Modular Quasi-pseudo Metrics and the Aggregation Problem

1. 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 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 the same set 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, for all $x,y\in X$, and for all $\lambda >0$, by $\tilde{w}(\lambda ,x,y)=\left(w_1(\lambda ,x,y),\dots ,w_n(\lambda ,x,y)\right).$

The next result was proved in [18]. 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 assertions are equivalent:

(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 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 Ftransforms 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 getting a characterization of 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

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

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. Whence we conclude that F is a modular (pseudo-)metric aggregation function. □

In the ligt of the preceding result, Theorems 5, 6 and 7 we immediately obtain the following statements.

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) 

$F\left(0_n\right)=0$, 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) 

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

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

Theorem 8. 

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

(1) 

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

(2) 

F is a modular pseudo-metric aggregation function.

(3) 

$F\left(0_n\right)=0$, 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) 

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

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 this end, 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 $i\in \{1,\dots ,n\}$. Whence 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 $i\in \{1,\dots ,n\}$. It follows 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. We still need to verify 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 always 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. 

It 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 in 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 0, satisfies all assumptions in the statement of Theorem 8 and, thus, it 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.

Next we provide several examples of modular quasi-pseudo-metric aggregation functions.

Example 1. 

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

(1) 

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

(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 which 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:

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

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

Example 3. 

Let the function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ defided 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 we face a refinement of the aforementioned problem. 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 extending Theorem 6.

The result below will play a crucial role in order to achieve our target. It must be stressed that it is an slight adaptation of a result given in [9]. 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 assertions are equivalent:

(1) 

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

(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). It 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 $a_i^i>0$. From the fact that F is subadditive we deduce that $F(a^1+\cdots +a^n)\le F\left(a^1\right)+\cdots +F\left(a^n\right)=0$. Whence we obtain the existence of $c\in {[0,\infty )}^n$ such that $c=a^1\cdots +a^n$ and, in addition, $F\left(c\right)=0$. Nevertheless, $c_i>0$ for all $i\in \{1,\dots ,n\}$, which is a contradiction because $min\{c_1,\cdots ,c_2\}=0$. □

Taking into account that Proposition 1 gives that every modular metric aggregation function is in fact a modular quasi-metric aggregation function we have the following characterization.

Theorem 9. 

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

(1) 

F is a modular quasi-metric aggregation function.

(2) 

F is a modular metric aggregation function.

(3) 

$F\left(0_n\right)=0$, 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). follows from Proposition 1. The equivalences (2) ⇒ (3) ⇔ (4) ⇔ (5) are provided 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. The same arguments to those applied to the prove of Theorem 8 gives that $F\circ \tilde{w}$ satisfies condition (MPQM2). It remains to prove that condition (MQM1) is hold.

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 we have 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 $n\in \mathbb{N}$. The following functions $F:{[0,+\infty ]}^n⟶[0,+\infty ]$ are, for all $w_1,\dots ,w_n\in ]0,\infty [$, modular quasi-metric aggregation functions:

(1) 

$F\left(a\right)={\sum }_{i=1}^n{w}_i{a}_i$. Observe that this instance 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 instance 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 the $a_1,\dots ,a_n$. 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 which is not a modular quasi-metric aggregation function, since it is not monotone. In the same way, the fuction 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.

Inspired by Example 2 we give a method to construct modular quasi-(pseudo-)metric aggregation functions in the following result.

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

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

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 that F satisfies the following property: if $a\in {[0,+\infty ]}^n$ and $F\left(a\right)=0$, then $a_i=0$ for some $i=1,\dots ,n.$

Proof. 

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

Case 1.

There exists $i\in \{1,\dots ,n\}$ such that $a_i=+\infty$. Then $b_i=+\infty$ 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, we distinguish two possible cases:

Case 2.

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 $G\left(0_n\right)=F\left(0_n\right)=0$ and, thus, Theorrem 8 gives that G is a modular quasi-pseudo-metric aggregation function. Finally, assume that F satisfies the property: if $a\in {[0,+\infty ]}^n$ and $F\left(a\right)=0$, then $a_i=0$ for some $i=1,\dots ,n.$. Suppose that $G\left(a\right)=0$. It follows that it must necessarily that $F\left(a\right)=0$. Then $a_i=0$ for some $i=1,\dots ,n.$. Consequently, by Theorem 9, we obtain that 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 us to explore the relationship between the aforementioned functions and the modular quasi-(pseudo-)metric aggregation functions.

First of all we show that there are quasi-(pseudo-)metric aggregation functions which are not modular quasi-(pseudo-)metric aggregation functions. The following example warranties such a statement.

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. Whence 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 which are not modular (pseudo-)metric aggregation functions. This fact was not studied in [18].

The example below gives an instance of modular quasi-(pseudo-)metric aggregation functions which 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 which are not a (pseudo-)metric aggregation function. This fact was not explored in [18].

The instances of modular quasi-metric aggregation function given in Example 4 inspires the following method to construct 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 $F:{[0,+\infty ]}^n\to [0,+\infty ]$ defined by $F\left(a\right)=g\left(W\right(a\left)\right)$ for all $a\in {[0,+\infty ]}^n$ is a modular quasi-metric aggregation function.

Proof. 

The subadditivity of $g\circ W$ gives that 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 $min\{a_1,\dots ,a_n\}=0$. By Theorem 9 we have that F is a modular quasi-metric aggregation function. □

The next example shows that the “separation” condition on g can not 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 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 function. We will say that a modular (quasi-)(pseudo-)metric aggregation function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is 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 the same set 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 assertions are equivalent.

(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). It 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 on X the collection ${\left\{w_i\right\}}_{i=1}^n$ 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. Since F is a modular quasi-pseudo-metric aggregation function we have that $F\circ \tilde{w}$ is a modular quasi-pseudo-metric on X. Moreover, we have, on the one hand, that $F\circ \tilde{w}(\lambda ,x,y)<+\infty$ for all $\lambda \in ]0,+\infty [$ and for all $x,y\in X$ and, on the other hand, that $F\circ \tilde{w}(\lambda ,x,y)=F(q_1(x,y)\dots ,q_n(x,y))$ for all $\lambda \in ]0,+\infty [$ and for all $x,y\in X$. Then $F\circ {\tilde{w}}_1$ 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 $x,y\in X$. Whence 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 eqiuvalence is guaranteed by the fact that $F{\upharpoonright }_{{[0,+\infty [}^n}$ is monotone and by Theorems 4 and 1.

(4) ⇒ (5). For the purpose of contradiction, assume that there exists $a\in {[0,+\infty ]}^n$ 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. It follows that $F\circ \tilde{d}$ 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 $u\ne v$, $+\infty =F\left(a\right)=F(d_1(u,v),\dots ,d_n(u,v))<+\infty$ which is a contradiction.

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

Similar arguments apply to the quasi-metic 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 assertions are equivalent.

(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 the light of the preceding result, it is clear that every finite modular quasi-(pseudo-)metric aggregation function merge a collection of modular quasi-(pseudo-)metrics which 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. The functions (2), (3), (4) and (5) given in Example 1 are instances of finite modular quasi-pseudo-metric aggregation functions. Nevertheless, the 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 which are not quasi-(pseudo-)metric aggregation functions as exposed in Section 0.

It seems interesting to stress that the function (5) in Example 1 and (5) and (6) in Example 4 are instances of modular (quasi-)(pseudo-)metric aggregation functions which 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 kind of functions.

In the following we characterize 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 assertions are equivalent.

(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 purpose of contradiction we assume that $F(+\infty ,\dots ,+\infty )=+\infty$. 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 $x\ne y$. Whence we have that $+\infty =F(+\infty ,\dots ,+\infty )=F\circ \tilde{w}(1,x,y)<+\infty$, 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 the fact that $w_i(\lambda ,x,y)\le +\infty$ for all $\lambda \in ]0,+\infty [$, for all $x,y\in X$ and for all $i\in \{1,\dots ,n\}$, we obtain that $F\circ \tilde{w}(\lambda ,x,y)=F(w_1(\lambda ,x,y),\dots ,w_n(\lambda ,x,y))\le F(+\infty ,\dots ,+\infty )<+\infty$. □

We end this section exploring a question that arise 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 wonder the following question: Do this type of functions 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 order to answer the posed question.

Given $n\in \mathbb{N}$, a function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ has $p\in [0,+\infty ]$ as an idempotent element if $F(p,\dots ,p)=p$ ([19]). In addition, F is said to be idempotent if it has every element in $[0,+\infty ]$ as an idempotent element, that is $F(p,\dots ,p)=p$ for all $p\in [0,+\infty ]$.

In the 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 modular (quasi-)(pseudo-)metric aggregation function, then the following assertions are equivalent:

(1) 

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

(2) ⇒ (1). Consider modular (quasi-)(pseudo-)metric w on a 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. □

2. 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 [6,21], that modular quasi-(pseudo-)metric aggregation functions enjoy. In some sense such properties allow us to develop a quick test for discarding candidate functions to aggregate modular quasi-(pseudo-)metrics.

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

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

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

In the light of the preceding notion we have the following result.

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 has not $u\in ]0,+\infty [$ as an absorbent element in at least two variables when F has $+\infty$ as an idempotent element.

(2) 

F has not 0 as an absorbent element in at least two variables when F has $p\in ]0,+\infty ]$ as an idempotent element.

(3) 

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

Proof. (1). Supose that F has $u\in ]0,+\infty [$ as an absorbent element in the two first variables. We have that $F(+\infty ,\dots ,+\infty ,)=+\infty$. Moreover, $2u=F(u,+\infty ,\dots ,+\infty )+F(+\infty ,u,\dots ,u)$. The subadditivity of F gives 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 in the two first variables. We have that $(p,\dots ,p)=(0,p,p,\dots ,p)+(p,0,\dots ,0)$ Since 0 is a an absorbent element in the two first variables we have that $F(0,p,p,\dots ,p)=F\left(\right(p,0,\dots ,0)=0$. Hence, by 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 absorbent element, for instance, in the two first 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 we have the following specific result.

Proposition 7. 

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

Proof. 

For the purpose of contradiction, we suppose that 0 is an absorbent element of F in its i-th and j-th variables with $i<j$. The subadditivity of F gives that

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

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, $0<F(1,\dots ,1)$. Whence we deduce that $0<F(1,\dots ,1)\le 0$, which is imposible. □

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 in 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, in addition, that $F(a_1,\dots ,0^{j)},\dots ,a_n)\le min\{a_1,\dots ,0^{j)},\dots ,a_n\}=0$. It follows that F has 0 as an absorbent element in 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 conjuctive.

Let us recall that, according to [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 $a_{i}e$ denotes the element of ${[0,+\infty ]}^n$ such that the i-th coordinate is a and the j-th coordinate with $j\ne i$ is e.

Next we discuss the neutral elements of modular quasi-pseudo-metric aggregation functions.

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 assertions hold:

(1) 

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

(2) 

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

Proof. (1). The case in which $+\infty$ is a neutral element, it is evident that $F(+\infty ,\dots ,+\infty )=+\infty$. Assume that 0 is 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 subaditivity 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. □

Following [22], a monotone and subadditive function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is an Aumann function whenever it satisfies that $F\left(a_{i}0\right)=a$ 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 neutral element, then F is an Aumann function.

Disjuntive functions play a distinguished role in aggregation theory. Let us recall that, according to [19], a function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ is disjunctive when $max\{a_1,\dots ,a_n\}\le F\left(a\right)$ for all $a\in {[0,+\infty ]}^n$.

In the 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 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 we have that $F\left(a_{i}0\right)=a_i$ for all $i\in \{1,\dots ,n\}$. The subadditivity of F implies that $F\left(a\right)\le {\sum }_{i=1}^{n}F\left(a_{i}0\right)={\sum }_{i=1}^n{a}_i$. Moreover, 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, for every collection of modular quasi-pseudo-metrics on a non-empty set X with 0 as neutral element, that 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 that $a_i\le F\left(a\right)$ 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 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 discussing a relevant property in aggregation theory, the so-called Lipschitz condition. Following [19], a function $F:{[0,+\infty ]}^n\to [0,+\infty ]$ will be 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 [23]) 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 $n\in \mathbb{N}$. 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 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, we have that $F\left(a\right)\le F\left(b\right)+F\left(c\right)$ and $F\left(b\right)\le F\left(a\right)+F\left(c\right)$. Whence we have that $|F\left(a\right)-F\left(b\right)|\le F\left(c\right)=F\left(\right|b_1-a_1|,\dots ,|b_n-a_n\left|\right)$. Now, 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. □

3. Conclusions and Future Work

In this paper we have exposed the modular quasi-(pseudo-)metric aggregation problem. A description of those functions that allow merging 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 to aggregate 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 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. In particular, we have shown that such functions are idempotent. Finally, the relationship between modular quasi-(pseudo-)metric aggregation functions and quasi-(pseudo-)metric aggregation functions has been discussed in such a way that significative differences have been evidenced.