Modular Quasi-Pseudo Metrics and the Aggregation Problem

Abstract

The applicability of the distance aggregation problem has attracted the interest of many authors. Motivated by this fact, in this paper, we face the modular quasi-(pseudo-)metric aggregation problem, which consists of analyzing the properties that a function must have to fuse a collection of modular quasi-(pseudo-)metrics into a single one. In this paper, we characterize such functions as monotone, subadditive and vanishing at zero. Moreover, a description of such functions in terms of triangle triplets is given, and, in addition, the relationship between modular quasi-(pseudo-)metric aggregation functions and modular (pseudo-)metric aggregation functions is discussed. Specifically, we show that the class of modular (quasi-)(pseudo-)metric aggregation functions coincides with that of modular (pseudo-)metric aggregation functions. The characterizations are illustrated with appropriate examples. A few methods to construct modular quasi-(pseudo-)metrics are provided using the exposed theory. By exploring the existence of absorbent and neutral elements of modular quasi-(pseudo-)metric aggregation functions, we find that every modular quasi-pseudo-metric aggregation function with 0 as the neutral element is an Aumann function, is majored by the sum and satisfies the 1-Lipschitz condition. Moreover, a characterization of those modular quasi-(pseudo-)metric aggregation functions that preserve modular quasi-(pseudo-)metrics is also provided. Furthermore, the relationship between modular quasi-(pseudo-)metric aggregation functions and quasi-(pseudo-)metric aggregation functions is studied. Particularly, we have proven that they are the same only when the former functions are finite. Finally, the usefulness of modular quasi-(pseudo-)metric aggregation functions in multi-agent systems is analyzed.

1. Introduction

The aggregation of different pieces of information that come from several sources is a common practice in applied sciences. The importance of the aggregation process is given by the fact that such pieces of information are transformed into a unique numerical value that is used to make a decision, which will allow solving the problem under consideration. In many problems, the aforementioned pieces of information correspond to distances between different elements in our space. Thus, the aggregation can be understood as a way to induce a global distance from a collection of particular distances. A typical situation occurs in path planning in robotics. Indeed, an autonomous robot must go from a start point to a target point in such a way that all obstacles encountered on the path are adequately avoided. The robot must simultaneously calculate distances to different obstacles and then merge them in order to make the best decision.

The applicability and the relevance of aggregating has motivated many authors to study the distance aggregation problem. Concretely, the problem can be formulated as follows: given $n\in \mathbb{N}$ ($\mathbb{N}$ stands for the set of positive integer numbers), a function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is said to be a distance aggregation function provided if, for each collection of distances ${\left\{d_i\right\}}_{i=1}^n$ defined on the (non-empty) set X, the function $F\circ \tilde{d}$ is a distance on X, where $\tilde{d}:X\times X\to {[0,+\infty ]}^n$ is defined by $\tilde{d}(x,y)=\left(d_1(x,y),\dots ,d_n(x,y)\right)$ for all $x,y\in X$. Notice that we allow that the distance can take the $+\infty$ value. Some authors call this type of distance extended distances (see [1,2]). In this problem, the objective is to find the conditions that a function F must satisfy in order to merge the collection of distances ${\left\{d_i\right\}}_{i=1}^n$ defined on the set X into a single one that is also defined on X.

1.1. The Classical Approach

The distance aggregation problem has been explored when the distance is exactly a pseudo-metric in [3,4]. Following [1], let us recall that a pseudo-metric on a (non-empty) set X is a function $d:X\times X\to [0,\infty [$ such that for all $x,y,z\in X$:

pm1

$d(x,x)=0$,

pm2

$d(x,y)=d(y,x)$,

pm3

$d(x,z)\le d(x,y)+d(y,z)$.

A pseudo-metric d on X is a metric when it satisfies the condition $\left(m1\right)$ given for all $x,y\in X$ as follows:

m1

$d(x,y)=0\iff x=y$.

A characterization of pseudo-metric aggregation functions was provided in [3,4]. With the aim of introducing such a characterization, let us recall that, following [5], $(a,b,c)\in {[0,+\infty ]}^n$ forms an n-triangular triplet whenever $a⪯b+c$, $b⪯a+c$ and $c⪯a+b$. Notice that we denote by ≤ the usual partial order on the extended set of real numbers, and that $a⪯b\iff a_i\le b_i$ for all $i=1,\dots ,n$. According to the nomenclature in [6], if $(a,b,c)\in {(0,+\infty ]}^n$ is a n-triangular triplet, then we will say that it is a positive n-triangular triplet. Moreover, a function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ transforms n-triangular triplets into a 1-triangular triplet provided that $\left(F\right(a),F(b),F(c\left)\right)$ is a 1-triangular triplet when $(a,b,c)$ is an n-triangular triplet. Hereafter, $0_n$ will denote the element of ${[0,+\infty ]}^n$ given by $0_n=(0,\dots ,0)$.

From now on, if the distances under consideration are not extended, then we will consider functions ${F:[0,+\infty [}^n\to [0,+\infty [$, which can be understood as functions $F:{[0,+\infty ]}^n\to [0,+\infty ]$ such that $F\left(\right[0,+\infty \left[\right)\subseteq [0,+\infty [$.

The following result was obtained in [3,4].

Theorem 1.

Let $n\in \mathbb{N}$, and let ${F:[0,+\infty [}^n\to [0,+\infty [$ be a function. Then the following statements are equivalent to each other.

(1) 

F is a pseudo-metric aggregation function.

(2) 

F satisfies the following conditions:

(2.1) 

$F\left(0_n\right)=0$;

(2.2) 

F transforms n-triangular triplets into a 1-triangular triplet.

The problem of metric aggregation has been explored in [6]. Notice that metric aggregation functions are a special type of distance aggregation function, where the distances to be aggregated are exact metrics. Those functions that merge a collection of metrics into a single one were characterized in the spirit of Theorem 1 as follows:

Theorem 2.

Let $n\in \mathbb{N}$ and let ${F:[0,+\infty [}^n\to [0,+\infty [$ be a function. The following statements are equivalent to each other.

(1) 

F is a metric aggregation function.

(2) 

F satisfies the following conditions:

(2.1) 

$F\left(0_n\right)=0$;

(2.2) 

F transforms positive n-triangular triplets into a positive 1-triangular triplet;

(2.3) 

If ${a\in [0,+\infty [}^n$ and $F\left(a\right)=0$, then $min\{a_1,\dots ,a_n\}=0$.

The relationship between pseudo-metric aggregation functions and metric aggregation functions has not been explored. However, it is not hard to see that there are pseudo-metric aggregation functions that are not metric aggregation functions as, for instance, the function constantly equals zero. Moreover, a slight modification of Example 10 in [6] shows that there are metric aggregation functions that are not pseudo-metric aggregation functions. Indeed, consider the function ${F:[0,+\infty [}^2\to [0,+\infty [$ given by $F\left(a\right)=0$ if $min\{a_1,a_2\}=0$ and $F\left(a\right)=2$ otherwise. Then F satisfies all the conditions in Theorem 2, and thus, it is a metric aggregation function. However, F does not transform 2-triangular triplets into a 1-triangular triplet because $(a,b,c\in [0,+\infty {[}^2)$ forms a 2-triangular triplet when we consider $a=(1,0)$, $b=(0,1)$ and $c=(1,1)$ but $\left(F\right(a),F(b),F(c\left)\right)$ is not a 1-triangular triple.

The symmetry inherent to pseudo-metrics limits their applicability for describing real problems, and for this reason, the notion of the quasi-pseudo metric was introduced in the literature (see, for instance, [7,8]). According to [7,8], a quasi-pseudo-metric on a (non-empty) set X is a function $q:X\times X\to [0,+\infty [$ such that for all $x,y,z\in X:$

qp1

$q(x,x)=0$;

qp2

$q(x,z)\le q(x,y)+q(y,z)$.

A quasi-pseudo-metric q on X is called a quasi-metric when it satisfies condition (qm1) given for all $x,y\in X$ as follows:

qm1

$q(x,y)=q(y,x)=0\iff x=y$.

It must be stressed that every (pseudo-)metric is a quasi-(pseudo-)metric that additionally satisfies the symmetry.

In [9], an extension of Theorem 2 was obtained in the framework of quasi-metrics (see Theorem 3 below). Observe that quasi-(pseudo-)metric aggregation functions are a special type of distance aggregation function, where the distances to be aggregated are exactly quasi-(pseudo-)metrics. Thus, several characterizations of quasi-metric aggregation functions have been provided. In order to state them, we recall that a function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is said to be monotone if $F\left(a\right)\le F\left(b\right)$ for each $a,b\in {[0,+\infty ]}^n$, with $a⪯b$. Moreover, a function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is said to be subadditive if $F(a+b)\le F\left(a\right)+F\left(b\right)$ for each $a,b\in {[0,+\infty ]}^n$, where we use the symbol + for the usual addition on ${[0,+\infty ]}^n$ and on $[0,+\infty ]$ simultaneously.

Theorem 3.

Let $n\in \mathbb{N}$, and let ${F:[0,+\infty [}^n\to [0,+\infty [$ be a function. Then the following statements are equivalent to each other.

(1) 

F is a quasi-metric aggregation function.

(2) 

F satisfies the following conditions:

(2.1) 

$F\left(0_n\right)=0$;

(2.2) 

If $F\left(a\right)=0$, then $min\{a_1,\dots ,a_n\}=0$;

(2.3) 

$F\left(a\right)\le F\left(b\right)+F\left(c\right)$ for each $a,b,c\in {[0,+\infty )}^n$, with $a⪯b+c$.

(3) 

F satisfies the following conditions:

(3.1) 

$F\left(0_n\right)=0$;

(3.2) 

If $F\left(a\right)=0$, then $min\{a_1,\dots ,a_n\}=0$;

(3.3) 

F is monotone and subadditive.

Although Theorem 1 was not extended to the context of quasi-pseudo-metrics in [9], it is not hard to check that the following characterization holds.

Theorem 4.

Let $n\in \mathbb{N}$, and let ${F:[0,+\infty [}^n\to [0,+\infty [$ be a function. Then the following statements are equivalent to each other.

(1) 

F is a quasi-pseudo-metric aggregation function.

(2) 

F satisfies the following conditions:

(2.1) 

$F\left(0_n\right)=0$;

(2.2) 

$F\left(a\right)\le F\left(b\right)+F\left(c\right)$ for each $a,b,c\in {[0,+\infty )}^n$, with $a⪯b+c$.

(3) 

F satisfies the following conditions:

(3.1) 

$F\left(0_n\right)=0$;

(3.2) 

F is monotone and subadditive.

It is clear that every quasi-(pseudo-)metric aggregation function is a (pseudo)-metric aggregation function. Nevertheless, the converse is not true, as Example 7 in [6] shows. Concretely, such an example provides a (pseudo-)metric aggregation function that is not monotone, and thus, it is not a quasi-(pseudo-)metric aggregation function.

1.2. The Modular Approach and the New Aggregation Problem

In certain physical applications, the classical notion of a (pseudo-)metric is not appropriate, and it is necessary to incorporate a positive parameter in the metric axioms. This gives rise to the concept of modular metrics. Following [2] (see also [10]), a modular (pseudo-)metric on a non-empty set X is a function $w:]0,+\infty [\times X\times X\to [0,+\infty ]$ that satisfies for all $x,y,z\in X$ the conditions below:

MPM1

$w(\lambda ,x,x)=0$ for all $\lambda \in ]0,+\infty [$;

MPM2

$w(\lambda ,x,y)=w(\lambda ,y,x)$ for all $\lambda \in ]0,+\infty [$;

MPM3

$w(\lambda +\mu ,x,z)\le w(\lambda ,x,y)+w(\mu ,y,z)$ for all $\lambda ,\mu \in ]0,+\infty [$.

When axiom (MPM1) is replaced by the following one,

MM1

$w(\lambda ,x,y)=0$ for all $\lambda \in ]0,+\infty [\iff x=y$,

then w is called a modular metric on X.

A typical example of a modular (pseudo-)metric $w_d$ can be constructed from a given (pseudo-)metric d on a non-empty set X in the following way: $w_d(\lambda ,x,y)=\frac{d(x,y)}{\lambda }$ for all $x,y\in X$ and for all $\lambda \in ]0,+\infty [$. Observe that the value $w_d(\lambda ,x,y)$ provided by the modular pseudo-metric $w_d$ can be interpreted as the mean velocity between the point x and y in time $\lambda$. Thus, considering the collection $\{w_{d,\lambda }:\lambda \in ]0,+\infty \left[\right\}$, where $w_{d,\lambda }(x,y)=w_d(\lambda ,x,y)$, one has the velocity field, which is nonlinear in general.

It must be pointed out that the value $w(\lambda ,x,y)$ can be understood in general as the distance between x and y with respect to the parameter $\lambda \in ]0,+\infty [$.

Many topological and metric studies on modular (pseudo-)metric spaces have been developed in the last years (see, for instance, [2,11,12] and references therein). Applications of modular (pseudo-)metrics to operator theory have been given in the last years. A few works in this direction can be found, for instance, in [10,13,14,15,16,17].

Motivated by the growing interest in the distance aggregation problem, a characterization of those functions that merge a collection of modular (pseudo-)metrics has been provided in [18]. With the aim of recalling such a result, we introduce the notion of a modular (pseudo-)metric aggregation function.

On account of [18], given $n\in \mathbb{N}$, a function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is a modular (pseudo-)metric aggregation function provided that, for each collection of modular (pseudo-)metrics ${\left\{w_i\right\}}_{i=1}^n$ defined on X, the function $F\circ \tilde{w}$ is a modular (pseudo-)metric on X, where $\tilde{w}:]0,+\infty [\times X\times X\to {[0,+\infty ]}^n$ is given by $\tilde{w}(\lambda ,x,y)=\left(w_1(\lambda ,x,y),\dots ,w_n(\lambda ,x,y)\right)$ for all $x,y\in X$ and for all $\lambda \in ]0,+\infty [$.

Notice that the aggregation problem in the context of modular (pseudo-)metrics is a particular case of the general distance aggregation problem stated at the beginning of this section.

In light of the preceding notions, we can recall the aforementioned characterization, the proof of which can be found in [18].

Theorem 5.

Let $n\in \mathbb{N}$, and let $F:{[0,+\infty ]}^n\to [0,+\infty ]$ be a function. Then the following statements are equivalent to each other.

(1) 

F is a modular pseudo-metric aggregation function.

(2) 

$F\left(0_n\right)=0$ and F is monotone and subadditive.

(3) 

$F\left(0_n\right)=0$ and $F\left(c\right)\le F\left(a\right)+F\left(b\right)$ for all $a,b,c\in {[0,+\infty ]}^n$, with $c⪯a+b$.

(4) 

$F\left(0_n\right)=0$ and F is monotone and transforms n-triangular triplets into a 1-triangular triplet.

According to [18], in the case of modular metrics, the preceding characterization can be stated as follows:

Theorem 6.

Let $n\in \mathbb{N}$, and let $F:{[0,+\infty ]}^n\to [0,+\infty ]$ be a function. The following statements are equivalent to each other.

(1) 

F is a modular metric aggregation function.

(2) 

$F\left(0_n\right)=0$ and F is monotone and subadditive. Moreover, if $a\in {[0,+\infty ]}^n$ and $F\left(a\right)=0$, then $a_i=0$ for some $i=1,\dots ,n.$

(3) 

$F\left(0_n\right)=0$, and in addition, $F\left(c\right)\le F\left(a\right)+F\left(b\right)$ for all $a,b,c\in {[0,+\infty ]}^n$, with $c⪯a+b$. Moreover, if $a\in {[0,+\infty ]}^n$ and $F\left(a\right)=0$, then $a_i=0$ for some $i=1,\dots ,n.$

(4) 

$F\left(0_n\right)=0$ and F is monotone and transforms n-triangular triplets into a 1-triangular triplet. Moreover, if $a\in {[0,+\infty ]}^n$ and $F\left(a\right)=0$, then $a_i=0$ for some $i=1,\dots ,n.$

Notice that Theorems 5 and 6 give methods for generating new modular (pseudo-)metrics from a given collection of modular (pseudo-)metrics. This could allow the development of new modular measurement tools that are better adapted to certain applied problems. Further, both theorems extend in some sense Theorems 1 and 2 to the modular framework.

Of course, every modular metric aggregation function is always a modular pseudo-metric aggregation function (this fact was stated as Corollary 1 in [18]). The following instance, which can be found in Example 1 in [18], shows that the reciprocal implication does not hold true. Consider the function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ defined by $F\left(a\right)=0$ for all $a\in {[0,+\infty ]}^n$. Then F satisfies all assumptions in the statement of Theorem 5 except $F\left(a\right)=0$ when ${a\in ]0,+\infty ]}^n$, which implies that it does not satisfy all conditions in the statement of Theorem 6.

According to [11], the fact that the value $w_d(\lambda ,x,y)$, yielded by the modular (pseudo-) metric $w_d$ introduced above, can be interpreted as the mean velocity between the points x and y in time $\lambda$, and in addition, the fact that such a velocity is typically asymmetric suggests that there is not sound reasoning to impose symmetry in the definition of modular distances (axiom (MPM2)). Hence, the notion of a modular quasi-(pseudo-)metric is introduced and explored in [11].

Following [11], a modular quasi-pseudo-metric on a non-empty set X is a function $w:]0,+\infty [\times X\times X\to [0,+\infty ]$ that satisfies for all $x,y,z\in X$ the conditions below:

MQPM1

$w(\lambda ,x,x)=0$ for all $\lambda \in ]0,+\infty [$;

MQPM2

$w(\lambda +\mu ,x,z)\le w(\lambda ,x,y)+w(\mu ,y,z)$ for all $\lambda ,\mu \in ]0,+\infty [$.

When axiom (MQPM1) is replaced by the following one,

MQPM1

$w(\lambda ,x,y)=w(\lambda ,y,x)=0$ for all $\lambda \in ]0,+\infty [\iff x=y$,

then w is called a modular quasi-metric on X.

Note that the concept of modular (pseudo-)metric space can be retrieved from the notion of a quasi-(pseudo-)metric when symmetry is imposed.

A paradigmatic example of a modular quasi-(pseudo-)metric that is not a modular (pseudo-)metric is given by the function $w_u:]0,+\infty [\times [0,+\infty [\to [0,+\infty [$ defined by $w_u(\lambda ,x,y)=\frac{max\{y-x,0\}}{\lambda }$ for all $\lambda \in ]0,+\infty [$ and for all $x,y\in [0,+\infty [$.

In light of all of the exposed facts, as a natural line of work, in this paper, we focus our efforts on the modular quasi-(pseudo-)metric aggregation problem. Thus, Section 2 is devoted to providing new characterizations, in the spirit of Theorems 5 and 6, of those functions that allow merging a collection of modular quasi-(pseudo-)metrics into a single one, extending to the modular framework those given for quasi-(pseudo-)metric aggregation functions in Theorems 3 and 4. In particular, a description of such functions in terms of triangle triplets is given, and in addition, the relationship between modular quasi-(pseudo-)metric aggregation functions and modular (pseudo-)metric aggregation functions is discussed. The aforementioned characterizations are illustrated with appropriate examples. The new results can allow the introduction of new modular measurement tools better adapted to certain applied problems. Hence, the section ends with a dissertation on the potential applicability of modular quasi-(pseudo-)metric aggregation functions to multi-agent systems. In Section 3, a few methods to construct modular quasi-(pseudo-)metrics are yielded. Several properties that are common in aggregation theory are explored and used to develop quick tests for discarding candidate functions to aggregate modular quasi-(pseudo-)metrics. Specifically, the existence of absorbent and neutral elements of modular quasi-(pseudo-)metric aggregation functions is analyzed. As a consequence of such a study, we have obtained that every modular quasi-pseudo-metric aggregation function that has 0 as a neutral element is always an Aumann function, is majored by the sum and satisfies the 1-Lipschitz condition. Moreover, a characterization of those modular quasi-(pseudo-)metric aggregation functions that preserve modular quasi-(pseudo-)metrics is also provided. To be specific, it is shown that such functions are those that are idempotent. Furthermore, as a natural question, the connection between modular quasi-(pseudo-)metric aggregation functions and quasi-(pseudo-)metric aggregation functions is studied. Particularly, we have proved that they are the same only when the former are finite. Finally, the conclusions of the developed work and an exposition of future research lines are detailed in Section 4.

2. The Modular Quasi-(Pseudo-)Metric Aggregation Problem

In this section, we face the problem of merging a collection of modular quasi-(pseudo-) metrics as a natural extension of the problems exposed in the preceding section of aggregating a collection of modular (pseudo-)metrics.

To this end, let us introduce the notion of a modular quasi-(pseudo-)metric aggregation function. Thus, given $n\in \mathbb{N}$, a function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is said to be a modular quasi-(pseudo-)metric aggregation function provided that for each collection of modular quasi-(pseudo-)metrics ${\left\{w_i\right\}}_{i=1}^n$ defined on X, the function $F\circ \tilde{w}$ is a modular quasi-(pseudo-)metric on X, where $\tilde{w}:]0,+\infty [\times X\times X\to {[0,+\infty ]}^n$ is given by $\tilde{w}(\lambda ,x,y)=\left(w_1(\lambda ,x,y),\dots ,w_n(\lambda ,x,y)\right)$ for all $x,y\in X$ and for all $\lambda >0$.

The next result was proved in ([18], Theorem 1). However, since it will be of great importance to our objective, we will include its proof for the sake of completeness.

Theorem 7.

Let $n\in \mathbb{N}$. If $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is a monotone function such that $F\left(0_n\right)=0$, then the following statements are equivalent to each other.

(1) 

F is subadditive.

(2) 

$F\left(c\right)\le F\left(a\right)+F\left(b\right)$ for all $a,b,c\in {[0,+\infty ]}^n$ with $c⪯a+b$.

(3) 

F transforms n-triangular triplets into a 1-triangular triplet.

Proof. 

(1) ⇒ (2). Let $a,b\in {[0,+\infty ]}^n$, with $c⪯a+b$. Then

$$F\left(c\right)\le F(a+b)\le F\left(a\right)+F\left(b\right).$$

Notice that the first inequality is derived from the monotony of F, and the second one is due to the subadditivity of F.

(2) ⇒ (3). It is a straightforward verification.

(3) ⇒ (1). Consider $a,b\in {[0,+\infty ]}^n$. Then $(a,b,c)$ forms an n-dimensional triangular triplet, where $c=a+b$. Since F transforms n-triangular triplets into a 1-triangular triplet, we deduce that $\left(F\right(a),F(b),F(c\left)\right)$ is a 1-triangular triplet. So $F\left(c\right)=F(a+b)\le F\left(a\right)+F\left(b\right)$, and thus, F is subadditve. □

Next, we focus our attention on characterizing modular quasi-(pseudo-) metric aggregation functions. With this aim, we note that every modular quasi-(pseudo-) metric aggregation function is a modular (pseudo-)metric aggregation function, such as the following result shows.

Proposition 1.

Let $n\in \mathbb{N}$. If $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is a modular quasi-(pseudo-)metric aggregation function, then F is a modular (pseudo-)metric aggregation function.

Proof. 

Consider a collection of modular (pseudo-)metrics ${\left\{w_i\right\}}_{i=1}^n$ on a non-empty set X. Then ${\left\{w_i\right\}}_{i=1}^n$ is a collection of modular quasi-(pseudo-)metrics on X, and thus, $F\circ \tilde{w}$ is a modular quasi-(pseudo-)metric on X. Moreover, for all $\lambda \in ]0,+\infty [$ and for all $x,y\in X$, we have that

$$\begin{array}{c}F\circ \tilde{w}(\lambda ,x,y)=F(w_1(\lambda ,x,y),\dots ,w_n(\lambda ,x,y))\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=F((w_1(\lambda ,y,x),\dots ,w_n(\lambda ,y,x))=F\circ \tilde{w}(\lambda ,y,x)\end{array}$$

since $w_i(\lambda ,x,y)=w_i(\lambda ,y,x)$ for all $i\in \{1,\dots ,n\}$. So $F\circ \tilde{w}$ is a modular (pseudo-)metric on X. From this, we conclude that F is a modular (pseudo-)metric aggregation function. □

In light of Proposition 1 and Theorems 5–7, the following statements can be immediately obtained.

Proposition 2.

Let $n\in \mathbb{N}$, and let $F:{[0,+\infty ]}^n\to [0,+\infty ]$ be a modular quasi-(pseudo-)metric aggregation function. Then the following assertions hold:

(1) 

$F\left(0_n\right)=0$ and F is monotone and subadditive.

(2) 

$F\left(0_n\right)=0$ and $F\left(c\right)\le F\left(a\right)+F\left(b\right)$ for all $a,b,c\in {[0,+\infty ]}^n$, with $c⪯a+b$.

(3) 

$F\left(0_n\right)=0$ and F is monotone and transforms n-triangular triplets into a 1-triangular triplet.

The characterization of modular quasi-pseudo-metric aggregation functions can be stated as follows:

Theorem 8.

Let $n\in \mathbb{N}$, and let $F:{[0,+\infty ]}^n\to [0,+\infty ]$ be a function. Then the following statements are equivalent to each other.

(1) 

F is a modular quasi-pseudo-metric aggregation function.

(2) 

F is a modular pseudo-metric aggregation function.

(3) 

$F\left(0_n\right)=0$ and F is monotone and subadditive.

(4) 

$F\left(0_n\right)=0$ and $F\left(c\right)\le F\left(a\right)+F\left(b\right)$ for all $a,b,c\in {[0,+\infty ]}^n$, with $c⪯a+b$.

(5) 

$F\left(0_n\right)=0$ and F is monotone and transforms n-triangular triplets into a 1-triangular triplet.

Proof. 

(1) ⇒ (2) follows from Proposition 1. The equivalences (2) ⇔ (3) ⇔ (4) ⇔ (5) are provided by Theorem 5. Now, we prove that (4) ⇒ (1). To prove this, consider a collection ${\left\{w_i\right\}}_{i=1}^n$ of modular quasi-pseudo-metrics on a non-empty set X. Then $w_i(\lambda +\mu ,x,z)\le w_i(\lambda ,x,y)+w_i(\mu ,y,z)$ for all $\lambda ,\mu \in ]0,+\infty [$, for all $x,y\in X$ and for all $i\in \{1,\dots ,n\}$. From this, we obtain that $a,b,c\in {[0,+\infty ]}^n$ satisfies that $c⪯a+b$, with $c_i=w_i(\lambda +\mu ,x,z)$, $a_i=w_i(\lambda ,x,y)$ and $b_i=w_i(\mu ,y,z)$ for all $i\in \{1,\dots ,n\}$. This implies that

$$\begin{array}{cc}F\circ \tilde{w}(\lambda +\mu ,x,z)=F(w_1(\lambda +\mu ,x,z),\dots ,w_n(\lambda +\mu ,x,z))=F\left(c\right)\le F\left(a\right)+F\left(b\right)& \\ =F(w_1(\lambda ,x,y),\dots ,w_n(\lambda ,x,y))+F(w_1(\mu ,y,z),\dots ,w_n(\mu ,y,z))& \\ =F\circ \tilde{w}(\lambda ,x,y)+F\circ \tilde{w}(\mu ,y,z).& \end{array}$$

Hence, condition (MQPM2) is satisfied. It remains to be verified that condition (MQPM1) is also satisfied. Since $w_i(\lambda ,x,x)=0$ for all $\lambda \in ]0,+\infty [$ and for all $x\in X$, we have that

$$F\circ \tilde{w}(\lambda ,x,x)=F(w_1(\lambda ,x,x),\dots ,w_n(\lambda ,x,x))=F\left(0_n\right)=0.$$

Therefore, $F\circ \tilde{w}$ is a modular quasi-(pseudo-)metric on X, and hence, F is a modular quasi-(pseudo-)metric aggregation function. □

The fact that every modular metric aggregation function is a modular pseudo-metric aggregation function provides the following consequence.

Corollary 1.

Let $n\in \mathbb{N}$, and let $F:{[0,+\infty ]}^n\to [0,+\infty ]$ be a function. If F is a modular metric aggregation function, then F is a modular quasi-pseudo-metric aggregation function.

Proof. 

The proof follows immediately from Theorems 6 and 8. □

It seems natural to wonder whether the converse of the preceding corollary is also true. Nevertheless, the answer to the posed question is negative. Indeed, observe that the classes of modular quasi-pseudo-metric aggregation functions and modular pseudo-metric aggregation functions are the same. Then the function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ constantly equals 0 (as commented in Section 1), satisfies all assumptions in the statement of Theorem 8 and, thus, is a modular quasi-pseudo-metric aggregation function. However, $F\left(a\right)=0$ when ${a\in ]0,+\infty ]}^n$, which implies that it does not satisfy all conditions in the statement of Theorem 6, and hence, it is not a modular metric aggregation function.

The following instances are examples of modular quasi-pseudo-metric aggregation functions.

Example 1.

Let $n\in \mathbb{N}$. For all $w_1,\dots ,w_n\in [0,\infty [$, the function $F:{[0,+\infty ]}^n⟶[0,+\infty ]$ defined below is a modular quasi-pseudo-metric aggregation function:

(1) 

$F\left(a\right)=\left\{\begin{array}{cc}0& \mathit{if}a=0_n,\hfill \\ +\infty & \mathit{otherwise}.\hfill \end{array}\right.$

(2) 

$F\left(a\right)={\sum }_{i=1}^n{w}_i{a}_i$.

(3) 

$F\left(a\right)=max\{w_1{a}_1,\dots ,w_n{a}_n\}$.

(4) 

$F\left(a\right)={\left({\sum }_{i=1}^n{\left(w_i{a}_i\right)}^p\right)}^{\frac{1}{p}}$ for all $p\in [1,\infty [$.

(5) 

$F\left(a\right)=min\{c,{\sum }_{i=1}^n{w}_i{a}_i\}$ with $c\in (0,\infty )$.

The following examples show functions that are not modular quasi-pseudo-metric aggregation functions.

Example 2.

Let $k\in (0,\infty )$. Define the functions $F_k:{[0,+\infty ]}^n\to [0,+\infty ]$ as follows:

$$F_k\left(a\right)=\left\{\begin{array}{cc}0\hfill & \mathit{if}a=0_n,\hfill \\ k\hfill & \mathit{if}min\{a_1,\dots a_n\}>0,\hfill \\ +\infty \hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

It is evident that $F_k$ is not monotone since $k=F_k\left(a\right)<F_k\left(b\right)=+\infty$, with $(1,0,\dots ,0)=b⪯a=(1,\dots ,1)$. Theorem 8 guarantees that $F_k$ is not a modular quasi-pseudo-metric aggregation function.

Example 3.

Let the function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ be defined as follows:

$$F\left(a\right)=\left\{\begin{array}{cc}\frac{1}{2}\hfill & ifa=(0,0),\hfill \\ 1\hfill & otherwise.\hfill \end{array}\right.$$

It is clear that $F(0,0)\ne 0$, so Theorem 8 ensures that F is not a modular quasi-pseudo-metric aggregation function.

Once the modular quasi-pseudo-metric aggregation problem has been studied, in the following, a refinement of the aforementioned problem is faced. Concretely, we try to describe those functions that are able to merge a collection of modular quasi-metrics into a single one. Accordingly, we are interested in getting an appropriate version of Theorem 8 that extends Theorem 6.

The result below will play a crucial role in order to achieve our target. It must be stressed that it is a slight adaptation of a result given in ([9], Lemma 2.5). However, we have decided to include the proof, which remains the same, in order to help the reader.

Lemma 1.

Let $n\in \mathbb{N}$, and let $F:{[0,+\infty ]}^n\to [0,+\infty ]$ be a subadditive function. Then the following statements are equivalent to each other.

(1) 

There exists $i_0\in \{1,\dots ,n\}$ such that $a_{i_0}=0$ for each $a\in {[0,+\infty ]}^n$, with $F\left(a\right)=0$.

(2) 

If $a\in {[0,+\infty ]}^n$ such that $F\left(a\right)=0$, then $min\{a_1,\dots ,a_n\}=0$.

Proof. 

(1) ⇒ (2). This is obvious.

(2) ⇒ (1). Suppose for the purpose of contradiction that for each $i\in \{1,\dots ,n\}$ there exists $a^i\in {[0,\infty )}^n$ with $F\left(a^i\right)=0$ and $a_i^i>0$. Since F is subadditive, it follows that $F(a^1+\cdots +a^n)\le F\left(a^1\right)+\cdots +F\left(a^n\right)=0$. Thus, there exists $c\in {[0,\infty )}^n$ such that $c=a^1\cdots +a^n$ and $F\left(c\right)=0$. Nevertheless, $c_i>0$ for all $i\in \{1,\dots ,n\}$, which is a contradiction because $min\{c_1,\cdots ,c_2\}=0$. □

Since Proposition 1 implies that every modular metric aggregation function is a modular quasi-metric aggregation function, the following characterization can be obtained.

Theorem 9.

Let $n\in \mathbb{N}$, and let $F:{[0,+\infty ]}^n\to [0,+\infty ]$ be a function. The following statements are equivalent to each other:

(1) 

F is a modular quasi-metric aggregation function.

(2) 

F is a modular metric aggregation function.

(3) 

$F\left(0_n\right)=0$ and F is monotone and subadditive. Moreover, if $a\in {[0,+\infty ]}^n$ and $F\left(a\right)=0$, then $a_i=0$ for some $i=1,\dots ,n.$

(4) 

$F\left(0_n\right)=0$, and, in addition, $F\left(c\right)\le F\left(a\right)+F\left(b\right)$ for all $a,b,c\in {[0,+\infty ]}^n$, with $c⪯a+b$. Moreover, if $a\in {[0,+\infty ]}^n$ and $F\left(a\right)=0$, then $a_i=0$ for some $i=1,\dots ,n.$

(5) 

$F\left(0_n\right)=0$, F is monotone and transforms n-triangular triplets into a 1-triangular triplet. Moreover, if $a\in {[0,+\infty ]}^n$ and $F\left(a\right)=0$, then $a_i=0$ for some $i=1,\dots ,n.$

Proof. 

(1) ⇒ (2). This follows from Proposition 1. The equivalences (2) ⇒ (3) ⇔ (4) ⇔ (5) are true by Theorem 6. Now we prove that (4) ⇒ (1). To this end, consider a collection ${\left\{w_i\right\}}_{i=1}^n$ of modular quasi-metrics on a non-empty set X. Similar to the arguments used in proving Theorem 8, one can show that $F\circ \tilde{w}$ satisfies condition (MPQM2). It remains to be proven that condition (MQM1) holds.

Since $w_i(\lambda ,x,x)=0$ for all $\lambda \in ]0,+\infty [$ and for all $x\in X$, we have that

$$F\circ \tilde{w}(\lambda ,x,x)=F(w_1(\lambda ,x,x),\dots ,w_n(\lambda ,x,x))=F\left(0_n\right)=0.$$

Now assume that $F\circ \tilde{w}(\lambda ,x,y)=F\circ \tilde{w}(\lambda ,y,x)=0$ for any $x,y\in X$ and for all $\lambda \in ]0,+\infty [$. Then $F(w_1(\lambda ,x,y),\dots ,w_n(\lambda ,x,y))=0$ and $F(w_1(\lambda ,y,x),\dots ,w_n(\lambda ,y,x))=0$. By Lemma 1, there exists $i_0\in \{1,\dots ,n\}$ such that $w_{i_0}(\lambda ,x,y)=w_{i_0}(\lambda ,y,x)=0$ for all $\lambda \in ]0,+\infty [$. The fact that $w_i$ is a modular quasi-metric on X yields that $x=y$. So F is a modular quasi-metric aggregation function. □

The following example gives instances of modular quasi-metric aggregation functions.

Example 4.

Let $n\in \mathbb{N}$ and $w_1,\dots ,w_n\in ]0,\infty [$. Then the following functions $F:{[0,+\infty ]}^n⟶[0,+\infty ]$ are modular quasi-metric aggregation functions:

(1) 

$F\left(a\right)={\sum }_{i=1}^n{w}_i{a}_i$. Observe that this class of functions contains the class of weighted arithmetic means and, thus, the arithmetic mean (see [19]).

(2) 

$F\left(a\right)=max\{w_1{a}_1,\dots ,w_n{a}_n\}$.

(3) 

$F\left(a\right)={\left({\sum }_{i=1}^n{\left(w_i{a}_i\right)}^p\right)}^{\frac{1}{p}}$ for all $p\in [1,\infty [$. This class of functions contains those root-mean-powers such that $p\ge 1$ (see [19]).

(4) 

$F\left(a\right)={\sum }_{i=1}^n{w}_i{a}_{\left(i\right)}$ with $w_i\ge w_j$ for $i<j$, where $a_{\left(i\right)}$ is the ith largest of $a_1,\dots ,a_n$. Of course, OWA operators with decreasing weights belong to this class of functions (see, for instance, [19,20]).

(5) 

$F\left(a\right)=min\{c,{\sum }_{i=1}^n{w}_i{a}_i\}$ with $c\in (0,\infty )$.

(6) 

$F\left(a\right)=\left\{\begin{array}{cc}0& ifmin\{a_1,\dots ,a_n\}=0,\hfill \\ c& otherwise,\hfill \end{array}\right.$ with $c\in (0,\infty )$.

Example 2 again shows a function that is not a modular quasi-metric aggregation function since it is not monotone. In the same way, the function exposed in Example 3 is not a modular quasi-metric aggregation function. Notice that in the aforementioned example, the image of $0_n$ is not zero.

In the next proposition, inspired by Example 2, we give a method to construct modular quasi-(pseudo-)metric aggregation functions.

Proposition 3.

Let $n\in \mathbb{N}$ and ${F:[0,+\infty [}^n\to [0,+\infty [$ be a monotone and subadditive function. Consider the function $G:{[0,+\infty ]}^n\to [0,+\infty ]$ defined by

$$G\left(a\right)=\left\{\begin{array}{cc}\hfill F\left(a\right)& {ifa\in [0,+\infty [}^n,\hfill \\ \hfill \infty & otherwise.\hfill \end{array}\right.$$

Then the following assertions hold:

(1) 

$G:{[0,+\infty ]}^n\to [0,+\infty ]$ is a modular quasi-pseudo-metric aggregation function provided that $F\left(0_n\right)=0$.

(2) 

$G:{[0,+\infty ]}^n\to [0,+\infty ]$ is a modular quasi-metric aggregation function provided that $F\left(0_n\right)=0$ and $F\left(a\right)=0$ implies $a_i=0$ for some $i=1,\dots ,n$.

Proof. 

We first show that G is monotone. Let $a,b\in {[0,+\infty ]}^n$ such that $a⪯b$. Indeed, let us distinguish two possible cases.

Case 1.

There exists $i\in \{1,\dots ,n\}$ such that $a_i=+\infty$. Then in this case, $b_i=+\infty$, and thus, $G\left(a\right)=G\left(b\right)=+\infty$. So $G\left(a\right)\le G\left(b\right)$.

Case 2.

$a_i\ne \infty$ for all $i\in \{1,\dots ,n\}$. Then $G\left(a\right)=F\left(a\right)\le G\left(b\right)$.

Next, we prove that G is subadditibve. With this aim, consider $a,b\in {[0,+\infty ]}^n$. Again, two possible cases are distinguished:

Case 1.

There exists $i\in \{1,\dots ,n\}$ such that either $a_i=+\infty$ or $b_i=+\infty$. Then $G(a+b)=+\infty$ and either $G\left(a\right)=+\infty$ or $G\left(b\right)=+\infty$. So $G(a+b)\le G\left(a\right)+G\left(b\right)$.

Case 2.

$a_i\ne \infty$ and $b_i\ne \infty$ for all $i\in \{1,\dots ,n\}$. Then ${a+b\in [0,+\infty [}^n$ and $G(a+b)=F(a+b)\le F\left(a\right)+F\left(b\right)=G\left(a\right)+G\left(b\right)$.

Therefore, G is subadditve.

Assume that $F\left(0_n\right)=0$. Then $G\left(0_n\right)=F\left(0_n\right)=0$. Thus, by Theorem 8, G is a modular quasi-pseudo-metric aggregation function. Finally, suppose F satisfies the property: if ${a\in [0,+\infty [}^n$ and $F\left(a\right)=0$, then $a_i=0$ for some $i=1,\dots ,n.$. Now if $G\left(a\right)=0$, then $F\left(a\right)=0$. Hence, $a_i=0$ for some $i=1,\dots ,n$. Consequently, by Theorem 9, F is a modular quasi-metric aggregation function. □

The fact that a function ${F:[0,+\infty [}^n\to [0,+\infty [$ satisfying all assumptions in the statement of Proposition 3 is either a quasi-pseudo-metric aggregation function or a quasi-metric aggregation function suggests that we explore the relationship between these functions and the modular quasi-(pseudo-)metric aggregation functions.

The following example guarantees that there are quasi-(pseudo-)metric aggregation functions that are not modular quasi-(pseudo-)metric aggregation functions.

Example 5.

Let ${F:[0,+\infty [}^2\to [0,+\infty [$ be the function defined by

$$F\left(a\right)=\left\{\begin{array}{cc}\hfill 0& ifa=(0,0),\hfill \\ \hfill 1& otherwise.\hfill \end{array}\right.$$

Clearly, F satisfies all assumptions in Theorem 3 and, thus, in Theorem 4. From this, we deduce that F is a quasi-(pseudo-)metric aggregation function. Now consider the collection of modular quasi-(pseudo-)metrics ${\left\{w_i\right\}}_1^n$ on $\mathbb{R}$, where $w_i=w$ for all $i\in \{1,\dots ,n\}$, with $w(\lambda ,x,x)=0$ for all $\lambda \in ]0,+\infty [$ and $w(\lambda ,x,y)=+\infty$ for all $\lambda \in ]0,+\infty [$ and for all $x,y\in \mathbb{R}$ such that $x\ne y$. Then $F\circ \tilde{w}$ is not a modular (quasi-)pseudo-metric aggregation function because $F\circ \tilde{w}(1,2,3)$ is not defined (observe that the value $F(+\infty ,\dots ,+\infty )$ is not defined).

Notice that Example 5 also shows that there are (pseudo-)metric aggregation functions that are not modular (pseudo-)metric aggregation functions. This fact was not studied in [18].

Below is an example of a modular quasi-(pseudo-)metric aggregation function that is not a quasi-(pseudo-)metric aggregation function.

Example 6.

Let $n\in \mathbb{N}$. Consider the function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ defined by $F\left(0_n\right)=0$ and $F\left(a\right)=+\infty$ for all $a\ne 0_n$. It is a simple matter to check that F is a modular quasi-(pseudo-)metric aggregation function but is not a quasi-(pseudo-)metric aggregation function.

Notice that Example 6 also shows that there are modular (pseudo-)metric aggregation functions that are not (pseudo-)metric aggregation functions. This fact was not explored in [18].

The instances of modular quasi-metric aggregation function given in Example 4 inspire the following method for constructing such functions.

Proposition 4.

Let $g:{[0,+\infty ]}^n⟶[0,+\infty ]$ be a subadditive, monotone function such that $g\left(a\right)=0$ if and only if $a=0_n$. Let $W:{[0,+\infty ]}^n\to {[0,+\infty ]}^n$ be a function such that $W\left(0_n\right)=0_n$ and satisfying the following conditions:

1 

If $W\left(a\right)=0_n$, then $min\{a_1,\dots ,a_n\}=0$.

2 

$W\left(a\right)⪯W\left(b\right)$ whenever $a⪯b$.

If the function $g\circ W:{[0,+\infty ]}^n⟶[0,+\infty ]$ is subadditive, then the function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ given by $F\left(a\right)=g\left(W\right(a\left)\right)$ for each $a\in {[0,+\infty ]}^n$ is a modular quasi-metric aggregation function.

Proof. 

The subadditivity of $g\circ W$ implies the subadditivity of F. Moreover, the monotony of F is directly derived from the monotony of g and condition (2). Furthermore, $F\left(0_n\right)=g\left(W\left(0_n\right)\right)=g\left(0_n\right)=0$. Now, assume that there is $a\in {[0,+\infty ]}^n$ such that $F\left(a\right)=0_n$. Then $g\left(W\left(a\right)\right)=0_n$. Hence, $W\left(a\right)=0_n$. It follows from condition (1) that $min\{a_1,\dots ,a_n\}=0$. Theorem 9 implies F is a modular quasi-metric aggregation function. □

The next example shows that $g\left(a\right)=0$ if and only if $a=0_n$ cannot be deleted from the statement of Proposition 4.

Example 7.

Consider the function $W:{[0,+\infty ]}^n\to {[0,+\infty ]}^n$ given by $W\left(a\right)=(a_1,\dots ,a_n)$. Then W satisfies all assumptions in the statement of Proposition 4. Fix $i_0\in \{1,\dots ,n\}$. Define the function $g:{[0,+\infty ]}^n⟶[0,+\infty ]$ by $g\left(a\right)=a_{i_0}$ for all $a\in {[0,+\infty ]}^n$. The term g is subadditive, monotone and satisfies that $g\left(0_n\right)=0$. However, $g\left(0_{i_0}1\right)=0$, but $0_{i_0}1\ne 0_n$, where $0_{i_0}1$ stands for the element of ${[0,+\infty ]}^n$ with the $i_0$th coordinate as 0 and the jth coordinate, with $j\ne i_0$, as 1. Clearly, the function $F:{[0,+\infty ]}^n⟶[0,+\infty ]$ given by $F\left(a\right)=g\left(W\right(a\left)\right)$ for all $a\in {[0,+\infty ]}^n$ fulfills that $F\left(0_{i_0}1\right)=g\left(0_{i_0}1\right)=0$, and as a consequence, it is not a modular quasi-metric aggregation function.

The next result clarifies when a modular (quasi-)pseudo-metric aggregation function is also a (quasi-)pseudo-metric aggregation function. In order to state it, we will make use of the notion of finite modular quasi-pseudo-metric aggregation functions. A modular (quasi-)(pseudo-)metric aggregation function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is said to be a finite modular (quasi-)(pseudo-)metric aggregation function provided that for each collection of modular (quasi-)(pseudo-)metrics ${\left\{w_i\right\}}_{i=1}^n$ defined on X such that $w_i(\lambda ,x,y)<+\infty$ for all $\lambda \in ]0,+\infty [$, for all $x,y\in X$ and for all $i\in \{1,\dots .n\}$, the function $F\circ \tilde{w}$ is a modular (quasi-)(pseudo-)metric on X, with $F\circ \tilde{w}(\lambda ,x,y)<+\infty$ for all $\lambda \in ]0,+\infty [$ and for all $x,y\in X$.

Theorem 10.

Let $n\in \mathbb{N}$, and let $F:{[0,+\infty ]}^n\to [0,+\infty ]$ be a modular quasi-pseudo-metric aggregation function. The following statements are equivalent to each other.

(1) 

F is a finite modular quasi-pseudo-metric aggregation function.

(2) 

F is a finite modular pseudo-metric aggregation function.

(3) 

$F{\upharpoonright }_{{[0,+\infty [}^n}$ is a quasi-pseudo-metric aggregation function.

(4) 

$F{\upharpoonright }_{{[0,+\infty [}^n}$ is a pseudo-metric aggregation function.

(5) 

$F\left(a\right)=+\infty$ for some $a\in {[0,+\infty ]}^n\iff a_i=+\infty$ for some $i\in \{1,\dots ,n\}$.

Proof. 

(1) ⇔ (2) is evident.

(1) ⇒ (3). Consider a collection of quasi-pseudo-metrics ${\left\{q_i\right\}}_{i=1}^n$ defined on a non-empty set X. Define the collection ${\left\{w_i\right\}}_{i=1}^n$ on X by $w_i(\lambda ,x,y)=q_i(x,y)$ for all $\lambda \in ]0,+\infty [$ and for all $x,y\in X$. Then ${\left\{w_i\right\}}_{i=1}^n$ is a collection of modular quasi-pseudo-metrics on X. The fact that F is a modular quasi-pseudo-metric aggregation function yields that $F\circ \tilde{w}$ is a modular quasi-pseudo-metric on X. On the one hand, $F\circ \tilde{w}(\lambda ,x,y)<+\infty$ for all $\lambda \in ]0,+\infty [$ and for all $x,y\in X$. On the other hand, $F\circ \tilde{w}(\lambda ,x,y)=F(q_1(x,y)\dots ,q_n(x,y))$ for all $\lambda \in ]0,+\infty [$ and for all $x,y\in X$. Thus, $F\circ {\tilde{w}}_1$ is a quasi-pseudo-metric on X, with $F\circ {\tilde{w}}_1(x,y)=F\circ \tilde{w}(1,x,y)=F(q_1(x,y)\dots ,q_n(x,y))$ for all $x,y\in X$. From this, we deduce that $F\circ \tilde{q}$ is a quasi-pseudo-metric on X. Therefore, $F{\upharpoonright }_{{[0,+\infty [}^n}$ is a quasi-pseudo-metric aggregation function.

(3) ⇔ (4). The equivalence is guaranteed by the fact that $F{\upharpoonright }_{{[0,+\infty [}^n}$ is monotone and by Theorems 1 and 4.

(4) ⇒ (5). By way of contradiction, suppose there is an $a\in {[0,+\infty ]}^n$ such that $F\left(a\right)=+\infty$ and $a_i\in [0,+\infty [$ for all $i\in \{1,\dots ,n\}$. Define the collection ${\left\{d_i\right\}}_{i=1}^n$ of pseudo-metrics on a non-empty set X by $d_i(x,y)=a_i{d}_D(x,y)$ for all $x,y\in X$, where $d_D$ is the discrete pseudo-metric on X. Since $F{\upharpoonright }_{{[0,+\infty [}^n}$ is a pseudo-metric aggregation function, $F\circ \tilde{d}$ is a pseudo-metric on X. Hence, $F\circ \tilde{d}(x,y)=F(d_1(x,y),\dots ,d_n(x,y))<+\infty$ for all $x,y\in X$. However, let $u,v\in X$, with $u\ne v$. Then $+\infty =F\left(a\right)=F(d_1(u,v),\dots ,d_n(u,v))<+\infty$, which is a contradiction.

(5) ⇒ (1). This is true since $F\left(\right[0,+\infty {[}^n)\subseteq [0,+\infty [$. □

Similar arguments apply to the quasi-metric case:

Theorem 11.

Let $n\in \mathbb{N}$, and let $F:{[0,+\infty ]}^n\to [0,+\infty ]$ be a modular quasi-metric aggregation function. The following statements are equivalent to each other.

(1) 

F is a finite modular quasi-metric aggregation function.

(2) 

F is a finite modular metric aggregation function.

(3) 

$F{\upharpoonright }_{{[0,+\infty [}^n}$ is a quasi-metric aggregation function.

(4) 

$F{\upharpoonright }_{{[0,+\infty [}^n}$ is a metric aggregation function.

(5) 

$F\left(a\right)=+\infty$ for some $a\in {[0,+\infty ]}^n\iff a_i=+\infty$ for some $i\in \{1,\dots ,n\}$.

In light of the preceding result, it is clear that every finite modular quasi-(pseudo-)metric aggregation function merges a collection of modular quasi-(pseudo-)metrics that do not take the $+\infty$ value into a modular quasi-(pseudo-)metric that also does not take the $+\infty$ value. This is the reason for the name. Functions (2), (3), (4) and (5) given in Example 1 are instances of finite modular quasi-pseudo-metric aggregation functions. Nevertheless, function (1) provided in the aforementioned example is a modular quasi-pseudo-metric aggregation function that is not finite.

It must be pointed out that Theorems 10 and 11 stated in the modular framework are surprising due to the fact that there are (pseudo-)metric aggregation functions that are not quasi-(pseudo-)metric aggregation functions, as exposed in Section 1.

It seems interesting to stress that function (5) in Example 1 as well as functions (5) and (6) in Example 4 are instances of modular (quasi-)(pseudo-)metric aggregation functions that always merge a collection of modular (quasi-)(pseudo-)metrics into a modular (quasi-)(pseudo-)metric that does not take the $+\infty$ value. This fact inspires the possibility of describing such kinds of functions.

Below is a characterization of such functions. Before stating the characterization, let us recall that, given $a\in [0,+\infty ]$, $a_{i}0$ will denote the element of ${[0,+\infty ]}^n$ with the ith coordinate as a and the jth coordinate, with $j\ne i$, as 0.

Proposition 5.

Let $n\in \mathbb{N}$, and let $F:{[0,+\infty ]}^n\to [0,+\infty ]$ be a modular (quasi-)(pseudo-)metric aggregation function. Then the following statements are equivalent to each other.

(1) 

If ${\left\{w_i\right\}}_{i=1}^n$ is a collection of modular (quasi-)(pseudo-)metrics defined on a non-empty set X, then $F\circ \tilde{w}(\lambda ,x,y)<+\infty$ for all $\lambda \in ]0,+\infty [$ and for all $x,y\in X$.

(2) 

$F(+\infty ,\dots ,+\infty )<+\infty$.

(3) 

$F(+{\infty }_{i}0)<+\infty$ for all $i\in \{1,\dots ,n\}$.

Proof. 

(1) ⇒ (2). For the sake of contradiction, suppose $F(+\infty ,\dots ,+\infty )=+\infty$. Now consider a non-empty set X (with at least two different elements) and the collection of modular (quasi-)(pseudo-)metrics ${\left\{w_i\right\}}_{i=1}^n$ on X such that $w_i=w$ for all $i\in \{1,\cdots ,n,\}$, where w is defined for all $\lambda \in ]0,+\infty [$ by $w(\lambda ,x,y)=0$ if $x=y$ and $w(\lambda ,x,y)=+\infty$ if $x\ne y$. Then $F\circ \tilde{w}$ is a modular (quasi-)(pseudo-)metric such that $F\circ \tilde{w}(1,x,y)=F(+\infty ,\dots ,+\infty )$ provided that $x\ne y$. From this, the following equality is obtained:

$$+\infty =F(+\infty ,\dots ,+\infty )=F\circ \tilde{w}(1,x,y)<+\infty ,$$

which is a contradiction. So $F(+\infty ,\dots ,+\infty )<+\infty$.

(2) ⇒ (3). Since F is a modular (quasi-)(pseudo-)metric aggregation function, we have that it is monotone. Thus, $F(+{\infty }_{i}0)\le F(+\infty ,\dots ,+\infty )<+\infty$ for all $i\in \{1,\dots ,n\}$.

(3) ⇒ (2). Since F is a modular (quasi-)(pseudo-)metric aggregation function, we have that it is subadditive. Hence, $F(+\infty ,\dots ,+\infty )\le {\sum }_{i=1}^{n}F(+{\infty }_{i}0)<n·+\infty =+\infty$.

(2) ⇒ (1). Let ${\left\{w_i\right\}}_{i=1}^n$ be a collection of modular (quasi-)(pseudo-)metrics defined on a non-empty set X. Then $F\circ \tilde{w}$ is a modular (quasi-)(pseudo)-metric on X. Since F is monotone, and considering that $w_i(\lambda ,x,y)\le +\infty$ for each $\lambda \in ]0,+\infty [$, for each $x,y\in X$ and for each $i\in \{1,\dots ,n\}$, the following inequality is deduced:

$$F\circ \tilde{w}(\lambda ,x,y)=F(w_1(\lambda ,x,y),\dots ,w_n(\lambda ,x,y))\le F(+\infty ,\dots ,+\infty )<+\infty .$$

□

We end this section by exploring a question that arises in a natural way. Since every modular (quasi-)(pseudo-)metric aggregation function fuses a collection of modular (quasi-)(pseudo-)metrics into a single one, it seems natural to ask the following question: Does this type of function preserve modular (quasi-)(pseudo-)metrics? Notice that by preserving, we mean that when all modular (quasi-)(pseudo-)metrics in the collection to be fused are the same, then the aggregation function gives such a modular (quasi-)(pseudo-) metric as the aggregated one.

The concept below plays a central role in answering the posed question.

Given $n\in \mathbb{N}$, $p\in [0,+\infty ]$ is an idempotent element of the function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ if $F(p,\dots ,p)=p$ (see [19]). In addition, F is idempotent if each element in $[0,+\infty ]$ is an idempotent element of F, i.e., $F(p,\dots ,p)=p$ for all $p\in [0,+\infty ]$.

In light of the preceding notion, the result below answers the query.

Theorem 12.

Let $n\in \mathbb{N}$, and let X be a non-empty set. If $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is a modular (quasi-)(pseudo-)metric aggregation function, then the following statements are equivalent to each other.

(1) 

$F\circ \tilde{w}=w$ for all modular (quasi-)(pseudo-)metrics on X.

(2) 

F is idempotent.

Proof. 

(1) ⇒ (2). Let $a\in [0,+\infty ]$. Fix ${\lambda }_0\in ]0,+\infty [$. Consider the modular (quasi-)(pseudo-) metric on a non-empty set X given by

$$w(\lambda ,x,y)=\left\{\begin{array}{cc}0& \mathrm{if}x=y\mathrm{and}\lambda >0,\hfill \\ a& \mathrm{if}x\ne y\mathrm{and}0<\lambda <{\lambda }_0,\hfill \\ 0& \mathrm{if}x\ne y\mathrm{and}\lambda \ge {\lambda }_0.\hfill \end{array}\right.$$

Then $F\circ \tilde{w}$ is a modular (quasi-)(pseudo-)metric on X, and $F\circ \tilde{w}=w$. So taking $0<\lambda <{\lambda }_0$, we obtain that $F(a,\dots ,a)=F(w(\lambda ,x,y),\dots ,w(\lambda ,x,y))=F\circ \tilde{w}(\lambda ,x,y)=w(\lambda ,x,y)=a$. From this, we conclude that F is idempotent.

(2) ⇒ (1). Consider modular (quasi-)(pseudo-)metric w on non-empty set X. Since F is idempotent, we have that $F\circ \tilde{w}(\lambda ,x,y)=F(w(\lambda ,x,y),\dots ,w(\lambda ,x,y))=w(\lambda ,x,y)$ for all $\lambda \in {[0,+\infty ]}^n$ and for all $x,y\in X$. So $F\circ \tilde{w}=w$, as claimed. □

Modular Quasi-Pseudo-Metrics and Multi-Agent Systems

Multi-agent systems are formed by a group of two or more autonomous agents that must perform a common mission. A typical problem in this context is the so-called multi-agent task allocation problem, in which each agent must select the best next task to carry out at any moment in time. An approach to address this problem is provided by response threshold methods (see, for instance, [21]). In such methods, there is a set of $n\in \mathbb{N}$ agents $A=\{a_1,\dots ,a_n\}$ and a set of $m\in \mathbb{N}$ tasks to carry out $T=\{t_1,\dots ,t_m\}$. Moreover, each agent $a_k\in A$ perceives, from each task $t_j$ to be performed, a stimulus ($s_{a_k,t_j}\in \mathbb{R}$). The stimulus indicates how appropriate task $t_j$ is for agent $a_k$. As an example, a stimulus could be the inverse of the distance between the task and the current location of the agent. According to [22], if agent $a_k$ is located at task $t_i$ and $s_{a_k,t_j}$ exceeds a threshold value ${\theta }_{a_k}$ (${\theta }_{a_k}\in ]0,+\infty [$), agent $a_k$ begins the execution of task $t_j$ according to the probability $p(a_k,ij)$ given as follows:

$$p(a_k,ij)=\frac{s_{a_k,t_j}^h}{s_{a_k,t_j}^h+{\theta }_{a_k}^h},$$

(1)

with $h\in \mathbb{N}$.

Observe that the value $p(a_k,ij)$ can be understood as the probability of leaving task $t_i$ with the aim of performing task $t_j$ when the stimulus takes the value $s_{a_k,t_j}$. Additionally, in (1), we assume that agent $a_k$ is located at task $t_i$. Notice that we assume that the threshold ${\theta }_{a_k}$ depends only on agent $a_k$ and that it is the same for all tasks. However, in many real missions, the distribution $\left\{p(a_k,ij)\right\}$ fails to satisfy the probability constraint ${\sum }_{j=1}^m{p}_{a_k,ij}=1$. To address this issue, the authors of [23] suggested that the values $\left\{p(a_k,ij)\right\}$ naturally follow a distribution of possibilities.

Recall that ${\left\{p_{a_k,ij}\right\}}_{j=1}^m$ is a (fuzzy) possibility distribution provided that ${max}_{j=1,\dots ,m}{p}_{a_k,ij}\le 1$ for every $1\le i\le m$ (see [24,25]). Observe that the numerical value ${max}_{j=1,\dots ,m}{p}_{a_k,ij}$ provides information about the most likely task for agent $a_k$ to go to from task $t_i$. It must be pointed out that the possibility of an event can be understood as the perception of the degree of feasibility of its occurrence instead of the probability, which is interpreted as the frequency of occurrence of the event.

Following [23], the expression given by (1) can be rewritten as follows:

$$p(a_k,ij)=\frac{1}{1+\frac{1}{{\theta }_{a_k}^h}{\left({\alpha }_u·max\{U_i-U_j,0\}+d_E(x_i,x_j)\right)}^h},$$

(2)

when the stimulus considered is exactly the inverse of the Euclidean distance $d_E$ between the current location $(x_i,y_i)$ of agent $a_k$ (task $t_i$) and the location $(x_j,y_j)$ of task $t_j$. Notice that $h\in \mathbb{N}$, and ${\alpha }_u\in ]0,+\infty [$ is a system parameter that ensures the utility value has the same dimension and scale as the distance while indicating how important the utility is in relation to the distance. Observe that each task $t_j$ is associated with a utility $U_j$ gained by the system when the agent performs the task. It must be pointed out that the value $max\{U_i-U_j,0\}$ measures the improvement in utility achieved when the agent moves from task $t_i$ to task $t_j$.

In view of the exposed facts, the agent will make a decision about the next task to perform following the possibilities given by (2). An extensive number of numerical experiments have shown that expression (2) allows the behavior of multi-agent systems to be adequately described (see [23]).

Notice that for a fixed agent $a_k$, Theorem 2 guarantees that the function $Q_k:{\left([0,+\infty [\times {\mathbb{R}}^2\right)}^2$$\to [0,+\infty [$ is a quasi-metric, where

$$Q_k(U_i,x_i,y_i),(U_j,x_j,y_j))=\frac{1}{{\theta }_{a_k}^h}\left({\alpha }_u·max\{U_i-U_j,0\}+d_E(x_i,x_j)\right).$$

Thus, $p(a_k,ij)=\frac{1}{1+Q_k^h(U_i,x_i,y_i),(U_j,x_j,y_j))}$, and quasi-metric aggregation functions have been shown to be a suitable tool for generating transition possibilities when modeling multi-agent systems. It is clear that each agent $a_k$ defines a quasi-metric $Q_k$. This makes it necessary to work with a family of distances when describing the behavior of these systems. However, (finite) modular quasi-metric aggregation functions could help to generate transition possibilities in the sense of (2) and, thus, to describe how multi-agent systems evolve by using a unique modular distance measure. Indeed, observe that Theorems 9 and 11 guarantee that the function $Q_{{\alpha }_u}:]0,+\infty [\times {\left([0,+\infty [\times {\mathbb{R}}^2\right)}^2\to [0,+\infty [$ is a modular quasi-metric such that $Q_{{\alpha }_u}(\lambda ,(U_i,x_i,y_i),(U_j,x_j,y_j))<+\infty$, where $Q_{{\alpha }_u}$ is given by

$$Q_{{\alpha }_u}(\lambda ,(U_i,x_i,y_i),(U_j,x_j,y_j))=\frac{1}{{\lambda }^h}\left({\alpha }_u·max\{U_i-U_j,0\}+d_E(x_i,x_j)\right).$$

Hence,

$$p(a_k,ij)=\frac{1}{1+Q_{{\alpha }_u}^h({\theta }_{a_k},(U_i,x_i,y_i),(U_j,x_j,y_j))}.$$

So modular quasi-metrics could be an appropriate tool to generate transition possibilities in the spirit of (2). In view of this fact, it seems natural to study the general problem of inducing possibilities of transitions from (finite) modular quasi-metric aggregation functions. This question will be addressed elsewhere.

3. The Aggregation Problem: Discarding Functions

In this section, we explore a few properties common in aggregation theory (see, for instance, [19]) and inspired by those explored in [6,26] that modular quasi-(pseudo-)metric aggregation functions must enjoy. In some sense, such properties enable us to develop a quick test for discarding candidate functions to aggregate modular quasi-(pseudo-)metrics.

On account of [19], a function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is said to have an element $u\in [0,+\infty ]$ as an absorbent (or annihilator)for its i-th variable when

$$F(a_1,\dots ,a_{i-1},u,a_{i+1},\dots ,a_n)=u$$

for all $a_1,\dots ,a_{i-1},a_i,\dots ,a_n\in [0,+\infty ]$.

We have the following result, which is inspired by those given in [6], in light of the previous concept.

Proposition 6.

Let $n\in \mathbb{N}$. If $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is a modular quasi-pseudo-metric aggregation function, then the following assertions hold:

(1) 

F does not have $u\in ]0,+\infty [$ as an absorbent element for at least two variables provided that F has $+\infty$ as an idempotent element.

(2) 

F does not have 0 as an absorbent element for at least two variables provided that F has $p\in ]0,+\infty ]$ as an idempotent element.

(3) 

F does not have $u\in ]0,+\infty [$ as an absorbent element for at least two variables provided that F has $p\in ]0,\infty [$ as an idempotent element, with $p>2u$.

Proof. 

(1). Suppose that F has $u\in ]0,+\infty [$ as an absorbent element for the first two variables. Then $F(+\infty ,\dots ,+\infty ,)=+\infty$. Moreover, $2u=F(u,+\infty ,\dots ,+\infty )+F(+\infty ,u,\dots ,u)$. The subadditivity of F implies that

$$+\infty =F(+\infty ,\dots ,+\infty )\le F(u,+\infty ,\dots ,+\infty )+F(+\infty ,u,\dots ,u)=2u,$$

which is a contradiction.

(2). Assume without loss of generality that 0 is an absorbent element for the first two variables. Then $(p,\dots ,p)=(0,p,p,\dots ,p)+(p,0,\dots ,0)$. Moreover, $F(0,p,p,\dots ,p)=F\left(\right(p,0,\dots ,0)=0$ since 0 is an absorbent element for the first two variables. Hence, by the subadditivity of F, $F(p,\dots ,p)\le F(0,p,p,\dots ,p)+F\left(\right(p,0,\dots ,0)=0$. So $p=F(p,\dots ,p)\le 0$, which is a contradiction.

(3). Suppose that F has u as an absorbent element, for instance, for the first two variables. Then $F(p,\dots ,p)=p>2u$. Moreover,

$$2u=F(u,p-u,p-u,\dots ,p-u)+F(p-u,u,u,\dots ,u)\ge F(p,\dots ,p)>2u,$$

which is a contradiction. □

In the quasi-metric case, the following result can be proved.

Proposition 7.

Let $n\in \mathbb{N}$. If $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is a modular quasi-metric aggregation function, then F does not have 0 as an absorbent element for at least two variables.

Proof. 

For the sake of contradiction, suppose 0 is an absorbent element of F for its ith and jth variables, with $i<j$. The subadditivity of F implies

$$\begin{array}{c}F(1,\dots ,1^{i)},\dots ,1^{j)},\dots ,1)\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le F(1,\dots ,0^{i)},\dots ,1^{j)},\dots ,1)+F(0,\dots ,1^{i)},\dots ,0^{j)},\dots ,0)=0.\end{array}$$

Since F is a modular quasi-metric aggregation function, we have that $F\left(a\right)>0$ for all ${a\in ]0,+\infty ]}^n$. Indeed, assume that there exists ${a\in ]0,+\infty ]}^n$ such that $F\left(a\right)=0$. Then $a_i=0$ for some $i\in \{1,\dots ,n\}$, which is a contradiction. Hence, $0<F(1,\dots ,1)$. So $0<F(1,\dots ,1)\le 0$, which is impossible. □

Proposition 7 guarantees that the functions $F:{[0,+\infty ]}^n\to [0,+\infty ]$ and $G:{[0,+\infty ]}^n\to [0,+\infty ]$ given by $F\left(a\right)={\prod }_{i=1}^n{a}_i$ and $G\left(a\right)=min\{a_1\dots ,a_n\}$ are not modular quasi-metric aggregation functions.

Following [19], a function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is said to be conjunctive provided that $F\left(a\right)\le min\{a_1,\dots ,a_n\}$ for all $a\in {[0,+\infty ]}^n$.

The following result will be useful later on.

Proposition 8.

Let $n\in \mathbb{N}$. If $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is a conjunctive function, then F has 0 as an absorbent element for at least two variables.

Proof. 

Consider $a\in {[0,+\infty ]}^n$. Let $i,j\in \{1,\dots ,n\}$, with $i<j$. We have that

$$F(a_1,\dots ,0^{i)},\dots ,a_n)\le min\{a_1,\dots ,0^{i)},\dots ,a_n\}=0$$

and

$$F(a_1,\dots ,0^{j)},\dots ,a_n)\le min\{a_1,\dots ,0^{j)},\dots ,a_n\}=0.$$

From this, we obtain that F has 0 as an annihilator element for at least two variables. □

As a consequence of the preceding result and Proposition 7, we obtain that every modular quasi-metric aggregation function is not conjunctive.

Following [19], a function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ has $e\in [0,+\infty ]$ as a neutral element if $F\left(a_{i}e\right)=a$ for all $a\in [0,+\infty ]$ and for all $i\in \{1,\dots ,n\}$, where $a_{i}e$ stands for the element of ${[0,+\infty ]}^n$ with the ith coordinate equal to a and the jth coordinate such that $j\ne i$ but is equal to e.

In the following, the existence of neutral elements of modular quasi-pseudo-metric aggregation functions is discussed.

Proposition 9.

Let $n\in \mathbb{N}$. If $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is a modular quasi-pseudo-metric aggregation function, then the following statements are true:

(1) 

$F(+\infty ,\dots ,+\infty )=+\infty$ provided that either 0 or $+\infty$ is a neutral element.

(2) 

F does not have $e\in ]0,+\infty [$ as a neutral element.

Proof. 

(1). For the case for which $+\infty$ is a neutral element, it is evident that $F(+\infty ,\dots ,+\infty )=+\infty$. Assume that 0 is a neutral element. Then $F(+\infty ,0,\dots ,0)=+\infty$. Since $F(+\infty ,0,\dots ,0)\le F(+\infty ,\dots ,+\infty )$, we conclude that $F(+\infty ,\dots ,+\infty )=+\infty$.

(2). Suppose for the purpose of contradiction that $e\in ]0,+\infty [$ is a neutral element. Set $a=(e,\dots ,e)$, $b=(0,e,\dots ,.e)$ and $c=(e,0,e,\dots ,e)$. Clearly $a=b+c$. Then, by the subadditivity of F, $F\left(a\right)\le F\left(b\right)+F\left(c\right)$. Since e is a neutral element, we obtain that $F\left(a\right)=e$ and $F\left(b\right)=F\left(c\right)=0$. Consequently $0<e\le 0$, which is a contradiction. □

Proposition 9 implies that the function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ given by

$$F\left(a\right)=\left\{\begin{array}{cc}max\{a_1,\dots ,a_n\}\hfill & \mathrm{if}a\in {[1,+\infty ]}^n,\hfill \\ min\{a_1,\dots ,a_n\}\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

is not a modular quasi-metric aggregation function. Notice that F has 1 as a neutral element.

Following [27], a monotone and subadditive function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is an Aumann function whenever $F\left(a_{i}0\right)=a$ for all $a\in [0,+\infty ]$ and for all $i\in \{1,\dots ,n\}$.

Corollary 2.

Let $n\in \mathbb{N}$. If $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is a modular quasi-pseudo-metric aggregation function with 0 as a neutral element, then F is an Aumann function.

In [19], a function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is called disjunctive if $max\{a_1,\dots ,a_n\}\le F\left(a\right)$ for all $a\in {[0,+\infty ]}^n$. These functions play a distinguished role in aggregation theory.

In light of the preceding notion, the next result guarantees, among other things, that every modular quasi-pseudo-metric aggregation function is disjunctive.

Proposition 10.

Let $n\in \mathbb{N}$. If $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is a modular quasi-pseudo-metric aggregation function that has 0 as a neutral element, then $\frac{1}{n}{\sum }_{i=1}^n{a}_i\le max\{a_1,\dots ,a_n\}\le F\left(a\right)\le {\sum }_{i=1}^n{a}_i$ for all $a\in {[0,+\infty ]}^n$.

Proof. 

Let $a\in {[0,+\infty ]}^n$. Then $a={\sum }_{i=1}^n{a}_{i}0$. Since 0 is a neutral element, we have that $F\left(a_{i}0\right)=a_i$ for all $i\in \{1,\dots ,n\}$. The subadditivity of F implies that $F\left(a\right)\le {\sum }_{i=1}^{n}F\left(a_{i}0\right)={\sum }_{i=1}^n{a}_i$. On the other hand, the monotony of F gives that $a_i=F\left(a_{i}0\right)\le F\left(a\right)$ for all $i\in \{1,\dots ,n\}$. Thus, $\frac{1}{n}{\sum }_{i=1}^n{a}_i\le max\{a_1,\dots ,a_n\}\le F\left(a\right)$. □

As a consequence of the preceding result, we deduce that, for every collection of modular quasi-pseudo-metrics on a non-empty set X with 0 as a neutral element, the following inequality holds for all ${\lambda }_0\in ]0,+\infty [$ and for all $x,y\in X$:

$$\frac{1}{n}\sum _{i=1}^n{w}_i(\lambda ,x,y)\le F\circ \tilde{w}(\lambda ,x,y)\le \sum _{i=1}^n{w}_i(\lambda ,x,y).$$

Observe that the proof of Proposition 10 shows that every modular quasi-pseudo-metric aggregation function with 0 as a neutral element satisfies $a_i\le F\left(a\right)$ for all $a\in {[0,+\infty ]}^n$ and for all $i\in \{1,\dots ,n\}$. This property allows us to prove the following one.

Proposition 11.

Let $n\in \mathbb{N}$. Let $F:{[0,+\infty ]}^n\to [0,+\infty ]$ be a modular quasi-pseudo-metric aggregation function that has 0 as a neutral element. If $G:{[0,+\infty ]}^n\to [0,+\infty ]$ is a conjunctive function, then $G\left(a\right)\le F\left(a\right)$ for all $a\in {[0,+\infty ]}^n$.

Proof. 

Let $a\in {[0,+\infty ]}^n$. Since G is conjunctive, $G\left(a\right)\le min\{a_1,\dots ,a_n\}$. The fact that $a_i\le F\left(a\right)$ yields that $G\left(a\right)\le min\{a_1,\dots ,a_n\}\le F\left(a\right)$. □

We end the paper by discussing a relevant property in aggregation theory: the so-called Lipschitz condition. Following [19], a function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is said to be k ($k\in ]0,\infty [$ ) Lipschitz with respect to an extended norm $\left|\right|·\left|\right|$ (which satisfies all axioms of classical norms and, in addition, the $+\infty$ value is allowed; see [28]) on ${[0,+\infty ]}^n$ when $\left|F\right(a)-F(b\left)\right|\le k\left|\right|a-b\left|\right|$ for all $a,b\in {[0,+\infty ]}^n$.

The result below shows that modular quasi-pseudo-metric aggregation functions are 1 Lipschitz with respect to the extended norm on ${[0,+\infty ]}^n$ defined as follows: $\left|\right|a\left|\right|={\sum }_{i=1}^n{a}_i$ for all $a\in {[0,+\infty ]}^n$.

Proposition 12.

Let $n\in \mathbb{N}$. If $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is a modular quasi-pseudo-metric aggregation function that has 0 as a neutral element, then for all $a,b\in {[0,+\infty ]}^n$, the following inequality holds:

$$|F\left(a\right)-F\left(b\right)|\le \sum _{i=1}^n|b_i-a_i|.$$

Proof. 

Let $a,b\in {[0,+\infty ]}^n$. Take $c\in {[0,+\infty ]}^n$ given by $c_i=|b_i-a_i|$ for all $i\in \{1,\dots ,n\}$. Clearly, $a⪯c+b$ and $b⪯a+c$. Then by Theorem 8, the inequalities $F\left(a\right)\le F\left(b\right)+F\left(c\right)$ and $F\left(b\right)\le F\left(a\right)+F\left(c\right)$ are satisfied. So $|F\left(a\right)-F\left(b\right)|\le F\left(c\right)=F\left(\right|b_1-a_1|,\dots ,|b_n-a_n\left|\right)$. Proposition 10 gives that $F\left(\right|b_1-a_1|,\dots ,|b_n-a_n\left|\right)\le {\sum }_{i=1}^n|b_i-a_i|$. Therefore, $|F\left(a\right)-F\left(b\right)|\le {\sum }_{i=1}^n|b_i-a_i|$, as claimed. □

4. Conclusions and Future Work

In this paper, we have exposed the modular quasi-(pseudo-)metric aggregation problem. A description of those functions that allow the merging of a collection of modular quasi-(pseudo-)metrics into a single one has been given in terms of triangle triplets. Moreover, the relationship between modular quasi-(pseudo-)metric aggregation functions and modular (pseudo-)metric aggregation functions has been discussed. The displayed characterizations are illustrated with appropriate examples. In addition, several methods to construct modular quasi-(pseudo-)metrics have been yielded. In order to develop quick tests for discarding candidate functions for aggregating modular quasi-(pseudo-)metrics, we have studied the existence of absorbent and neutral elements of modular quasi-(pseudo-)metric aggregation functions. As a consequence of such a study, we have obtained that every modular quasi-pseudo-metric aggregation function that has 0 as a neutral element is always an Aumann function, is majored by the sum and satisfies the 1-Lipschitz condition. Moreover, a characterization of those modular quasi-(pseudo-)metric aggregation functions that preserve modular quasi-(pseudo-)metrics has been also provided. To be specific, we have shown that such functions are idempotent. Furthermore, the relationship between modular quasi-(pseudo-)metric aggregation functions and quasi-(pseudo-)metric aggregation functions has been discussed. Particularly, we have proven that they are the same only when the former are finite. New problems could be addressed in the study of distance aggregation functions: for example, the description of the relationship between the modular quasi-pseudo-metric aggregation problem and the problem of merging fuzzy relations depending on a parameter as fuzzy quasi-metrics in the spirit of [29]. Finally, the usefulness of modular quasi-(pseudo-)metric aggregation functions in multi-agent systems has been analyzed. As future research, it seems natural to face the general problem of inducing possibilities of transitions from modular quasi-metric aggregation functions.