfrak{m}}_U\left(X\right)={\displaystyle \frac{\#X}{n}}$;

6. 

The Dirac measure [21] for a given $k\in N$: ${\mathfrak{m}}_D\left(X\right)=\left\{\begin{array}{cc}1\hfill & if k\in X\hfill \\ 0\hfill & if k\notin X;\hfill \end{array}\right.$

7. 

The weakest and strongest fuzzy measures [21]: ${\mathfrak{m}}_{\perp }\left(X\right)=\left\{\begin{array}{cc}1\hfill & if X=N\hfill \\ 0\hfill & if X\ne N\hfill \end{array}\right.$ and

${\mathfrak{m}}_{\top }\left(X\right)=\left\{\begin{array}{cc}1\hfill & if X\ne \varnothing \hfill \\ 0\hfill & if X=\varnothing ;\hfill \end{array}\right.$

8. 

Let $\alpha \in (0,1)$. The 3-valued measure ${\mathfrak{m}}_{3v}\left(X\right)=\left\{\begin{array}{cc}1\hfill & if X=N\hfill \\ 0\hfill & if X=\varnothing \hfill \\ \alpha \hfill & otherwise;\hfill \end{array}\right.$

9. 

Maximum (introduced here): ${\mathfrak{m}}_{max}\left(X\right)=\left\{\begin{array}{cc}{\displaystyle \frac{maxX}{n}}\hfill & if X\ne \varnothing \hfill \\ 0\hfill & if X=\varnothing .\hfill \end{array}\right.$

Observe that if $\mathfrak{m}$ is a fuzzy measure on N, then ${\mathfrak{m}}_{\perp }\le \mathfrak{m}\le {\mathfrak{m}}_{\top }$, ${\mathfrak{m}}_{PM}^p\le {\mathfrak{m}}_{PM}^q$ when $1\le p\le q$ or $q\le p\le 1$, ${\mathfrak{m}}_{PM}^q\le {\mathfrak{m}}_{max}$ if $q\ge 1$.

Table 1. The fuzzy measures of Example 2 and the properties they satisfy.

Fuzzy Measure	Symmetric	Additive	Boolean	Self Dual
${\mathfrak{m}}_{PM}^q$ for $0<q\ne 1$	✓	×	×	×
${\mathfrak{m}}_R$	×	✓	×	✓
${\mathfrak{m}}_{{max}_k}$ for $k\in N\setminus \{1,n\}$	×	×	✓	×
${\mathfrak{m}}_{\lambda D}$ for $k\in N\setminus \{1,n\}$ and $\lambda \in (0,5)$	×	×	×	✓
${\mathfrak{m}}_U$	✓	✓	×	✓
${\mathfrak{m}}_D$	×	✓	✓	✓
${\mathfrak{m}}_{\perp }$	✓	×	✓	×
${\mathfrak{m}}_{\top }$	✓	×	✓	×
${\mathfrak{m}}_{3v}$ for $\alpha \in (0,0.5)\cup (0.5,1)$	✓	×	×	×
${\mathfrak{m}}_{3v}$ for $\alpha =0.5$	✓	×	×	✓
${\mathfrak{m}}_{max}$	×	×	×	×

Table 1. The fuzzy measures of Example 2 and the properties they satisfy.

Fuzzy Measure	Symmetric	Additive	Boolean	Self Dual
${\mathfrak{m}}_{PM}^q$ for $0<q\ne 1$	✓	×	×	×
${\mathfrak{m}}_R$	×	✓	×	✓
${\mathfrak{m}}_{{max}_k}$ for $k\in N\setminus \{1,n\}$	×	×	✓	×
${\mathfrak{m}}_{\lambda D}$ for $k\in N\setminus \{1,n\}$ and $\lambda \in (0,5)$	×	×	×	✓
${\mathfrak{m}}_U$	✓	✓	×	✓
${\mathfrak{m}}_D$	×	✓	✓	✓
${\mathfrak{m}}_{\perp }$	✓	×	✓	×
${\mathfrak{m}}_{\top }$	✓	×	✓	×
${\mathfrak{m}}_{3v}$ for $\alpha \in (0,0.5)\cup (0.5,1)$	✓	×	×	×
${\mathfrak{m}}_{3v}$ for $\alpha =0.5$	✓	×	×	✓
${\mathfrak{m}}_{max}$	×	×	×	×

2.3. Discrete Sugeno Integrals

The term “integral” in mathematics generally refers to the idea of aggregating values of a function based on some “measure” of importance. For example, in the Lebesgue integral, values of a function are added together by considering the size of the sets in which the function assumes values. The Sugeno integral generalizes this concept, but instead of a traditional additive measure, it uses a fuzzy measure (not necessarily additive), which represents the “degree of importance” or “weight” of subsets of variables—such as the relevance of different criteria in a decision, for example. When the set of values is finite, the Sugeno integral is called discrete and always corresponds to an aggregation function in the sense of Definition 1. The discrete Sugeno integrals include several important classes of aggregation functions, such as medians, weighted minimum, and weighted maximum. Furthermore, these integral operators satisfy several important properties such as Lipschitz continuity, idempotency, and some comonotonicities which are useful in some applications [3,5,22,42].

Definition 5

(Definition 4.76 [22]). Let $\mathfrak{m}:\wp \left(N\right)\to [0,1]$ be a fuzzy measure. The discrete Sugeno integral is the function ${\mathcal{S}}_{\mathfrak{m}}:{[0,1]}^n\to [0,1]$ defined by (for all $\overrightarrow{x}=(x_1,\dots ,x_n)\in {[0,1]}^n$)

$${\mathcal{S}}_{\mathfrak{m}}\left(\overrightarrow{x}\right)=\underset{i=1}{\overset{n}{max}}min(x_{\left(i\right)},\mathfrak{m}\left(A_{\left(i\right)}\right)),$$

(1)

where $(x_{\left(1\right)},\dots ,x_{\left(n\right)})$ is an increasing permutation on the input $\overrightarrow{x}\in {[0,1]}^n$, that is, $x_{\left(1\right)}\le \dots \le x_{\left(n\right)}$ and $A_{\left(i\right)}=\{\left(i\right),\dots ,\left(n\right)\}$ is the subset of indices corresponding to the $n-i+1$ largest components of $\overrightarrow{x}$.

Furthermore, in [5,40], authors show how to compute the general Sugeno integral of a positive simple function $f:T\to \mathbb{R}$, where T is a nonempty set based on its cumulative distribution functions with respect to a capacity. Clearly, this approach can also be suited for the discrete Sugeno integrals as defined in Definition 5. For that, take any simple function $f:N\to [0,1]$, $\alpha \in [0,1]$ and the set $F_{\alpha }\left(f\right)=\{i\in N|f\left(i\right)\ge \alpha \}$. Function $g:(0,1]\to [0,1]$ defined as $g\left(\alpha \right)=\mathfrak{m}\left(F_{\alpha }\left(f\right)\right)$ is called the cumulative distribution function of f with respect to a fuzzy measure $\mathfrak{m}$. Then, for any non-empty and finite $M\subseteq (0,1]$, it is possible to compute the discrete Sugeno integral of f as follows:

$${\mathcal{S}}_{\mathfrak{m}}\left(f\right)=\underset{\alpha \in M}{max}min(\alpha ,g\left(\alpha \right)).$$

Remark 1

([22] page 174). An aggregation function is a Sugeno integral if and only if it is min and max-homogeneous. In addition, each Sugeno integral ${\mathcal{S}}_{\mathfrak{m}}$ is idempotent, comonotone maxitive, comonotone minimitive, and Kernel, that is,

1. 

min-homogeneous: ${\mathcal{S}}_{\mathfrak{m}}\left(\overrightarrow{x}\right)\wedge r={\mathcal{S}}_{\mathfrak{m}}(a_1\wedge r,\dots ,x_n\wedge r)$ for each $\overrightarrow{x}\in {[0,1]}^n$ and $r\in [0,1]$;

2. 

max-homogeneous: ${\mathcal{S}}_{\mathfrak{m}}\left(\overrightarrow{x}\right)\vee r={\mathcal{S}}_{\mathfrak{m}}(a_1\vee r,\dots ,x_n\vee r)$ for each $\overrightarrow{x}\in {[0,1]}^n$ and $r\in [0,1]$;

3. 

Idempotent: ${\mathcal{S}}_{\mathfrak{m}}(x,\dots ,x)=x$ for each $x\in [0,1]$;

4. 

Comonotone maxitive: ${\mathcal{S}}_{\mathfrak{m}}\left(\overrightarrow{x}\right)\vee {\mathcal{S}}_{\mathfrak{m}}\left(\overrightarrow{y}\right)={\mathcal{S}}_{\mathfrak{m}}(x_1\vee y_1,\dots ,x_n\vee y_n)$ for each pair of comonotone vectors $\overrightarrow{x},\overrightarrow{y}\in {[0,1]}^n$;

5. 

Comonotone minimitive: ${\mathcal{S}}_{\mathfrak{m}}\left(\overrightarrow{x}\right)\wedge {\mathcal{S}}_{\mathfrak{m}}\left(\overrightarrow{y}\right)={\mathcal{S}}_{\mathfrak{m}}(x_1\wedge y_1,\dots ,x_n\wedge y_n)$ for each pair of comonotone vectors $\overrightarrow{x},\overrightarrow{y}\in {[0,1]}^n$;

6. 

Kernel: $|{\mathcal{S}}_{\mathfrak{m}}\left(\overrightarrow{x}\right)-{\mathcal{S}}_{\mathfrak{m}}\left(\overrightarrow{y}\right)|\le \underset{i=1}{\overset{n}{max}}|x_i-y_i|$ for each $\overrightarrow{x},\overrightarrow{y}\in {[0,1]}^n$.

where two vectors $\overrightarrow{x},\overrightarrow{y}\in {[0,1]}^n$ are comonotone if there exists a permutation σ on N such that $x_{\sigma \left(1\right)}\le \dots \le x_{\sigma \left(n\right)}$ and $y_{\sigma \left(1\right)}\le \dots \le y_{\sigma \left(n\right)}$;

Sugeno integrals have already been generalized in different ways (see, for example, [43,44,45]). Here, we are particularly interested in the generalization where min is replaced by an arbitrary t-norm which is a particular case of the Sugeno-like q-integrals based on left conjunctions proposed in [46] taking a t-norm as conjunction. In this work, we consider pt-norms instead of t-norms.

Definition 6.

Let $\mathfrak{m}:\wp \left(N\right)\to [0,1]$ be a fuzzy measure and $PT$ be a partial t-norm. The discrete Sugeno integral based on $PT$ is function ${\mathcal{S}}_{\mathfrak{m}}^{PT}:{[0,1]}^n\to [0,1]$ defined for all $\overrightarrow{x}=(x_1,\dots ,x_n)\in {[0,1]}^n$ by

$${\mathcal{S}}_{\mathfrak{m}}^{PT}\left(\overrightarrow{x}\right)=\underset{i=1}{\overset{n}{max}}PT(x_{\left(i\right)},\mathfrak{m}\left(A_{\left(i\right)}\right)),$$

(2)

where $\left(x_{\left(1\right)},\dots ,x_{\left(n\right)}\right)$ is an increasing permutation on the input $\overrightarrow{x}$, that is, $x_{\left(1\right)}\le \dots \le x_{\left(n\right)}$, and $A_{\left(i\right)}=\{\left(i\right),\dots ,\left(n\right)\}$ is the subset of indices corresponding to the $n-i+1$ largest components of $\overrightarrow{x}$.

It should be noted that discrete Sugeno integrals based on a pt-norm also embed Pan integrals [47] which substitute the minimum in the Sugeno integral by the product.

3. Interval-Valued pt-Norms

Let $L\left(\right[0, 1\left]\right)=\left\{\right[a,b\left]\right|0\le a\le b\le 1\}$. For each arbitrary interval $X\in L\left(\right[0, 1\left]\right)$, its lower and upper extrema are indicated by $\underline{X}$ and $\overline{X}$, respectively. Note that $L\left(\right[0, 1\left]\right)$ is closed under the dot product, that is, for any $X,Y\in L\left(\right[0, 1\left]\right)$ and $\lambda \in [0, 1]$, we have $\lambda X=[\lambda \underline{X},\lambda \overline{X}]\in L\left([0,1]\right)$ and $X·Y=[\underline{X}·\underline{Y},\overline{X}·\overline{Y}]\in L\left([0,1]\right)$. We consider the following two partial orders on $L\left(\right[0, 1\left]\right)$, the product, and the subset inclusion orders

$$X{\le }_{p}Y \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if}\underline{X}\le \underline{Y} \mathrm{and} \overline{X}\le \overline{Y}$$

$$X\subseteq Y \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if}\underline{Y}\le \underline{X} \mathrm{and} \overline{X}\le \overline{Y}.$$

In addition, for the strict version of orders ${\le }_p$ and ⊆ (denoted, respectively, by ${<}_p$ and ⊂), we introduce the following auxiliary strict order:

$$X⋐Y \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if}\underline{Y}<\underline{X} \mathrm{and} \overline{X}<\overline{Y}$$

Remark 2.

X and Y are not comparable with respect to ${\le }_p$, that is, neither $X{\le }_{p}Y$ nor $Y{\le }_{p}X$ if and only if $X⋐Y$ or $Y⋐X$. In addition, if $X⋐Y$, then $X\subseteq Y$ and there is $Z\in L\left(\right[0,1\left]\right)$ such that $X⋐Z⋐Y$.

Furthermore, a partial order ⪯ on $L\left(\right[0, 1\left]\right)$ is an admissible order if it is a linear (total) order and refines ${\le }_p$, i.e., $X⪯Y$ when $X{\le }_{p}Y$ [27]. In some cases, when an infix notation is clearer, we use $X{\vee }_{⪯}Y$ and $X{\wedge }_{⪯}Y$ instead of ${max}_{⪯}(X,Y)$ and ${min}_{⪯}(X,Y)$, respectively.

Observe that for any two different admissible orders ${⪯}_1$ and ${⪯}_2$ and each $X,Y\in L\left(\right[0, 1\left]\right)$ such that $X{\prec }_{1}Y$ and $Y{\prec }_{2}X$, if $X{\le }_{p}Y$, then $X{⪯}_{2}Y$, which is a contradiction, and if $Y{\le }_{p}X$, then $Y{⪯}_{1}X$, which is also a contradiction. Then, by Remark 2, either $X⋐Y$ or $Y⋐X$. From now on, ⪯ always represents an arbitrary admissible order on $L\left(\right[0, 1\left]\right)$ and ≺ the strict order associated to ⪯, i.e., $X\prec Y$ if and only if $X\ne Y$ and $X⪯Y$. For specific admissible orders, a different notation is used as the following examples of admissible orders:

• Lexical 1: $X{⪯}_{Lex1}Y$ if and only if $\underline{X}<\underline{Y}$ or ($\underline{X}=\underline{Y}$ and $\overline{X}\le \overline{Y}$);
• Lexical 2: $X{⪯}_{Lex2}Y$ if and only if $\overline{X}<\overline{Y}$ or ($\overline{X}=\overline{Y}$ and $\underline{X}\le \underline{Y}$);
• Xu-Yager: $X{⪯}_{XY}Y$ if and only if $\underline{X}+\overline{X}<\underline{Y}+\overline{Y}$ or ($\underline{X}+\overline{X}=\underline{Y}+\overline{Y}$ and $\overline{X}-\underline{X}\le \overline{Y}-\underline{Y}$);
• $X_0$ order for a fixed $X_0\in L\left([0,1]\right)$:

  $$X{⪯}_{X_0}Y\iff \left\{\begin{array}{cc}X{\le }_{p}Y\hfill & \mathrm{or}\hfill \\ X⋐Y \mathrm{and} Y{\prec }_{Lex2}{X}_0\hfill & \mathrm{or}\hfill \\ Y⋐X \mathrm{and} X_0{⪯}_{Lex2}X\hfill \end{array}\right.$$
• $(r,s)$ order for $r,s\in {\mathbb{R}}^{+}$: $X{⪯}_s^{r}Y$ if and only if ${\overline{X}}^r+{\underline{X}}^s<{\overline{Y}}^r+{\underline{Y}}^s$ or ( ${\overline{X}}^r+{\underline{X}}^s={\overline{Y}}^r+{\underline{Y}}^s$ and $\overline{X}\le \overline{Y}$).

First, observe that indeed ${⪯}_{X_0}$ is an admissible order. Given $X,Y\in L\left(\right[0, 1\left]\right)$, directly from the definition of ${⪯}_{X_0}$, if $X{\le }_{p}Y$, then $X{⪯}_{X_0}Y$, and if $Y{\le }_{p}X$, then $Y{⪯}_{X_0}X$. So ${⪯}_{X_0}$ refines ${\le }_p$. Now, if X is not comparable with Y with respect to ${\le }_p$, by Remark 2, either $X⋐Y$ or $Y⋐X$. In the first case, if $Y{\prec }_{Lex2}{X}_0$, then by the definition of ${⪯}_{X_0}$ we have $X{⪯}_{X_0}Y$, and if $X_0{⪯}_{Lex2}Y$ then by definition of ${⪯}_{X_0}$ it holds that $Y{⪯}_{X_0}X$. In the second case, $Y⋐X$ is analogous. Therefore, ${⪯}_{X_0}$ is total and hence is an admissible order.

The first three admissible orders given above were defined in [27]. The $X_0$ order is a new proposed order and the $(r,s)$ order is a new family of admissible orders. Other families of admissible orders can be found in [23,32,48].

Remark 3.

If $X⋐Y$, then $X{⪯}_{Lex2}Y$ and $Y{⪯}_{Lex1}X$.

Remark 4.

For any two different admissible orders ${⪯}_1$ and ${⪯}_2$, there exist $X,Y\in L\left(\right[0,1\left]\right)$ such that $X{\prec }_{1}Y$ and $Y{\prec }_{2}X$. Then, since both ${⪯}_1$ and ${⪯}_2$ are admissible, if $X{<}_{p}Y$, then we have $X{\prec }_{2}Y$, which is a contradiction, and if $Y{<}_{p}X$, then $Y{\prec }_{1}X$, which is also a contradiction. Therefore, by Remark 2, either $X⋐Y$ or $Y⋐X$.

Definition 7.

An admissible order ⪯ is convex compatible combination, or just convex, if $\lambda X_1+(1-\lambda )X_2⪯\lambda Y_1+(1-\lambda )Y_2$ for each $\lambda \in [0,1]$ and $X_1,X_2,Y_1,Y_2\in L\left([0,1]\right)$ such that $X_1⪯Y_1$ and $X_2⪯Y_2$.

For instance, the admissible orders ${⪯}_{Lex1}$, ${⪯}_{Lex2}$ and ${⪯}_{XY}$ are convex, but ${⪯}_{X_0}$ is not convex when $X_0⋐[0,1]$. Indeed, for $X_1=\left[\underline{X_0}+{\displaystyle \frac{\overline{X_0}-\underline{X_0}}{4}},\underline{X_0}+{\displaystyle \frac{\overline{X_0}-\underline{X_0}}{3}}\right]$, $Y_1=\left[\underline{X_0},\underline{X_0}+{\displaystyle \frac{\overline{X_0}-\underline{X_0}}{2}}\right]$, $X_2=\left[{\displaystyle \frac{\underline{X_0}}{4}},1\right]$ and $Y_2=\left[{\displaystyle \frac{\underline{X_0}}{2}},\overline{X_0}+{\displaystyle \frac{1-\overline{X_0}}{4}}\right]$, we have that $X_1{⪯}_{X_0}{Y}_1$ since $X_1⋐Y_1$ and $Y_1{⪯}_{Lex2}{X}_0$ and $X_2{⪯}_{X_0}{Y}_2$ because $Y_2⋐X_2$ and $X_0{⪯}_{Lex2}{X}_2$. But, for $\lambda =0.5$, $\lambda X_1+(1-\lambda )X_2={\displaystyle \frac{X_1+X_2}{2}}={\displaystyle \frac{[12\underline{X_0}+3\overline{X_0},12+8\underline{X_0}+4\overline{X_0}]}{24}}$ and $\lambda Y_1+(1-\lambda )Y_2={\displaystyle \frac{Y_1+Y_2}{2}}={\displaystyle \frac{[18\underline{X_0},3+6\underline{X_0}+15\overline{X_0}]}{24}}$. That is, if $0.45<\underline{X_0}$ and $0.375<\overline{X_0}<0.9$, then $\lambda Y_1+(1-\lambda )Y_2⋐\lambda X_1+(1-\lambda )X_2$ and $X_0{\prec }_{Lex2}\lambda X_1+(1-\lambda )X_2$. Therefore, $\lambda Y_1+(1-\lambda )Y_2{\prec }_{X_0}\lambda X_1+(1-\lambda )X_2$.

Definition 8

([48]). Function $\mathbb{A}:L{\left([0,1]\right)}^n\to L\left([0,1]\right)$ is called an n ary interval-valued aggregation function with respect to ⪯ (⪯-IVAF in short) if it satisfies conditions

($\mathbb{A}$1) 

$\mathbb{A}$ is non-decreasing in each argument: for each $i\in \{1,\dots ,n\}$, if $X_i⪯Y$, then $\mathbb{A}(X_1,\dots ,X_n)⪯\mathbb{A}(X_1,\dots ,X_{i-1},Y,X_{i+1},\dots ,X_n)$;

($\mathbb{A}$2) 

$\mathbb{A}$ satisfies the boundary conditions: (i) $\mathbb{A}\left(\right[0,0],\dots ,[0,0\left]\right)=[0,0]$ and

(ii) $\mathbb{A}\left(\right[1,1],\dots ,[1,1\left]\right)=[1,1]$.

Interval-valued t-norms have been widely investigated as, for example, considering the product order in [49,50] and taking into account an arbitrary total order in [51]. Here, we consider an interval-valued generalization of pt-norms taking into account an arbitrary admissible order.

Definition 9.

Let ⪯ be an admissible order and $\mathcal{PT}:L{\left([0,1]\right)}^2\to L\left([0,1]\right)$ a function. Then, $\mathcal{PT}$ is an interval-valued partial t-norm with respect to ⪯ (⪯-IV pt-norm) if it satisfies the following conditions:

($\mathcal{PT}$1) 

Symmetry: $\mathcal{PT}(X,Y)=\mathcal{PT}(Y,X)$;

($\mathcal{PT}$2) 

⪯-Increasing: if $Y⪯Z$, then $\mathcal{PT}(X,Y)⪯\mathcal{PT}(X,Z)$;

($\mathcal{PT}$3) 

Boundary condition: $\mathcal{PT}\left(\right[1,1],X)=X$.

Example 3.

Some examples of IV pt-norms for an arbitrary admissible order (and also for ${\le }_p$ as in [49]) for $X,Y\in L\left(\right[0,1\left]\right)$ are given as follows:

• $${\mathcal{PT}}_0(X,Y)=\left\{\begin{array}{ll}X·Y\hfill & if X=[1,1] or Y=[1,1]\hfill \\ [0,0]& otherwise;\end{array}\right.$$
• ${\mathcal{PT}}_{mi{n}_{⪯}}(X,Y)={min}_{⪯}(X,Y)$;
• Let $X_0\in L\left([0,1]\right)$,

  $${\mathcal{PT}}_{X_0}(X,Y)=\left\{\begin{array}{ll}\underset{⪯}{min}(X,Y)\hfill & if X_0⪯\underset{⪯}{max}(X,Y)\hfill \\ [0,0]& otherwise.\end{array}\right.$$

Furthermore, for each member of the family ${⪯}_s^r$ of admissible orders, ${\mathcal{PT}}_P(X,Y)=X·Y$ is a ${⪯}_s^r$-IV pt-norm if it fulfills $1\le s\le r$ or $s\le r\le 1$.

It is worth noting that ${\mathcal{PT}}_0$ in Example 3 is an ⪯-IV pt-norm for any admissible order ⪯, i.e., it does not depend on the admissible order considered.

Proposition 2.

Let $\mathcal{PT}:L{\left([0,1]\right)}^2\to L\left([0,1]\right)$ be a ⪯-IV pt-norm. Then,

1. 

$\mathcal{PT}(X,[0,0\left]\right)=\mathcal{PT}\left(\right[0,0],X)=[0,0]$ for each $X\in L\left(\right[0,1\left]\right|)$;

2. 

${\mathcal{PT}}_0⪯\mathcal{PT}⪯{\mathcal{PT}}_{{min}_{⪯}}$.

Proof. 

From ($\mathcal{PT}1$), ($\mathcal{PT}2$), and ($\mathcal{PT}3$), it follows that $\mathcal{PT}\left(\right[0,0],X)=\mathcal{PT}(X,[0,0\left]\right)⪯\mathcal{PT}\left(\right[1,1],[0,0\left]\right)=[0,0]$ and therefore $\mathcal{PT}\left(\right[0,0],X)=\mathcal{PT}(X,[0,0\left]\right)=[0,0]$.

Let $X,Y\in L\left(\right[0,1\left]\right)$. If X or Y are $[0,0]$ or $[1,1]$, then from ($\mathcal{PT}1$), ($\mathcal{PT}3$) and the previous item ${\mathcal{PT}}_0(X,Y)=\mathcal{PT}(X,Y)={\mathcal{PT}}_{{min}_{⪯}}(X,Y)$. Elsewhere, ${\mathcal{PT}}_0(X,Y)=[0,0]⪯\mathcal{PT}(X,Y)$. Furthermore, from ($\mathcal{PT}1$), ($\mathcal{PT}2$), and ($\mathcal{PT}3$), it holds that $\mathcal{PT}(X,Y)⪯\mathcal{PT}(X,[1,1])=X={\mathcal{PT}}_{{min}_{⪯}}(X,Y)$ case $X⪯Y$ and $\mathcal{PT}(X,Y)⪯\mathcal{PT}([1,1],Y)=Y={\mathcal{PT}}_{{min}_{⪯}}(X,Y)$ case $Y⪯X$; then, in both cases, $\mathcal{PT}(X,Y)⪯{\mathcal{PT}}_{{min}_{⪯}}(X,Y)$. Therefore, ${\mathcal{PT}}_0(X,Y)⪯\mathcal{PT}(X,Y)⪯{\mathcal{PT}}_{{min}_{⪯}}(X,Y)$ for any $X,Y\in L\left(\right[0,1\left]\right)$. □

Corollary 1.

Let ${⪯}_1$ and ${⪯}_2$ be two different admissible orders. Then ${\mathcal{PT}}_{{min}_{{⪯}_1}}$ is not a ${⪯}_2$-IV pt-norm.

Proof. 

Since ${⪯}_1$ and ${⪯}_2$ are different, there are $X,Y\in L\left(\right[0,1\left]\right)$ such that $X{\prec }_{1}Y$ and $Y{\prec }_{2}X$. So, ${\mathcal{PT}}_{{min}_{{⪯}_2}}(X,Y)=Y{\prec }_{2}X={\mathcal{PT}}_{{min}_{{⪯}_1}}(X,Y)$ and therefore, by Proposition 2, ${\mathcal{PT}}_{{min}_{{⪯}_1}}$ is not a ${⪯}_2$-IV pt-norm. □

An important class of IV t-norms with respect to the product order are those defined by using two t-norms, one applied to the left bounds whereas the other one to the right bounds, called t-representable IV t-norms in [49]. In ([50], Corollary 33), it was proven that t-representable IV t-norms and inclusion monotonic IV t-norms are equivalent concepts. In general, the property fails for the case of ⪯-IV pt-norms. For example, $\widehat{T_L}([0.3,0.6],[0.2.0.8])=[0,0.4]{⪯}_{Lex1}[0,0.5]=\widehat{T_L}([0.2,0.7],[0.4.0.8])$ and $\widehat{T_{min}}([0.2,0.8],[0.4.0.6])=[0.2,0.6]{⪯}_{Lex2}[0.3,0.6]=\widehat{T_{min}}([0.3,0.6],[0.4.0.6])$, but $[0.2,0.7]{⪯}_{Lex1}[0.3,0.6]$ and $[0.3,0.6]{⪯}_{Lex2}[0.2,0.8]$. Given $P{T}_1,P{T}_2\in \mathbb{PT}$, $P{T}_1\le P{T}_2$, and either $P{T}_1$ or $P{T}_2$ are strict, the following propositions establish a sufficient and necessary condition for $\widehat{P{T}_{1}P{T}_2}(X,Y)=[P{T}_1(\underline{X},\underline{Y}),P{T}_2(\overline{X},\overline{Y})]$ to be a ⪯-IV pt-norm for some admissible order ⪯.

Proposition 3.

Let $P{T}_1,P{T}_2:{[0,1]}^2\to [0,1]$ be two pt-norms such that $P{T}_2$ is strict and $P{T}_1\le P{T}_2$. Then, $\widehat{P{T}_{1}P{T}_2}(X,Y)=[P{T}_1(\underline{X},\underline{Y}),P{T}_2(\overline{X},\overline{Y})]$ for all $X,Y\in L\left(\right[0,1\left]\right)$ is an ⪯-IV pt-norm if and only if $⪯={⪯}_{Lex2}$.

Proof. 

First, note that $\widehat{P{T}_{1}P{T}_2}$ is well defined, since $P{T}_1\le P{T}_2$.

$(\Rightarrow$) We prove this by contraposition. Suppose that $⪯\ne {⪯}_{Lex2}$; then, there are $Y,Z\in L\left(\right[0,1\left]\right)$ such that $Y\prec Z$ and $Z{\prec }_{Lex2}Y$. So, by Remarks 3 and 4, $Z⋐Y$ and thereafter $\widehat{P{T}_{1}P{T}_2}([0,1],Z)=[0,\overline{Z}]<[0,\overline{Y}]=\widehat{P{T}_{1}P{T}_2}([0,1],Y)$, i.e. $\widehat{P{T}_{1}P{T}_2}([0,1],Z)\prec \widehat{P{T}_{1}P{T}_2}([0,1],Y)$. Therefore, $\widehat{P{T}_{1}P{T}_2}$ is not an ⪯-IV pt-norm.

$(\Leftarrow )$ As $P{T}_1$ and $P{T}_2$ are symmetric and have 1 as a neutral element, it is immediate that $\widehat{P{T}_{1}P{T}_2}$ satisfies $\left(\mathcal{PT}1\right)$ and $\left(\mathcal{PT}3\right)$. If $Y⪯Z$, then, by the definition of ${⪯}_{Lex2}$, we have two cases:

1.

If $\overline{Y}<\overline{Z}$, then $P{T}_2(\overline{X},\overline{Y})<P{T}_2(\overline{X},\overline{Z})$ because $P{T}_2$ is strict. Therefore, $\widehat{P{T}_{1}P{T}_2}(X,Y)=[P{T}_1(\underline{X},\underline{Y}),P{T}_2(\overline{X},\overline{Y})]{⪯}_{Lex2}[P{T}_1(\underline{X},\underline{Z}),P{T}_2(\overline{X},\overline{Z})]$$=\widehat{P{T}_{1}P{T}_2}(X,Z)$.

2.

If $\overline{Y}=\overline{Z}$ and $\underline{Y}\le \underline{Z}$, then we have $P{T}_2(\overline{X},\overline{Y})=P{T}_2(\overline{X},\overline{Z})$ and $P{T}_1(\underline{X},\underline{Y})\le P{T}_1(\underline{X},\underline{Z})$. Hence, $\widehat{P{T}_{1}P{T}_2}(X,Y)=[P{T}_1(\underline{X},\underline{Y}),P{T}_2(\overline{X},\overline{Y})]{⪯}_{Lex2}[P{T}_1(\underline{X},\underline{Z}),P{T}_2(\overline{X},\overline{Z})]$$=\widehat{P{T}_{1}P{T}_2}(X,Z)$.

Thus, $\widehat{P{T}_{1}P{T}_2}$ satisfies $\left(\mathcal{PT}2\right)$. □

Proposition 4.

Let $P{T}_1,P{T}_2:{[0,1]}^2\to [0,1]$ be two pt-norms such that $P{T}_1$ is strict and $P{T}_1\le P{T}_2$. Then, $\widehat{P{T}_{1}P{T}_2}$ is a ⪯-IV pt-norm for some admissible order ⪯ if and only if $⪯={⪯}_{Lex1}$.

Proof. 

Analogous to Proposition 3. □

Note that, except in trivial cases, the convex combination of t-norms is not a t-norm [20,52]. Fortunately, pt-norms are trivially closed under convex combination, since it does not require associativity property. However, the convex combination of ⪯-IV pt-norms may not result in a ⪯-IV pt-norm for some admissible orders, which motivates the following definition.

Definition 10.

Let ${\mathcal{PT}}_1$ and ${\mathcal{PT}}_2$ be two ⪯-IV pt-norms. The convex combination of ${\mathcal{PT}}_1$ and ${\mathcal{PT}}_2$ for $\lambda \in [0,1]$ is the function $\mathcal{PT}:L{\left([0,1]\right)}^2\to L\left([0,1]\right)$ defined by

$$\mathcal{PT}(X,Y)=\lambda {\mathcal{PT}}_1(X,Y)+(1-\lambda ){\mathcal{PT}}_2(X,Y).$$

Theorem 1.

Let ⪯ be an admissible order. If ⪯ is convex, then ⪯-IV pt-norms are closed for convex combinations.

Proof. 

Let ${\mathcal{PT}}_1$ and ${\mathcal{PT}}_2$ be two ⪯-IV pt-norms, $\lambda \in [0,1]$ and $\mathcal{PT}$ the convex combination of ${\mathcal{PT}}_1$ and ${\mathcal{PT}}_2$ for $\lambda$. Then, it is easy to see that $\mathcal{PT}$ is symmetric and has $[1,1]$ as a neutral element. Let $X_1,X_2,Y_1,Y_2\in L\left([0,1]\right)$ such that $X_1⪯Y_1$ and $X_2⪯Y_2$; then, ${\mathcal{PT}}_1(X_1,X_2)⪯{\mathcal{PT}}_1(Y_1,Y_2)$ and ${\mathcal{PT}}_2(X_1,X_2)⪯{\mathcal{PT}}_2(Y_1,Y_2)$, and therefore, because ⪯ is convex, then $\mathcal{PT}(X_1,X_2)=\lambda {\mathcal{PT}}_1(X_1,X_2)+(1-\lambda ){\mathcal{PT}}_2(X_1,X_2)⪯\lambda {\mathcal{PT}}_1(Y_1,Y_2)+(1-\lambda ){\mathcal{PT}}_2(Y_1,Y_2)=\mathcal{PT}(Y_1,Y_2)$. Thereafter, $\mathcal{PT}$ is a ⪯-IV pt-norm. □

Corollary 2.

Let ${\mathcal{PT}}_1$ and ${\mathcal{PT}}_2$ be two ${⪯}_{Lex2}$-IV pt-norms. Then, the convex combination of ${\mathcal{PT}}_1$ and ${\mathcal{PT}}_2$ also is a ${⪯}_{Lex2}$-IV pt-norm.

Proof. 

Straightforward from Theorem 1. □

Corollary 3.

Let $P{T}_1,P{T}_2,P{T}_3,P{T}_4:{[0,1]}^2\to [0,1]$ be four pt-norms such that $P{T}_2$ and $P{T}_4$ are strict, $P{T}_1\le P{T}_2$ and $P{T}_3\le P{T}_4$. Then, for any $\lambda \in [0,1]$, the convex combination of $\widehat{P{T}_{1}P{T}_2}$ and $\widehat{P{T}_{3}P{T}_4}$ is a ${⪯}_{Lex2}$-IV pt-norm.

Proof. 

Straightforward from Proposition 3 and Theorem 1. □

4. Interval-Valued Fuzzy Measures

In [31,33], the notion of fuzzy measures was generalized in a natural way to fuzzy measures with interval values, where the counter domain is $L\left(\right[0, 1\left]\right)$ equipped with an admissible order. In this work, the product order is considered instead of an admissible order.

Definition 11.

Let $\wp \left(N\right)$ be the power set of N. Function $\mu :\wp \left(N\right)\to L\left(\right[0,1\left]\right)$ is called an interval-valued fuzzy measure if for all $X,Y\subseteq N$ the following conditions hold:

1. 

$\mu (\varnothing )=[0,0]$;

2. 

$\mu \left(N\right)=[1,1]$;

3. 

If $X\subseteq Y$, then $\mu \left(X\right){\le }_p\mu \left(Y\right)$.

Note that any interval-valued fuzzy measure in the sense of Definition 11 is also an interval-valued fuzzy measure with respect to any admissible order, that is, in the sense of [31,33].

Analogously to the punctual case, there are several types of interval-valued fuzzy measures, the following among them:

Symmetric if $\mu \left(X\right)=\mu \left(Y\right)$ for all $X,Y\subseteq N$ such that $\#X=\#Y$;

Additive if $\mu (X\cup Y)=\mu \left(X\right)+\mu \left(Y\right)$ for all $X,Y\subseteq N$ such that $X\cap Y=\varnothing$;

Boolean if $\mu \left(X\right)\in \left\{\right[0,0],[1,1\left]\right\}$ for each $X\subseteq N$;

Crisp if $\mu \left(X\right)$ is a degenerate interval for each $X\subseteq N$;

Self-dual if $\mu \left(X\right)=[1,1]-\mu \left(X^c\right)$ for each $X\subseteq N$.

Example 4.

In the following, some interval-valued fuzzy measures are listed with their respective type if it is the case. Let $X\subseteq N$.

1. 

The power interval measure: ${\mu }_{PM}^{[p,q]}\left(X\right)=\left[{\left({\displaystyle \frac{\#X}{n}}\right)}^q,{\left({\displaystyle \frac{\#X}{n}}\right)}^p\right]$ for a fixed $[p,q]\in L\left(\right[0,1\left]\right)$ such that $p>0$ is symmetric, and is additive and self-dual just when $p=q=1$, in which case it is called uniform;

2. 

The Dirac interval measure for a given $k\in N$: ${\mu }_D\left(X\right)=\left\{\begin{array}{ll}[1,1]\hfill & if k\in X\hfill \\ [0,0]& if k\notin X\end{array}\right.$ is Boolean, crisp, and self-dual;

3. 

The weakest and strongest interval fuzzy measures: ${\mu }_{\perp }\left(X\right)=\left\{\begin{array}{ll}[1,1]\hfill & if X=N\hfill \\ [0,0]& if X\ne N\end{array}\right.$ and ${\mu }_{\top }\left(X\right)=\left\{\begin{array}{ll}[1,1]\hfill & if X\ne \varnothing \hfill \\ [0,0]& if X=\varnothing \end{array}\right.$ both are Boolean, crisp, and symmetric;

4. 

Let $a,b\in (0,1]$ such that $a\le b$. The 3-interval-valued measure:

${\mu }_{3iv}\left(X\right)=\left\{\begin{array}{ll}[1,1]\hfill & if X=N\hfill \\ [0,0]& if X=\varnothing is crisp if and only if a=b\\ [a,b]& otherwise\end{array}\right.$;

5. 

Uniform-Maximum (introduced here): ${\mu }_{Umax}\left(X\right)=\left\{\begin{array}{ll}\left[{\displaystyle \frac{\#X}{n}},{\displaystyle \frac{maxX}{n}}\right]\hfill & if X\ne \varnothing \hfill \\ [0,0]& Otherwise.\end{array}\right.$

Theorem 2.

Function $\mu :\wp \left(N\right)\to L\left(\right[0,1\left]\right)$ is an interval-valued fuzzy measure if and only if there are two fuzzy measures ${\mathfrak{m}}_1$ and ${\mathfrak{m}}_2$ such that for each $X\subseteq N$, $\mu \left(X\right)=[{\mathfrak{m}}_1\left(X\right),{\mathfrak{m}}_2\left(X\right)]$.

Proof. 

(⇒) Let $\underline{\mu },\overline{\mu }:\wp \left(N\right)\to [0,1]$ be the functions defined for each $X\subseteq N$ by $\underline{\mu }\left(X\right)=\underline{\mu \left(X\right)}$ and $\overline{\mu }\left(X\right)=\overline{\mu \left(X\right)}$. Then, $\underline{\mu }(\varnothing )=\underline{\mu (\varnothing )}=\underline{[0,0]}=0$, and it is proven analogously that $\underline{\mu }\left(N\right)=1$, $\overline{\mu }(\varnothing )=0$ and $\overline{\mu }\left(N\right)=1$. Furthermore, if $X\subseteq Y\subseteq N$, then $\mu \left(X\right){\le }_p\mu \left(Y\right)$, i.e., $\underline{\mu \left(X\right)}\le \underline{\mu \left(Y\right)}$ and $\overline{\mu \left(X\right)}\le \overline{\mu \left(Y\right)}$, and therefore, $\underline{\mu }\left(X\right)\le \underline{\mu }\left(Y\right)$ and $\overline{\mu }\left(X\right)\le \overline{\mu }\left(Y\right)$. Thereafter, $\underline{\mu }$ and $\overline{\mu }$ are fuzzy measures and for each $X\subseteq N$, $\mu \left(X\right)=[\underline{\mu }\left(X\right),\overline{\mu }\left(X\right)]$.

(⇐) $\mu (\varnothing )=[{\mathfrak{m}}_1(\varnothing ),{\mathfrak{m}}_2(\varnothing )]=[0,0]$, $\mu \left(N\right)=[{\mathfrak{m}}_1\left(N\right),{\mathfrak{m}}_2\left(N\right)]=[1,1]$ and if $X\subseteq Y\subseteq N$, then ${\mathfrak{m}}_1\left(X\right)\le {\mathfrak{m}}_1\left(Y\right)$ and ${\mathfrak{m}}_2\left(X\right)\le {\mathfrak{m}}_2\left(Y\right)$, and thereafter, $\mu \left(X\right)=[{\mathfrak{m}}_1\left(X\right),{\mathfrak{m}}_2\left(X\right)]{\le }_p[{\mathfrak{m}}_1\left(Y\right),{\mathfrak{m}}_2\left(Y\right)]=\mu \left(Y\right)$. So, $\mu$ is an interval-valued fuzzy measure. □

Definition 12.

Let ${\mu }_1,{\mu }_2:\wp \left(N\right)\to L\left([0,1]\right)$ be two interval-valued fuzzy measures. Then, a convex combination of ${\mu }_1$ and ${\mu }_2$ for a $\lambda \in [0,1]$ is the function $\mu :\wp \left(N\right)\to L\left(\right[0,1\left]\right)$ defined for each $X\subseteq N$ by

$$\mu \left(X\right)=\lambda {\mu }_1\left(X\right)+(1-\lambda ){\mu }_2\left(X\right).$$

Proposition 5.

Let ${\mu }_1,{\mu }_2:\wp \left(N\right)\to L\left([0,1]\right)$ be two interval-valued fuzzy measures and $\lambda \in [0,1]$. Then, the convex combination of ${\mu }_1$ and ${\mu }_2$ for any $\lambda \in [0,1]$ is also an interval-valued fuzzy measure.

Proof. 

Straightforward. □

5. Interval-Valued Discrete Sugeno-like Integrais Based on ⪯-IV pt-Norms

There are only a few interval-valued versions of the discrete Sugeno integral and the discrete Sugeno-like integral. In particular, there is Aumann’s approach, where a Sugeno integral is applied to the lower ends of the intervals and also to the upper ends of the intervals [30,53]. In [30], an interval Sugeno integral definition was also introduced based on a preference value $\lambda \in [0, 1]$ to represent the size of the optimistic and pessimistic preferences over interval information that decision makers have. In [29], the authors propose an interval version of Choquet–Sugeno integrals which generalizes the Choquet and Sugeno integrals to interval-valued functions with an admissible order. Finally, Fumanal et al. [33] replace the maximum and minimum by two interval functions, G and F, and also consider an interval-valued fuzzy measure with respect to an admissible order such that these triples satisfy a specific condition to ensure that different increasing permutations of the inputs give the same results. It is worth noting that this approach is very general; indeed, if $F(X,Y)=X$ for each $X,Y\in L\left(\right[0, 1\left]\right)$, then any symmetric function $G:L{\left([0,1]\right)}^n\to L\left([0,1]\right)$ is an interval-valued Sugeno integral in the sense of Fumanal et al. In this section, we propose an alternative notion for the interval-valued version of the discrete Sugeno integral, utilizing an interval-valued pt-norm instead of the minimum and an interval-valued fuzzy measure based on the product order.

Given vector $\overrightarrow{X}=(X_1,\dots ,X_n)\in L{\left([0,1]\right)}^n$ and an admissible order ⪯ on $L\left(\right[0, 1\left]\right)$, a bijection $(·)$ of $N=\{1,\dots ,n\}$ is called ⪯-increasing permutation of $\overrightarrow{X}$ whenever $X_{\left(1\right)}⪯\dots ⪯X_{\left(n\right)}$.

Definition 13.

Let $\mathcal{PT}$ be a ⪯-IV pt-norm and $\mu :\wp \left(N\right)\to L\left(\right[0,1\left]\right)$ be an IV fuzzy measure. The interval-valued discrete Sugeno integral based on $\mathcal{PT}$ is the function ${\mathcal{S}}_{\mu }^{\mathcal{PT}}:L{\left([0,1]\right)}^n\to L\left([0,1]\right)$ defined for each $\overrightarrow{\mathbf{X}}=(X_1,\dots ,X_n)\in L{\left([0,1]\right)}^n$ by

$${\mathcal{S}}_{\mu }^{\mathcal{PT}}\left(\overrightarrow{\mathbf{X}}\right)=\underset{i=1}{\overset{n}{max}}⪯\mathcal{PT}(X_{\left(i\right)},\mu \left(H_{\left(i\right)}\right))$$

where $(X_{\left(1\right)},\dots ,X_{\left(n\right)})$ is an ⪯-increasing permutation of $\overrightarrow{X}$ and $H_{\left(i\right)}=\{\left(i\right),\dots ,\left(n\right)\}$.

Remark 5.

When there are repetitions of intervals in the input vector, there are different ⪯-increasing permutations with the same results. For example, the permutations $(2,1,3,5,4)$ and $[2,5,1,3,4]$ are ${⪯}_{XY}$-increasing permutations of the vector $\overrightarrow{X}=([0.3,0.4],[0.2,0.5],[0.3,0.4],$$[0.1,0.7],[0.3,0.4])$. However, using the first permutation, the result is

$$\begin{array}{cc}{\mathcal{S}}_{{\mu }_{Umax}}^{{\mathcal{PT}}_P}\left(\overrightarrow{\mathbf{X}}\right)\hfill & =\underset{i=1}{\overset{n}{max}}{⪯}_{XY}{\mathcal{PT}}_P(X_{\left(i\right)},{\mu }_{Umax}\left(H_{\left(i\right)}\right))\hfill \\ & =[0.2,0.5]·[1,1]{\vee }_{{⪯}_{XY}}[0.3,0.4]·[0.8,1]{\vee }_{{⪯}_{XY}}[0.3,0.4]·[0.6,1]{\vee }_{{⪯}_{XY}}\hfill \\ & [0.3,0.4]·[0.4,1]{\vee }_{{⪯}_{XY}}[0.1,0.7]·[0.2,0.8]\hfill \\ & =[0.2,0.5]\hfill \end{array}$$

and taking the second one, the result is

$$\begin{array}{cc}{\mathcal{S}}_{{\mu }_{Umax}}^{{\mathcal{PT}}_P}\left(\overrightarrow{\mathbf{X}}\right)\hfill & =\underset{i=1}{\overset{n}{max}}{⪯}_{XY}{\mathcal{PT}}_P(X_{\left[i\right]},{\mu }_{Umax}\left(H_{\left[i\right]}\right))\hfill \\ & =[0.2,0.5]·[1,1]{\vee }_{{⪯}_{XY}}[0.3,0.4]·[0.8,1]{\vee }_{{⪯}_{XY}}[0.3,0.4]·[0.6,0.8]{\vee }_{{⪯}_{XY}}\hfill \\ & [0.3,0.4]·[0.4,0.8]{\vee }_{{⪯}_{XY}}[0.1,0.7]·[0.2,0.8]\hfill \\ & =[0.2,0.5].\hfill \end{array}$$

Definition 14.

Let $\mathcal{PT}$ be a ⪯-interval-valued pt-norm and $\mu :\wp \left(N\right)\to L\left(\right[0,1\left]\right)$ an interval-valued fuzzy measure. Then, the interval-valued discrete Sugeno integral based on $\mathcal{PT}$ and μ is

1. 

Symmetric if ${\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1,\dots ,X_n)={\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_{\sigma \left(1\right)},\dots ,X_{\sigma \left(n\right)})$ for each $X_1,\dots ,X_n\in L\left([0,1]\right)$ and permutation σ of N;

2. 

Averaging if ${min}_{⪯}⪯{\mathcal{S}}_{\mu }^{\mathcal{PT}}⪯{max}_{⪯}$;

3. 

${min}_{⪯}$-homogeneous if ${\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1{\wedge }_{⪯}R,\dots ,X_n{\wedge }_{⪯}R)={\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1,\dots ,X_n){\wedge }_{⪯}R$ for each $X_1,\dots ,X_n,R\in L\left([0,1]\right)$;

4. 

${max}_{⪯}$-homogeneous if ${\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1{\vee }_{⪯}R,\dots ,X_n{\vee }_{⪯}R)={\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1,\dots ,X_n){\vee }_{⪯}R$ for each $X_1,\dots ,X_n,R\in L\left([0,1]\right)$;

5. 

Idempotent if ${\mathcal{S}}_{\mu }^{\mathcal{PT}}(X,\dots ,X)=X$ for each $X\in L\left(\right[0,1\left]\right)$;

6. 

Comonotone maxitive whenever ${\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1{\vee }_{⪯}{Y}_1,\dots ,X_n{\vee }_{⪯}{Y}_n)={\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1,\dots ,X_n){\vee }_{⪯}$

${\mathcal{S}}_{\mu }^{\mathcal{PT}}(Y_1,\dots ,Y_n)$ for each ⪯-comonotone pair of vectors $\overrightarrow{X},\overrightarrow{Y}\in L{\left([0,1]\right)}^n$;

7. 

Comonotone minimitive if ${\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1{\wedge }_{⪯}{Y}_1,\dots ,X_n{\wedge }_{⪯}{Y}_n)={\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1,\dots ,X_n){\wedge }_{⪯}{\mathcal{S}}_{\mu }^{\mathcal{PT}}(Y_1,\dots ,Y_n)$ for each ⪯-comonotone pair of vectors $\overrightarrow{X},\overrightarrow{Y}\in L{\left([0,1]\right)}^n$;

Proposition 6.

Let $\mathcal{PT}$ be an ⪯-interval-valued pt-norm and $\mu :\wp \left(N\right)\to L\left(\right[0,1\left]\right)$ be an interval-valued fuzzy measure. Then, the discrete Sugeno integral with interval values based on $\mathcal{PT}$ and μ is an idempotent and averaging function. In addition, if μ is symmetric, then ${\mathcal{S}}_{\mu }^{\mathcal{PT}}$ is also symmetric.

Proof. 

Idempotency: ${\mathcal{S}}_{\mu }^{\mathcal{PT}}(X,\dots ,X)=\mathcal{PT}(X,[1,1]){\vee }_{⪯}\underset{i=2}{\overset{n}{max}}⪯\mathcal{PT}(X_{\left(i\right)},\mu \left(H_{\left(i\right)}\right))=X$ for each $X\in L\left(\right[0,1\left]\right)$.

Averaging: $mi{n}_{⪯}(X_1,\dots ,X_n)=X_{\left(1\right)}=\mathcal{PT}(X_{\left(1\right)},[1,1])⪯\mathcal{PT}(X_{\left(1\right)},[1,1]){\vee }_{⪯}\underset{i=2}{\overset{n}{max}}⪯\mathcal{PT}(X_{\left(i\right)},\mu \left(H_{\left(i\right)}\right))={\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1,\dots ,X_n)$. Furthermore, since by Proposition 2, $\mathcal{PT}(X_{\left(i\right)},\mu \left(H_{\left(i\right)}\right))⪯{\mathcal{PT}}_{mi{n}_{⪯}}(X_{\left(i\right)},\mu \left(H_{\left(i\right)}\right))⪯X_{\left(i\right)}⪯X_{\left(n\right)}$, it holds that ${\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1,\dots ,X_n)$$=\underset{i=1}{\overset{n}{max}}⪯\mathcal{PT}(X_{\left(i\right)},\mu \left(H_{\left(i\right)}\right))⪯X_{\left(n\right)}=ma{x}_{⪯}(X_1,\dots ,X_n)$ for any $X_1,\dots ,X_n\in L\left([0,1]\right)$.

Symmetry: Let $X_1,\dots ,X_n\in L\left([0,1]\right)$, $[·]$ a permutation of N and $Y_i=X_{\left[i\right]}$ for each $i\in N$. Then, if $(·)$ is an increasing permutation of $(X_1,\dots ,X_n)$ and $\langle ·\rangle$ an increasing permutation of $(Y_1,\dots ,Y_n)$, then $X_{\left(i\right)}=Y_{\langle i\rangle }$ and, because $\mu$ is symmetric, $\mu \left(H_{\left(i\right)}\right)=\mu \left(H_{\langle i\rangle }\right)$. So, ${\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1,\dots ,X_n)=\underset{i=1}{\overset{n}{max}}⪯\mathcal{PT}(X_{\left(i\right)},\mu \left(H_{\left(i\right)}\right))=\underset{i=1}{\overset{n}{max}}⪯\mathcal{PT}(Y_{\langle i\rangle },\mu \left(H_{\langle i\rangle }\right))={\mathcal{S}}_{\mu }^{\mathcal{PT}}(Y_1,\dots ,Y_n)$. □

Theorem 3.

Let $\mathcal{PT}$ be a ⪯-interval-valued pt-norm and $\mu :\wp \left(N\right)\to L\left(\right[0,1\left]\right)$ be an interval-valued fuzzy measure. Then, the interval-valued discrete Sugeno integral based on $\mathcal{PT}$ and μ is ${min}_{⪯}$-homogeneous, ${max}_{⪯}$-homogeneous, comonotone maxitive, and comonotone minitive.

Proof. 

Let $X_1,\dots ,X_n,R\in L\left([0,1]\right)$ and $(·)$ be an ⪯-increasing permutation of $\overrightarrow{X}=(X_1,\dots ,X_n)$, i.e., $X_{\left(1\right)}⪯\dots ⪯X_{\left(n\right)}$. By convention, consider $X_{(n+1)}=[1,1]$.

Then, on the one hand, either $R\prec X_{\left(1\right)}$, in which case, because ${\mathcal{S}}_{\mu }^{\mathcal{PT}}$ is idempotent and an averaging function (proven in Proposition 6), it holds that

$$\begin{array}{cc}{\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1{\wedge }_{⪯}R,\dots ,X_n{\wedge }_{⪯}R)\hfill & ={\mathcal{S}}_{\mu }^{\mathcal{PT}}(R,\dots ,R)\hfill \\ & =R\hfill \\ & ={\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1,\dots ,X_n){\wedge }_{⪯}R\hfill \end{array},$$

or there is $k\in N$ such that $X_{\left(k\right)}⪯R⪯X_{(k+1)}$; then, $Y_{\left(1\right)}=X_{\left(1\right)},\dots ,Y_{\left(k\right)}=X_{\left(k\right)}$, $Y_{(k+1)}=\dots =Y_{\left(n\right)}=R$. So,

$\begin{array}{cc}{\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1{\wedge }_{⪯}R,\dots ,X_n{\wedge }_{⪯}R)\hfill & =\underset{i=1}{\overset{n}{max}}⪯\mathcal{PT}(Y_{\left(i\right)},\mu \left(H_{\left(i\right)}\right))\hfill \\ & =\mathcal{PT}(R,\mu \left(H_{(k+1)}\right)){\vee }_{⪯}\underset{i=1}{\overset{k}{max}}⪯\mathcal{PT}(X_{\left(i\right)},\mu \left(H_{\left(i\right)}\right))\hfill \\ & =\underset{i=1}{\overset{n}{max}}⪯\left(\mathcal{PT}(X_{\left(i\right)},\mu \left(H_{\left(i\right)}\right)){\wedge }_{⪯}R\right)\hfill \\ & =(\underset{i=1}{\overset{n}{max}}⪯\mathcal{PT}(X_{\left(i\right)},\mu \left(H_{\left(i\right)}\right))){\wedge }_{⪯}R\hfill \\ & ={\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1,\dots ,X_n){\wedge }_{⪯}R.\hfill \end{array}$

Therefore, ${\mathcal{S}}_{\mu }^{\mathcal{PT}}$ is ${min}_{⪯}$-homogeneous. The proof that it is also ${max}_{⪯}$-homogeneous is analogous.

On the other hand, let $\overrightarrow{X},\overrightarrow{Y}\in L{\left([0,1]\right)}^n$ be a ⪯-comonotone pair of vectors and $\overrightarrow{Z}=\overrightarrow{X}{\vee }_{⪯}\overrightarrow{Y}=(X_1{\vee }_{⪯}{Y}_1,\dots ,X_n{\vee }_{⪯}{Y}_n)$. If $(·):N\to N$ is a permutation such that $(X_{\left(1\right)},\dots ,x_{\left(n\right)})$ is a ⪯-increascing permutation of $\overrightarrow{X}$, then, by the comonotonicity of $\overrightarrow{X}$ with $\overrightarrow{Y}$, $(Y_{\left(1\right)},\dots ,Y_{\left(n\right)})$ and $(Z_{\left(1\right)},\dots ,Z_{\left(n\right)})$ are ⪯-increascing permutation of $\overrightarrow{Y}$ and $\overrightarrow{Z}$. Notice that $Z_{\left(i\right)}$ is either $X_{\left(i\right)}$ or $Y_{\left(i\right)}$, and therefore $\mathcal{PT}(Z_{\left(i\right)},\mu \left(H_{\left(i\right)}\right))=\mathcal{PT}(X_{\left(i\right)},\mu \left(H_{\left(i\right)}\right)){\vee }_{⪯}\mathcal{PT}(Y_{\left(i\right)},\mu \left(H_{\left(i\right)}\right))$. Thereafter, ${\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1{\vee }_{⪯}{Y}_1,\dots ,X_n{\vee }_{⪯}{Y}_n)=\underset{i=1}{\overset{n}{max}}⪯\mathcal{PT}(Z_{\left(i\right)},\mu \left(H_{\left(i\right)}\right))=\underset{i=1}{\overset{n}{max}}⪯\mathcal{PT}(X_{\left(i\right)},\mu \left(H_{\left(i\right)}\right)){\vee }_{⪯}\mathcal{PT}(Y_{\left(i\right)},\mu \left(H_{\left(i\right)}\right))$. Hence, ${\mathcal{S}}_{\mu }^{\mathcal{PT}}$ is comonotone maxitive. The proof that ${\mathcal{S}}_{\mu }^{\mathcal{PT}}$ also is comonotone minitive is analogous. □

The next theorem presents a sufficient condition to guarantee that the convex combination of two interval-valued discrete Sugeno integrals, as proposed in this paper, is also an interval-valued discrete Sugeno integral with respect to the same admissible order.

Theorem 4.

Let ⪯ be a convex admissible order, ${\mathcal{PT}}_1$ and ${\mathcal{PT}}_2$ be two ⪯-interval-valued pt-norms with respect to ⪯ and μ be an interval-valued fuzzy measure. Then, if for each $X,Y\in L\left(\right[0,1\left]\right)$ and $U,V\subseteq N$, we have that

$${\mathcal{PT}}_1(X,\mu \left(U\right))⪯{\mathcal{PT}}_1(Y,\mu \left(V\right))\iff {\mathcal{PT}}_2(X,\mu \left(U\right))⪯{\mathcal{PT}}_2(Y,\mu \left(V\right))$$

(3)

then the convex combination of ${\mathcal{S}}_{\mu }^{{\mathcal{PT}}_1}$ and ${\mathcal{S}}_{\mu }^{{\mathcal{PT}}_2}$ for $\lambda \in [0,1]$ is an interval-valued discrete Sugeno integral based on an ⪯-IV pt-norm.

Proof. 

Let $\lambda \in [0,1]$. By Theorem 1, the convex combination of ${\mathcal{PT}}_1$ and ${\mathcal{PT}}_2$, denoted by $\mathcal{PT}$, is an interval-valued pt-norm. Furthermore, since ${\mathcal{S}}_{\mu }^{{\mathcal{PT}}_1}(X_1,\dots ,X_n)={\mathcal{PT}}_1(X_{\left(k\right)},\mu \left(H_{\left(k\right)}\right))$ for some $k\in N$, then, by Equation (3), ${\mathcal{S}}_{\mu }^{{\mathcal{PT}}_2}(X_1,\dots ,X_n)={\mathcal{PT}}_2(X_{\left(k\right)},\mu \left(H_{\left(k\right)}\right))$. Therefore, because ⪯ is convex,

$$\begin{array}{cc}\lambda {\mathcal{PT}}_1(X_{\left(k\right)},\mu \left(H_{\left(k\right)}\right))+(1-\lambda ){\mathcal{PT}}_2(X_{\left(k\right)},\mu \left(H_{\left(k\right)}\right))\hfill & =\mathcal{PT}(X_{\left(k\right)},\mu \left(H_{\left(k\right)}\right))\hfill \\ & =\underset{i=1}{\overset{n}{max}}⪯\mathcal{PT}(X_{\left(i\right)},\mu \left(H_{\left(i\right)}\right)).\hfill \end{array}$$

Hence, ${\mathcal{S}}_{\mu }^{\mathcal{PT}}(X_1,\dots ,X_n)=\lambda {\mathcal{S}}_{\mu }^{{\mathcal{PT}}_1}(X_1,\dots ,X_n)+(1-\lambda ){\mathcal{S}}_{\mu }^{{\mathcal{PT}}_2}(X_1,\dots ,X_n).$ □

6. Final Remarks

The Sugeno integral is an important average function that has been used in various decision-making methods, artificial intelligence, and data sciences, as one can see in [3].

Several studies have proposed variants of Sugeno integrals, replacing the minimum with alternative aggregation functions, such as multiplication [47], overlap functions [7], left conjunctions [46], and t-norms [11,12]. Given the significance of the t-norm class as conjunctive aggregation functions, we focus on Suarez integrals. However, since the associativity property of t-norms does not impact the properties of Sugeno-like integrals, it is reasonable to require only the other properties of t-norms. This motivated us to introduce the concept of partial triangular norms (pt-norms) as a substitute for t-norms in defining a new variant of Sugeno integrals.

On the other hand, Sugeno integrals and Sugeno-like integrals have been extended to the context of interval-valued fuzzy set theory. Specifically, the proposals in [29,30,33] employ an admissible order for ordering intervals and interval-valued measurements, which must be isotonic with respect to the inclusion sets and the admissible order. In contrast, the interval-valued Sugeno-like integrals proposed in this paper use an admissible order for ranking input intervals, but the interval-valued fuzzy measure is based on the product order, making it an interval-valued fuzzy measure for any admissible order. From a practical perspective, this approach is easier to define and more flexible, as it does not require changes when the admissible order is modified.

Furthermore, it was also proven that the proposed interval-valued Sugeno-like integrals with respect to an admissible order ⪯ are an interval-valued averaging function that satisfies some good properties, such as idempotency, symmetry, ${min}_{⪯}$-homogeneity, ${max}_{⪯}$-homogeneity, maxitive comonotonicity, and minitive comonotonicity.

In future studies, we want to explore some applications of the Least Squares Method proposed in [42] but considering discrete Sugeno integrals based on pt-norms (and some new Sugeno-like integrals) as the aggregation operator and analyze performance. From there, we will adapt the methods to work with interval data sets.