On Modularity, Indistinguishability and Generalized Metrics: Duality and Aggregation

Abstract

In this paper, we prove that, on the one hand, a duality relationship between different types of modular T-transitive relations and the reciprocal modular generalized metrics exists and, on the other hand, that, based on this duality, a construction of functions that aggregate modular T-transitive relations can be made from functions aggregating generalized metrics. Furthermore, we provide a guide for the families of that type of functions that can be used for the aforementioned purpose. Finally, illustrative examples of how to create such functions via the duality are given.

1. Introduction

One of the most common problems when working in applied sciences is the need to combine different pieces of information into a single one. This process is crucial to extract conclusions and to make decisions in order to solve the problem under consideration. In many cases, such information is related to the similarity or dissimilarity between objects. It is for this reason that both the notions of the T-indistinguishability relation and metric (or any generalized version of them) and the functions that are able to aggregate them, preserving their mathematical properties, have been extensively studied. Some instances where such notions have been of great utility in obtaining applications can be found, for instance, in [1,2,3,4,5,6].

Based on the fact that similarity and dissimilarity are dual concepts, it is natural to ask whether this duality also exists between T-indistinguishability relations and metrics. The answer to this question was given in [7] (see also [8]), proving that a pseudo-metric could be obtained from a T-indistinguishability relation by means of the use of an additive generator of the t-norm (T). Moreover, in the same reference, it was shown that a T-indistinguishability relation could be generated trough a pseudo-metric using an additive generator of the t-norm. Later on, in [9], the construction given in [7] was extended to the metric case, and it was also proven that the t-norm needs to be continuous when a T-indistinguishability relation is induced from a (pseudo-)metric.

In view of the mentioned studies, in [10,11], the aforementioned existing duality for T-indistinguishability relations and metrics was extended to the quasi-pseudo-metric and relaxed metric frameworks, respectively. It must be noted that, in this paper, relaxed metrics are considered in the sense of [12,13]. For more details, check Section 2.

Furthermore, in [14], it was proven that functions that aggregate T-indistinguishability relations can be constructed by means of functions that aggregate pseudo-metrics involving the use of additive generators and their associated pseudo-inverses. Inspired by this specific construction, similar results were explored for pseudo-metrics, quasi-pseudo-metrics and relaxed metrics in [15,16,17].

It is worth noting that when the measure is relative to a certain parameter, the notions of an indistinguishability relation and metric may not be appropriate in order to capture the similarity or dissimilarity between objects. In order to overcome such a handicap, the notion of a modular pseudo-metric was introduced in [18], and later on, the concept of modular indistinguishability was introduced in [19]. In particular, the existence of a duality relationship between these two notions, in a similar way as in the non-modular framework, was also proven in [19]. Some instances of recent applications of both modular metrics and modular indistinguishability relations can be found in [6,20,21,22,23,24].

Motivated by the potential applications of modular metrics and modular indistinguishability relations, many articles have been published regarding their aggregation (see [6,25,26,27,28,29]). Along those lines, in [30], the technique introduced in [14] for generating functions that aggregate pseudo-metrics by means of functions that aggregate T-indistinguishability relations involving the use of additive generators and their associated pseudo-inverses was extended to the context of modular pseudo-metrics and modular T-indistinguishability relations. In particular, descriptions of both types of function was provided.

In this paper, we will expand the work started in [19] and later continued in [30] to the modular versions of quasi-pseudo metrics and relaxed metrics, two kinds of generalized modular metrics. Hence, on the one hand, we will introduce a technique to generate a modular T-pre-order from a modular quasi-pseudo metric and a modular relaxed T-indistinguishability relation from a modular relaxed metric via the use of an additive generator of the t-norm. Moreover, the reciprocal technique will also be developed using the pseudo-inverse of the aforementioned additive generator. In view of these results, we will explore the relationship between functions aggregating these generalized modular indistinguishability relations and their generalized modular metric counterpart, taking advantage of the previous specific duality construction extending that given in [14]. In particular, we will provide a specific construction of the second family of functions through the first one, making use of both additive generators and their associated pseudo-inverses. Finally, we will provide some illustrative examples of how the mentioned transformation processes can be applied.

It is worth mentioning that the work carried out in this paper is only possible thanks to the use of many previous results distributed across the literature. More precisely, the proofs carried out in the upcoming sections are heavily dependent on the results from [15,16,17,19,30] regarding the duality relationship between measures and the results from [6,25,26,27,28,29] regarding aggregation techniques. The specific parts that are relevant to our contribution are extensively detailed in Section 2 and during the proofs in which they are used.

The rest of the paper is structured as follows. To begin with, a more in-depth explanation of the notions and previous results that set the basis for our research are provided in Section 2. Section 3 is devoted to the extension of the results from [19], as detailed in Section 2, where the construction of the aforementioned generalized modular metrics using additive generators is provided and the construction of the above generalized modular indistinguishability relations using pseudo-inverse is also given. Next, in Section 4, we prove that functions that aggregate each one of the aforementioned modular generalized metrics can be obtained by means of a specific construction of functions that merge some type of modular T-transitive fuzzy relations making use of both additive generators and their associated pseudo-inverses. All results regarding aggregation are compiled in Section 4.4, which summarizes the type of function that can be used to aggregate each kind of T-transitive relation (modular or non modular) to obtain each kind of generalized metric (modular or non modular). In Section 5, we exemplify how the theorems and, thusm the mentioned techniques exposed in Section 4 can be used to obtain the desired type of function. Finally, in Section 6, we expose some conclusions and ideas for future work.

2. Preliminaries

Let us recall the notions that have worked as a basis for our research and which will be used in the subsequent sections. On the one hand, the concept of a T-indistinguishability relation was introduced in [31] by E. trillas in 1982 in the following way.

Definition 1.

Given a non empty set (X) and a t-norm (T), a fuzzy set $(E:X\times X\to [0,1]$) is defined as a T-indistinguishability relation whenever it fulfills, for all $x,y,z\in X$, the following conditions:

• (E.1) $E(x,x)=1$ (reflexivity),
• (E.2) $E(x,y)=E(y,x)$ (symmetry),
• (E.3) $T\left(E\right(x,y),E(y,z\left)\right)\le E(x,z)$ (T-transitivity).

Moreover, we say that E is a T-equality (see [5]) in such a case that it also satisfies the following:

• (E.4) $E(x,y)=1$ implies that $x=y$ (point separation).

Such a notion can be used to measure the similarity (or even indistinguishability) between two elements from a given set. In this sense, the higher the value of $E(x,y)$ is, the more similar (or indistinguishable) x and y are to (or from) each other. In the succeeding, we assume that the reader is familiar with the basics of t-norms. If this is not the case, we suggest referencing [32].

On the other hand, according to [33], an extended pseudo-metric can be defined as follows:

Definition 2.

Given a non-empty set (X), a function ($d:X\times X\to [0,\infty ]$) is called an extended pseudo-metric on X if it satisfies the following for each $x,y,z\in X$:

• (D.1) $d(x,x)=0$ (self distance 0),
• (D.2) $d(x,y)=d(y,x)$ (symmetry),
• (D.3) $d(x,y)+d(y,z)\le d(x,z)$ (triangular inequality).

Moreover, an extended metric is an extended pseudo-metric that also fulfills the following:

• (D.4) $d(x,y)=0$ implies that $x=y$ (metric point separation).

In the particular case where $d(x,y)\le b$ for all $x,y\in X$ and some $b\in ]0,\infty ]$, we say that d is a b-bounded (pseudo-)metric. It follows from the definition that an extended (pseudo-)metric is a ∞-bounded (pseudo-)metric.

A remarkable property of T-indistinguishability relations is their duality relationship with extended pseudo-metrics. More precisely, in [8], it was shown that an extended pseudo-metric on a non-empty set (X) can be obtained from a T-indistinguishability relation (E) on X by means of an additive generator of the Archimedean t-norm (T; Proposition 6 in [8]) and reciprocally but using the pseudo-inverse of the additive generator and taking the t-norm (T) as continuous (Proposition 7 in [8]). Thus, given a non-empty set (X), a T-indistinguishability relation (E) on X and an additive generator (f) of T, the function defined for all $x,y\in X$ by

$$d^{E,f}(x,y)=f\left(E(x,y)\right)$$

(1)

is an extended pseudo-metric on X. Similarly, given a non-empty set (X), an extended pseudo-metric (d) on X and the pseudo-inverse ($f^{(-1)}$) of an additive generator (f) of a continuous and Archimedean t-norm (T), the function defined, for all $x,y\in X$, by

$$E^{d,f}(x,y)=f^{(-1)}\left(d(x,y)\right)$$

(2)

is a T-indistinguishability relation on X.

Recall that, according to [32], an additive generator of a triangular norm is defined as follows.

Definition 3.

An additive generator ($f:[0,1]\to [0,\infty ]$) of a t-norm (T) is a strictly decreasing function that is also right continuous at 0 and satisfies $f\left(1\right)=0$ such that for all $x,y\in {[0,1]}^2$, we have

$$f\left(x\right)+f\left(y\right)\in Ran\left(f\right)\cup [f\left(0\right),\infty ]andT(x,y)=f^{(-1)}(f\left(x\right)+f\left(y\right)),$$

where the pseudo-inverse $f^{(-1)}$ is the mapping ($f^{(-1)}:[0,\infty ]\to [0,1]$) defined by $f^{(-1)}\left(y\right)=\inf \{t:t\in [0,1]\mathrm{and}f\left(t\right)\le y\}$. Observe that in the particular case in which an additive generator (f) is continuous (or, equivalently, the t-norm (T) is continuous and Archimedean), its pseudo-inverse is given by

$$f^{(-1)}\left(y\right)=f^{-1}\left(\min \{f\left(0\right),y\}\right).$$

Analogously, in [9], the duality relationship given by expressions (1) and (2) was proven between T equalities and metrics. Thus, an extended metric is induced considering a T-equality in (1), and reciprocally, a T equality is generated considering an extended metric in (2).

As a matter of fact, this duality relationship exists whenever the considered functions in (1) and (2) are obtained by relaxing reciprocal properties from equalities and extended metrics (see [10,11]). In particular, if E is a fuzzy relation from Table 1 and f is an additive generator of a t-norm (T), then $d^{E,f}$ is the generalized extended metric from Table 2 located in the same row as E. Conversely, if d is a generalized extended metric from Table 2 and $f^{(-1)}$ is the pseudo-inverse of an additive generator (f) of a continuous t-norm (T), then $E^{d,f}$ is the fuzzy relation from Table 1 located in the same row as d. Given that all considered fuzzy relations from Table 1 satisfy condition (E.3), we will refer to them as T-transitive fuzzy relations. Notice that a fuzzy set ($E:X\times X\to [0,1]$) satisfies the asymmetric point separation property, provided that $E(x,y)=E(y,x)=1\iff x=y$. In a similar way, a function ($d:X\times X\to [0,\infty ]$) fulfills the distance asymmetric point separation when $d(x,y)=d(y,x)=0\iff x=y$.

Table 1. List of fuzzy relations and the properties they fulfill. Note that the “Y” symbol means that the fuzzy relation in that row fulfills the condition, and the “−” symbol means that such a fuzzy relation does not necessarily fulfill that condition (however, it could). Also note that relaxed indistinguishability relations are abbreviated as “Relaxed Indist”.

Fuzzy Relation (E)	Reflexivity (E.1)	Symmetry (E.2)	T-Transitivity (E.3)	Point Separation (E.4)	Asymmetric Point Separation
Equality	Y	Y	Y	Y	Y
Indistinguishability	Y	Y	Y	−	−
Partial Order	Y	−	Y	−	Y
Pre-order	Y	−	Y	−	−
Relaxed Indist.	−	Y	Y	−	−

Table 2. List of generalized extended metrics and the properties they fulfill. Notice that the “Y” symbol means that the generalized extended metric in that row fulfills the condition, and the “−” symbol means that such a generalized extended metric does not necessarily fulfill that condition (however, it could).

Generalized Extended Metric (d)	Self Dist. 0 (D.1)	Symmetry (D.2)	Triang. Ineq. (D.3)	Metric Point Separation (D.4)	Distance Asymmetric Point Separation
Metric	Y	Y	Y	Y	Y
Pseudo-metric	Y	Y	Y	−	−
Quasi-metric	Y	−	Y	−	Y
Quasi-psuedo-metric	Y	−	Y	−	−
Relaxed pseudo-metric	−	Y	Y	−	−

Inspired by this duality relationship, many studies regarding the construction of functions that merge generalized extended metrics via functions merging T-transitive relations have arisen (see [14,15,16,17]). In particular, Pradera, Trillas and Castiñeira proved Theorem 1 (below) in [14].

In order to fully understand the formulation of such a theorem, let us recall that, given a collection of t-norms ($\mathcal{T}={\left\{T_i\right\}}_{i=1}^n$) and a collection of fuzzy relations (${\left\{E_i\right\}}_{i=1}^n$), we say that we have a collection of $\mathcal{T}$-indistinguishability relations if each $E_i$ is a $T_i$-indistinguishability relation on X for all $i\in \{1,\dots ,n\}$. Moreover, given a collection of $\mathcal{T}$-indistinguishability relations (${\left\{E_i\right\}}_{i=1}^n$), a function ($F:{[0,1]}^n\to [0,1]$) is said to aggregate $\mathcal{T}$-indistinguishability relations into a T-indistinguishability relation, provided that $F(E_1,\dots ,E_n)$, where $F(E_1,\dots ,E_n)(x,y)=F(E_1(x,y),\dots ,E_n(x,y))$ for all $x,y\in X$, is a T-indistinguishability relation on X. The notion of an aggregation function for the rest of the T-transitive relations in Table 1 can be defined in a similar way. Furthermore, if we have a collection of ${\left(b_i\right)}_{i=1}^n$-bounded pseudo-metrics (each $d_i$ is a $b_i$-bounded pseudo-metric on X for all $i\in \{1,\dots ,n\}$), then a function( H: ${[0,b_i]}^n\to [0,c]$, with $c\in ]0,\infty ]$)aggregates a collection of ${\left(b_i\right)}_{i=1}^n$-bounded pseudo-metrics ${\left\{d_i\right\}}_{i=1}^n$ on X, provided that $H(d_1,\dots ,d_n)$ is a c-bounded pseudo-metric on X, where $H(d_1,\dots ,d_n)(x,y)=H(d_1(x,y),\dots ,d_n(x,y))$ for all $x,y\in X$. Of course, when the ${\left(b_i\right)}_{i=1}^n$-bounded pseudo-metrics are exactly a collection of ∞-bounded pseudo-metrics (extended pseudo-metrics) on X, that is, $b_i=\infty$ for all $i\in \{1,\dots ,n\}$, we say that a function (G: ${[0,\infty ]}^n\to [0,\infty ]$) aggregates extended pseudo-metrics into an extended pseudo-metric, given that $G(d_1,\dots ,d_n)$ is an extended pseudo-metric on a non-empty (X), where $G(d_1,\dots ,d_n)(x,y)=G(d_1(x,y),\dots ,d_n(x,y))$ for all $x,y\in X$.

The notions of a function aggregating $\mathcal{T}$-indistinguishability relations into a T-indistinguishability relation can be adapted as expected when the collection of transitive relationships corresponds to the rest of those included in Table 1. The same occurs for the case of functions aggregating bounded pseudo-metrics (extended pseudo-metrics) according to Table 2.

Theorem 1.

Let $n\in \mathbb{N}$, let $\mathcal{T}={\left\{T_i\right\}}_{i=1}^n$ be a collection of continuous Archimedean t-norms, let $f_{T_i}$ an additive generator of $T_i$ and let $f_{T_i}^{(-1)}$ be the pseudo-inverse of $f_{T_i}$. If T is a continuous Archimedean t-norm, $f_T$ is an additive generator of T and $F:{[0,1]}^n\to [0,1]$ is a function, then the following assertions are equivalent:

(1) 

F aggregates $\mathcal{T}$-indistinguishability relations into a T-indistinguishability relation.

(2) 

The function expressed as $H:{\prod }_{i=1}^n{[0,f_{T_i}\left(0\right)]}^n\to [0,f_T\left(0\right)]$, where $H=f_T\circ F\circ (f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)})$, aggregates every collection of ${\left(f_{T_i}\left(0\right)\right)}_{i=1}^n$-bounded pseudo-metrics ${\left\{d_i\right\}}_{i=1}^n$ into a $f_T\left(0\right)$-bounded pseudo-metric.

Notice that if, in the preceding result, all continuous t-norms are strict, then assertion $\left(2\right)$ matches with the following: The function expressed as $G:{\prod }_{i=1}^n[0,\infty ]\to [0,\infty ]$, where $G=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, aggregates every collection of extended pseudo-metrics (${\left\{d_i\right\}}_{i=1}^n$) into an extended pseudo-metric.

Following the idea from Theorem 1, analogous versions for the rest of the pairs of functions (generalized indistinguishability relation and generalized extended metric) corresponding to Table 1 and Table 2 have been proven. To be more precise, in [15], it was proven that those functions aggregating extended metrics (or the bounded version) can be obtained from functions aggregating fuzzy equalities. Later on, in [16], an extension of the same result was proven with respect to to obtaining functions that are able to merge relaxed extended pseudo-metrics (or their bounded versions) from those merging relaxed indistinguishability relations. Finally, in [17], an analogous result was proven for functions that merge pre-orders and extended quasi-pseudo-metrics, as well as for functions that aggregate partial orders and extended quasi-metrics.

The notion of an indistinguishability relation may fall short of fully describing situations where the degree of similarity between objects is variable. In other words, whenever the degree of similarity depends on a parameter, indistinguishability relations are not the best option to capture such information. Motivated by this fact, Miñana and Valero introduced, in [19], the concept of a modular indistinguishability relation.

Definition 4.

Given a non-empty set (X) and a t-norm (T), the fuzzy set expressed as $E:X\times X\times ]0,\infty [\to [0,1]$ is said to be a modular indistinguishability relation whenever the following properties are fulfilled for all $x,y,z\in X$ and all $t,s>0$:

• (M.E.1) $E(x,x,t)=1$,
• (M.E.2) $E(x,y,t)=E(y,x,t)$,
• (M.E.3) $T\left(E\right(x,y,t),E(y,z,s\left)\right)\le E(x,z,t+s)$.

In particular, a modular equality is a modular indistinguishability relation that also satisfies the following:

• (M.E.4) $E(x,y,t)=1$ for all $t>0$ implies that $x=y$.

It is worth noting that properties (M.E.1), (M.E.2), (M.E.3) and (M.E.4) can be understood as modular versions of conditions (E.1), (E.2), (E.3) and (E.4). In this sense, this relation captures how similar two elements are, depending on the considered value of the parameter.

As for indistinguishability relations, there is a dual notion for their modular version. Such a concept is denoted as a modular pseudo-metric and was introduced in [18]. Let us recall its definition.

Definition 5.

Given a non-empty set (X), a function ($d:X\times X\times ]0,\infty [\to [0,\infty ]$) is called a modular pseudo-metric whenever it fulfills, for all $x,y,z\in X$ and all $t,s>0$, the following conditions:

• (M.D.1) $d(x,x,t)=0$,
• (M.D.2) $d(x,y,t)=d(y,x,t)$,
• (M.D.3) $d(x,z,t+s)\le d(x,y,t)+d(y,z,s)$ (modular triangular inequality).

In particular, a modular metric is a modular pseudo-metric that also satisfies the following:

• (M.D.4) $d(x,y,t)=0$ for all $t>0$ implies that $x=y$.

Notice that the notion of a bounded pseudo-metric can be extended to the modular context as follows (according to [30]): a modular pseudo-metric (d) on X, given $b\in ]0,\infty [$, is said to be b-bounded, provided that $d(x,y,t)\le b$ for all $x,y\in X$ and for all $t>0$.

In a similar way as in the non modular context, the aforementioned duality relationship between modular indistinguishability relations and modular pseudo-metrics was proven in [19]. As a matter of fact, the authors also proved the duality relationship between modular equalities and modular metrics. A possibility of obtaining modular pseudo-metrics (modular metrics) from modular indistinguishability relations (modular equalities) was provided with the following theorem.

Theorem 2.

Let X be a non-empty set, and let $T^{*}$ be a t-norm with an additive generator ($f_{T^{*}}$). If T is a t-norm, then the following assertions are equivalent:

(1) 

$T^{*}$ is weaker than T, i.e., $T^{*}\le T$.

(2) 

For any modular indistinguishability relation (E) on X for T, the function expressed as $d^{E,f_{T^{*}}}:X\times X\times ]0,\infty [\to [0,\infty ]$, defined, for each $x,y\in X$ and $t>0$, by

$$d^{E,f_{T^{*}}}(x,y,t)=f_{T^{*}}\left(E(x,y,t)\right),$$

is a modular pseudo-metric on X.

(3) 

For any modular equality (E) on X for T, the function expressed as $d^{E,f_{T^{*}}}:X\times X\times ]0,\infty [\to [0,\infty ]$, defined, for each $x,y\in X$ and $t>0$, by

$$d^{E,f_{T^{*}}}(x,y,t)=f_{T^{*}}\left(E(x,y,t)\right),$$

is a modular metric on X.

Conversely, a way to obtain a modular indistinguishability relation (modular equality) from a modular pseudo-metric (modular metric) is stated in the next theorem.

Theorem 3.

Let X be a non-empty set, and let $T^{*}$ be a continuous t-norm with an additive generator ($f_{T^{*}}$). If d is a modular pseudo-metric on X, then the function expressed as $E^{d,f_{T^{*}}}:X\times X\times ]0,\infty [\to [0,1]$, defined, for all $x,y\in X$ and $t>0$, by

$$E^{d,f_{T^{*}}}(x,y,t)=f_{T^{*}}^{(-1)}\left(d(x,y,t)\right)$$

is a modular indistinguishability relation for $T^{*}$. Moreover, $E^{d,f_{T^{*}}}$ is a modular equality if and only if d is a modular metric on X.

It is relevant to mention that the continuity assumed in the statement of Theorem 3 cannot be deleted, as shown in [19].

Theorems 2 and 3 inspired a modular version of Theorem 1, which was presented in [30]. The formulation of this result takes advantage of the fact that the aggregation problem for modular indistinguishability relations was solved in [29] and for modular pseudo-metrics in [25]. Next, we provide such a theorem.

It should be noted that the concept of a collection of $\mathcal{T}$-indistinguishability relations, as well as the concept of a function that aggregates $\mathcal{T}$-indistinguishability relations into a T-indistinguishability relation, can be adapted to the modular context in a manner similar to that in the non-modular case. The same applies to functions that merge bounded pseudo-metrics and extended pseudo-metrics.

Theorem 4.

Let $n\in \mathbb{N}$, let $\mathcal{T}={\left\{T_i\right\}}_{i=1}^n$ be a collection of continuous Archimedean t-norms, and let ${\left\{f_{T_i}\right\}}_{i=1}^n$ be a collection of additive generators of $\mathcal{T}$. If T is a continuous Archimedean t-norm and $F:{[0,1]}^n\to [0,1]$ is a function, then the following assertions are equivalent:

(1) 

F aggregates modular $\mathcal{T}$-indistinguishability relations into a modular T-indistinguishability relation.

(2) 

$F\left(1_n\right)=1$ and $T\left(F\right(a),F(b\left)\right)\le F\left(c\right)$ whenever $a,b,c\in {[0,1]}^n$ such that $T_i(a_i,b_i)\le c_i$ for all $i\in \{1,\dots ,n\}$, where $1_n=(1,\dots ,1)$.

(3) 

The function expressed as $G:{[0,\infty ]}^n\to [0,f_T\left(0\right)]$, where $G=f_T\circ F\circ (f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)})$, satisfies the following:

(a) 

$G\left(0_n\right)=0$, where $0_n=(0,\dots ,0)$.

(b) 

$G\left(c\right)\le G\left(a\right)+G\left(b\right)$ for all $a,b,c\in {[0,\infty ]}^n$ such that $c_i\le a_i+b_i$ for all $i\in \{1,\dots ,n\}$.

(4) 

The function expressed as $G:{[0,\infty ]}^n\to [0,f_T\left(0\right)]$, where $G=f_T\circ F\circ (f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)})$, aggregates every collection of modular pseudo-metrics ${\left\{d_i\right\}}_{i=1}^n$ into a $f_T\left(0\right)$-bounded modular pseudo-metric.

(5) 

The function expressed as $H:{\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]\to [0,f_T\left(0\right)]$, where $H=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, aggregates every collection of ${\left(f_{T_i}\left(0\right)\right)}_{i=1}^n$-bounded modular pseudo-metrics ${\left\{d_i\right\}}_{i=1}^n$ into a $f_T\left(0\right)$-bounded modular pseudo-metric.

(6) 

The function expressed as $H:{\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]\to [0,f_T\left(0\right)]$, where $H=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, satisfies the following:

(a) 

$H\left(0_n\right)=0$.

(b) 

$H\left(c\right)\le H\left(a\right)+H\left(b\right)$ for all $a,b,c\in {\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]$ such that $c_i\le a_i+b_i$ for all $i\in \{1,\dots ,n\}$.

In the succeeding sections, we introduce the outcome of our research, which takes advantage of all exposed results. With this purpose, on account of [26,27,28,29], notice that all T-transitive relations included in Table 1 and all generalized extended metrics included in Table 2 can be adapted to the modular framework extending Definitions 4 and 5 in an obvious way. Nevertheless, in order to help the reader, let us recall the asymmetric point separation property in the modular context. A fuzzy set ($E:X\times X\times ]0,\infty [$) fulfills the asymmetric point separation, provided that $E(x,y,t)=E(y,x,t)=1$ for all $t>0$$\iff x=y$. Similarly, a function ($d:X\times X\times ]0,\infty [\to [0,\infty ]$) verifies the distance asymmetric point separation property if $d(x,y,t)=d(y,x,t)=0$ for all $t>0$$\iff x=y$.

3. The Duality Relationship in the Modular Framework

In this section, our goal is to study the duality relationship between each modular T-transitive relation and its reciprocal modular generalized metric. An extension of the reasoning exposed in [19] to prove Theorem 2 can be used to obtain a more general version of this theorem. For the purpose of completion, in the succeeding, we will include an adapted version of such reasoning to prove the desired statements. In particular, we will start by proving Lemma 1.

Lemma 1.

Let X be a non empty set and $T^{*}$ be a t-norm with an additive generator ($f_{T^{*}}:[0,1]\to [0,\infty ]$). Assume that $E:X\times X\times ]0,\infty [\to [0,1]$ is a fuzzy relation. For all $x,y\in X$ and for $t\ge 0$, the function expressed as $d_{E,f_{T^{*}}}:X\times X\times ]0,\infty [\to [0,\infty ]$ is defined as follows:

$$d_{E,f_{T^{*}}}(x,y,t)=f_{T^{*}}\left(E(x,y,t)\right).$$

Under these conditions, the following statements hold:

(a) 

If E fulfills the reflexivity condition (M.E.1), then $d_{E,f_{T^{*}}}$ satisfies the reflexivity condition for metrics (M.D.1).

(b) 

If E fulfills the symmetry condition (M.E.2), then $d_{E,f_{T^{*}}}$ satisfies the symmetry condition for metrics (M.D.2).

(c) 

If E fulfills the point separation condition (M.E.4), then $d_{E,f_{T^{*}}}$ satisfies the point separation condition for metrics (M.D.4).

(d) 

If E fulfills the asymmetric point separation condition, then $d_{E,f_{T^{*}}}$ satisfies the distance asymmetric point separation condition.

Proof. 

(a)

Assume that $E(x,x,t)=1$ for all $x\in X$ and all $t>0$. Also recall that $f_{T^{*}}\left(1\right)=0$ (since $f_{T^{*}}$ is an additive generator of $T^{*}$). Consequently,

$$d_{E,f_{T^{*}}}(x,x,t)=f_{T^{*}}\left(E(x,x,t)\right)=f_{T^{*}}\left(1\right)=0$$

for all $x\in X$ and all $t>0$, as we wanted.

(b)

Based on the hypothesis that $E(x,y,t)=E(y,x,t)$ for all $t>0$, we can deduce that

$$d_{E,f_{T^{*}}}(x,y,t)=f_{T^{*}}\left(E(x,y,t)\right)=f_{T^{*}}\left(E(y,x,t)\right)=d_{E,f_{T^{*}}}(y,x,t),$$

obtaining the desired result.

(c)

Assume that, given some $x,y\in X$, we have $d_{E,f_{T^{*}}}(x,y,t)=0$ for all $t>0$. In such a case, $f_{T^{*}}\left(E(x,y,t)\right)=0$ for all $t>0$. Moreover, provided that $f_{T^{*}}$ is an additive generator of $T^{*}$, $f_{T^{*}}$ is strictly decreasing; therefore, $f_{T^{*}}$ is injective. Consequently, we find that $E(x,y,t)=1$ for all $t>0$ and, hence, that $x=y$.

(d)

Assume that, given some $x,y\in X$, we have $d_{E,f_{T^{*}}}(x,y,t)=d_{E,f_{T^{*}}}(y,x,t)=0$ for all $t>0$. Then, $f_{T^{*}}\left(E(x,y,t)\right)=0$ and $f_{T^{*}}\left(E(y,x,t)\right)=0$ for all $t>0$. The fact that $f_{T^{*}}$ is strictly decreasing means $E(x,y,t)=E(y,x,t)=1$ for all $t>0$. It follows that $x=y$, as claimed.

 □

Making use of this result, we are ready to prove Theorem 5, which extends and improves Theorem 2.

Theorem 5.

Let X be a non-empty set and $T^{*}$ be a t-norm with an additive generator ($f_{T^{*}}:[0,1]\to [0,\infty ]$). Assume that T is a t-norm and $E:X\times X\times ]0,\infty [\to [0,1]$ is a fuzzy relation. If $d_{E,f_{T^{*}}}:X\times X\times ]0,\infty [\to [0,\infty ]$ is a function defined by

$$d_{E,f_{T^{*}}}(x,y,t)=f_{T^{*}}\left(E(x,y,t)\right)$$

for all $x,y\in X$ and all $t>0$, then the following assertions are equivalent:

(1) 

$T^{*}\le T$

(2) 

$d_{E,f_{T^{*}}}$ satisfies the modular triangular inequality on X, provided that E is a modular T-transitive relation on X.

(3) 

$d_{E,f_{T^{*}}}$ is a modular quasi-pseudo-metric on X, provided that E is a modular T-fuzzy pre-order on X.

(4) 

$d_{E,f_{T^{*}}}$ is a modular relaxed pseudo-metric on X, provided that E is a modular T-relaxed indistinguishability relation on X.

(5) 

$d_{E,f_{T^{*}}}$ is a modular pseudo-metric on X, provided that E is a modular T-indistinguishability relation on X.

(6) 

$d_{E,f_{T^{*}}}$ is a modular quasi-metric on X, provided that E is a modular T-fuzzy partial order on X.

(7) 

$d_{E,f_{T^{*}}}$ is a modular metric on X, provided that E is a modular T-equality on X.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$. Assume that E is a modular T-transitive relation on X. Then, for all $x,y\in X$ and for all $t>0$, the following inequality is fulfilled:

$$T\left(E\right(x,z,t),E(y,z,s\left)\right)\le E(x,y,t+s).$$

Based on the fact that $T^{*}\le T$, it is also true that

$$\begin{array}{cc}\hfill f_{T^{*}}^{(-1)}(f_{T^{*}}\left(E(x,z,t)\right)+f_{T^{*}}\left(E(z,y,s)\right))& =T^{*}(E(x,z,t),E(y,z,s))\hfill \\ \hfill & \le T\left(E\right(x,z,t),E(y,z,s\left)\right).\hfill \end{array}$$

Hence, we get

$$f_{T^{*}}^{(-1)}(f_{T^{*}}\left(E(x,z,t)\right)+f_{T^{*}}\left(E(z,y,s)\right))\le E(x,y,t+s).$$

Since $f_{T^{*}}$ is an additive generator, $f_{T^{*}}$ is a strictly decreasing function. Thus, we obtain

$$f_{T^{*}}\left(f_{T^{*}}^{(-1)}(f_{T^{*}}\left(E(x,z,t)\right)+f_{T^{*}}\left(E(z,y,s)\right))\right)\ge f_{T^{*}}\left(E(x,y,t+s)\right)$$

Based on the fact that $f_{T^{*}}^{(-1)}$ is the pseudo-inverse of the additive generator ($f_{T^{*}}$), two different cases need to be considered:

(i)

$f_{T^{*}}\left(E(x,z,t)\right)+f_{T^{*}}\left(E(z,y,s)\right)\in \mathrm{Ran}\left(f_{T^{*}}\right)$. Then, there exists $w\in [0,1]$ with $f_{T^{*}}\left(w\right)=f_{T^{*}}\left(E(x,z,t)\right)+f_{T^{*}}\left(E(z,y,s)\right)$.

From the definition of $f_{T^{*}}^{(-1)}$, it is clear that

$$\begin{array}{cc}\hfill f_{T^{*}}\left(f_{T^{*}}^{(-1)}(f_{T^{*}}\left(E(x,z,t)\right)+f_{T^{*}}\left(E(z,y,s)\right))\right)& =f_{T^{*}}\left(f_{T^{*}}^{(-1)}\left(f_{T^{*}}\left(w\right)\right)\right)=f_{T^{*}}\left(w\right)\hfill \\ \hfill & =f_{T^{*}}\left(E(x,z,t)\right)+f_{T^{*}}\left(E(z,y,s)\right).\hfill \end{array}$$

So

$$f_{T^{*}}\left(E(x,y,t+s)\right)\le f_{T^{*}}\left(E(x,z,t)\right)+f_{T^{*}}\left(E(z,y,s)\right).$$

(ii)

$f_{T^{*}}\left(E(x,z,t)\right)+f_{T^{*}}\left(E(z,y,s)\right)\notin \mathrm{Ran}\left(f_{T^{*}}\right)$. Observe that $0\le E(x,y,t+s)$. Hence, we find that

$$f_{T^{*}}\left(E(x,y,t+s)\right)\le f_{T^{*}}\left(0\right).$$

Furthermore, since $f_{T^{*}}\left(E(x,z,t)\right)+f_{T^{*}}\left(E(z,y,s)\right)\notin \mathrm{Ran}\left(f_{T^{*}}\right)$ we have

$$f_{T^{*}}\left(0\right)\le f_{T^{*}}\left(E(x,z,t)\right)+f_{T^{*}}\left(E(z,y,s)\right).$$

Thus, we find that

$$f_{T^{*}}\left(E(x,y,t+s)\right)\le f_{T^{*}}\left(E(x,z,t)\right)+f_{T^{*}}\left(E(z,y,s)\right),$$

obtaining the desired results, i.e., that $d_{E,f_{T^{*}}}(x,y,t+s)\le d_{E,f_{T^{*}}}(x,z,t)+d_{E,f_{T^{*}}}(z,y,s)$ for all $x,y,z\in X$ and for all $t>0$.

$\left(2\right)\Rightarrow \left(3\right)$. Let E be a modular fuzzy T-pre-order on X. Notice that, in particular, E is a modular T-transitive relation on X. Therefore, $d_{E,f_{T^{*}}}$ satisfies the modular triangular inequality on X.

In addition, provided that E is a modular fuzzy T-pre-order, we know that $E(x,x,t)=1$ for all $t>0$. Then, assertion (a) in Lemma 1 warrants that $d_{E,f_{T^{*}}}$ is a modular quasi-pseudo-metric on X.

$\left(2\right)\Rightarrow \left(4\right)$. Let E be a modular T-relaxed indistinguishability relation on X. Notice that, in particular, E is a modular T-transitive relation on X. Therefore, $d_{E,f_{T^{*}}}$ satisfies the modular triangular inequality on X.

From the definition of a modular T-relaxed indistinguishability relation on X, we know that $E(x,y,t)=E(y,x,t)$ for all $t>0$. Then, according to assertion (b) in Lemma 1, we know that E is a modular relaxed pseudo-metric.

$\left(3\right)\Rightarrow \left(5\right)$. Let E be a modular T-indistinguishability relation on X. It can be deduced that E is also a modular T-fuzzy pre-order on X and, as a consequence, that $d_{E,f_{T^{*}}}$ is a quasi-pseudo-metric on X.

Based on the fact that E is a modular T-indistinguishability relation, we find that $E(x,y,t)=E(y,x,t)$ for all $t>0$. Then, assertion (b) in Lemma 1 ensures that $d_{E,f_{T^{*}}}$ is a pseudo-metric on X.

$\left(3\right)\Rightarrow \left(6\right)$. Let E be a modular fuzzy T-partial order on X. It can be deduced that E is also a modular fuzzy T-pre-order on X and, as a consequence, that $d_{E,f_{T^{*}}}$ is a quasi-pseudo-metric on X.

Based on the fact that E fulfills the asymmetric point separation condition, assertion (d) in Lemma 1 guarantees that $d_{E,f_{T^{*}}}$ is a modular quasi-metric on X.

$\left(4\right)\Rightarrow \left(5\right)$. Since E is a modular T-indistinguishability relation on X, we find that E is a modular T-relaxed indistinguishability relation on X. Hence, we immediately find that $d_{E,f_{T^{*}}}$ is a modular relaxed pseudo-metric on X. Moreover, E satisfies the reflectivity and symmetry conditions. Assertions (a) and (b) in Lemma 1 show that $d_{E,f_{T^{*}}}$ fulfills the reflexivity and symmetry conditions for metrics. So we conclude that $d_{E,f_{T^{*}}}$ is a modular pseudo-metric on X.

$\left(5\right)\Rightarrow \left(7\right)$. The desired proof can be obtained using reasoning analogous to that from $\left(3\right)\Rightarrow \left(6\right)$ but now applying assertion (c) in Lemma 1.

$\left(6\right)\Rightarrow \left(7\right)$. Let E be a modular T-equality on X. It can be deduced that E is also a modular fuzzy T-partial order on X and, as a consequence, that $d_{E,f_{T^{*}}}$ is a modular quasi-metric on X.

Additionally, the symmetry of E and assertion (b) in Lemma 1 provide the symmetry of $d_{E,f_{T^{*}}}$, showing that $d_{E,f_{T^{*}}}$ is a modular metric on X.

$\left(7\right)\Rightarrow \left(1\right)$. This implication matches with implication $\left(3\right)\Rightarrow \left(1\right)$ in Theorem 3 of [19].

To sum up, we have proven that $\left(1\right)\Rightarrow \left(2\right)\Rightarrow \left(3\right)\Rightarrow \left(5\right)\Rightarrow \left(7\right)\Rightarrow \left(1\right)$, as well as that $\left(3\right)\Rightarrow \left(6\right)\Rightarrow \left(7\right)\Rightarrow \left(1\right)\Rightarrow \left(3\right)$ and that $\left(2\right)\Rightarrow \left(4\right)\Rightarrow \left(5\right)\Rightarrow \left(7\right)\Rightarrow \left(1\right)\Rightarrow \left(2\right)$. Therefore, we can conclude that all the statements are equivalent. □

It must be pointed out that the preceding result extends Propositions 2 and 3 in [8] to the modular framework and improves Theorem 2.

Conversely, it can also be shown that all the modular T-transitive relations studied in Theorem 5 can be constructed from some of the generalized modular metrics presented in that theorem. In the same way as before, these results can be obtained by extending Theorem 3, taking inspiration from the arguments from [19]. It is worth mentioning that, in this case, the continuity of the considered t-norm is required to guarantee that the modular triangular transitivity $(M.E.4)$ is fulfilled. An explanation of this can be found in ([8] Example 3). In order reduce the complexity of the proof, we will first introduce an auxiliary result.

Lemma 2.

Let X be a non-empty set, and let $T^{*}$ be a continuous Archimedean t-norm with an additive generator ($f_{T^{*}}:[0,1]\to [0,\infty ]$). Assume that $d:X\times X\times ]0,\infty [\to [0,\infty ]$ is a function. For all $x,y\in X$ and $t>0$, the fuzzy relation expressed as $E^{d,f_{T^{*}}}:X\times X\times ]0,\infty [\to [0,1]$ is defined by

$$E^{d,f_{T^{*}}}(x,y,t)=f_{T^{*}}^{(-1)}\left(d(x,y,t)\right).$$

Under these conditions, the following statements hold:

(a) 

If d satisfies condition (M.D.1), then $E^{d,f_{T^{*}}}$ satisfies condition (M.E.1).

(b) 

If d satisfies condition (M.D.2), then $E^{d,f_{T^{*}}}$ satisfies condition (M.E.2).

(c) 

If d satisfies condition (M.D.4), then $E^{d,f_{T^{*}}}$ satisfies condition (M.E.4).

(d) 

If d satisfies the distance asymmetric point separation condition, then $E^{d,f_{T^{*}}}$ satisfies the asymmetric point separation condition.

Proof. 

(a)

Assume that $d(x,x,t)=0$ for all $x\in X$ and for all $t>0$. Then,

$$E^{d,f_{T^{*}}}(x,x,t)=f_{T^{*}}^{(-1)}\left(d(x,x,t)\right)=f_{T^{*}}^{-1}\left(\min \{f_{T^{*}}\left(0\right),0\}\right)=f_{T^{*}}^{-1}\left(0\right)=1$$

for all $x\in X$ and for all $t>0$.

(b)

By hypothesis, $d(x,y,t)=d(y,x,t)$ for all $x\in X$ and for all $t>0$. Hence,

$$E^{d,f_{T^{*}}}(x,y,t)=f_{T^{*}}^{(-1)}\left(d(x,y,t)\right)=f_{T^{*}}^{(-1)}\left(d(y,x,t)\right)=E^{d,f_{T^{*}}}(y,x,t)$$

for all $x,y\in X$ and for all $t>0$.

(c)

Let $x,y\in X$ such that, for all $t>0$, we have $E^{d,f_{T^{*}}}(x,y,t)=1$. Then, we find that

$$1=E^{d,f_{T^{*}}}(x,y,t)=f_{T^{*}}^{(-1)}\left(d(x,y,t)\right).$$

Provided that $f_{T^{*}}$ is strictly decreasing and that, thus, $f_{T^{*}}\left(0\right)>0$, we deduce that

$$\begin{array}{cc}\hfill 0& =f_{T^{*}}\left(1\right)=f_{T^{*}}{f}_{T^{*}}^{(-1)}\left(d(x,y,t)\right)=f_{T^{*}}(f_{*}^{-1}\left(\min \{f_{T^{*}}\left(0\right),d(x,y,t)\}\right)\hfill \\ \hfill & =\min \{f_{T^{*}}\left(0\right),d(x,y,t)\}=d(x,y,t)\hfill \end{array}$$

for all $t>0$. Hence, $x=y$.

(d)

The proof follows arguments similar to those given in the proof of assertion (c).

 □

With these results in mind, next, we provide the desired theorem.

Theorem 6.

Let X be a non-empty set, and let $T^{*}$ be a continuous Archimedean t-norm with an additive generator ($f_{T^{*}}:[0,1]\to [0,\infty ]$). Consider the function expressed as $d:X\times X\times ]0,\infty [\to [0,\infty ]$; for all $x,y\in X$ and all $t>0$, the fuzzy relation expressed as $E^{d,f_{T^{*}}}:X\times X\times ]0,\infty [\to [0,1]$ is defined by

$$E^{d,f_{T^{*}}}(x,y,t)=f_{T^{*}}^{(-1)}\left(d(x,y,t)\right).$$

In such a case, the following assertions are equivalent:

(1) 

$T^{*}\ge T$

(2) 

$E^{d,f_{T^{*}}}$ is a modular T-transitive relation on X, provided that d satisfies the modular triangular inequality on X.

(3) 

$E^{d,f_{T^{*}}}$ is a modular T-fuzzy pre-order on X, provided that d is a modular quasi-pseudo-metric on X.

(4) 

$E^{d,f_{T^{*}}}$ is a modular T-relaxed indistinguishability relation on X, provided that d is a modular relaxed pseudo-metric on X.

(5) 

$E^{d,f_{T^{*}}}$ is a modular T-indistinguishability relation on X, provided that d is a modular pseudo-metric on X.

(6) 

$E^{d,f_{T^{*}}}$ is a modular T-fuzzy partial order on X, provided that d is a modular quasi-metric on X.

(7) 

$E^{d,f_{T^{*}}}$ is a modular T-equality on X, provided that d is a modular metric on X.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$. Let d be a function that satisfies the modular triangular inequality. Thus, for all $x,y,z\in X$ and $t,s\ge 0$, we have the following:

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

Given that $f_{T^{*}}$ is a strictly decreasing function, we have

$$E^{d,f_{T^{*}}}(x,y,t+s)=f_{T^{*}}^{(-1)}\left(d(x,y,t+s)\right)\ge f_{T^{*}}^{(-1)}(d(x,z,t)+d(z,y,s)).$$

Furthermore,

$$\begin{array}{cc}\hfill T(E^{d,f_{T^{*}}}(x,z,t),& E^{d,f_{T^{*}}}(z,y,s))\le T^{*}(E^{d,f_{T^{*}}}(x,z,t),E^{d,f_{T^{*}}}(z,y,s))\hfill \\ \hfill & =f_{T^{*}}^{(-1)}(f_{T^{*}}\left(E^{d,f_{T^{*}}}(x,z,t)\right)+f_{T^{*}}\left(E^{d,f_{T^{*}}}(z,y,s)\right))\hfill \\ \hfill & =f_{T^{*}}^{(-1)}(\min \{f_{T^{*}}\left(0\right),d(x,z,t)\}+\min \{f_{T^{*}}\left(0\right),d(z,y,s)\}).\hfill \end{array}$$

Since

$$\min \{f_{T^{*}}\left(0\right),d(x,z,t)\}+\min \{f_{T^{*}}\left(0\right),d(z,y,s)\}\le d(x,z,t)+d(z,y,s),$$

based on the fact that $f_{T^{*}}^{(-1)}$ is decreasing, it is clear that

$$f_{T^{*}}^{(-1)}(\min \{f_{T^{*}}\left(0\right),d(x,z,t)\}+\min \{f_{T^{*}}\left(0\right),d(z,y,s)\})\ge f_{T^{*}}^{(-1)}(d(x,z,t)+d(z,y,s)).$$

Now, it is enough to prove that

$$f_{T^{*}}^{(-1)}(d(x,z,t)+d(z,y,s))\ge f_{T^{*}}^{(-1)}(\min \{f_{T^{*}}\left(0\right),d(x,z,t)\}+\min \{f_{T^{*}}\left(0\right),d(z,y,s)\}).$$

In order to do so, we will consider two possible cases:

(i)

$\max \{d(x,z,t),d(z,y,s)\}\ge f_{T^{*}}\left(0\right)$: In such a case, it is clear that

$$d(x,z,t)+d(z,y,s)\ge f_{T^{*}}\left(0\right).$$

Recall that $f_{T^{*}}^{(-1)}\left(x\right)=f_{T^{*}}^{-1}\left(\min \{f_{T^{*}}\left(0\right),x\}\right)$ (in the case of $f_{T^{*}}\left(0\right)=\infty$, $f_{T^{*}}^{(-1)}\left(x\right)=f_{T^{*}}^{-1}\left(x\right)$). As a consequence,

$$f_{T^{*}}^{(-1)}(d(x,z,t)+d(z,y,s))=f_{T^{*}}^{-1}\left(\min \{f_{T^{*}}\left(0\right),d(x,z,t)+d(z,y,s)\}\right)=0.$$

Moreover,

$$\min \{f_{T^{*}}\left(0\right),d(x,z,t)\}+\min \{f_{T^{*}}\left(0\right),d(z,y,s)\}\ge f_{T^{*}}\left(0\right)$$

is satisfied. The fact that $f_{T^{*}}^{(-1)}$ is decreasing yields

$$\begin{array}{cc}\hfill f_{T^{*}}^{(-1)}(d(x,z,t)+d(z,y,s))& =0=f_{T^{*}}^{(-1)}\left(f_{T^{*}}\left(0\right)\right)\hfill \\ \hfill & \ge f_{T^{*}}^{(-1)}(\min \{f_{T^{*}}\left(0\right),d(x,z,t)\}+\min \{f_{T^{*}}\left(0\right),d(z,y,s)\}).\hfill \end{array}$$

(ii)

$\max \{d(x,z,t),d(z,y,s)\}<f_{T^{*}}\left(0\right)$: In such a case,

$$\min \{f_{T^{*}}\left(0\right),d(x,z,t)\}+\min \{f_{T^{*}}\left(0\right),d(z,y,s)\}=d(x,z,t)+d(z,y,s),$$

and the desired equality is obtained.

We have proven that

$$f_{T^{*}}^{(-1)}(d(x,z,t)+d(z,y,s))=f_{T^{*}}^{(-1)}(\min \{f_{T^{*}}\left(0\right),d(x,z,t)\}+\min \{f_{T^{*}}\left(0\right),d(z,y,s)\}).$$

Thus, $E^{d,f_{T^{*}}}$ is T transitive, i.e.,

$$E^{d,f_{T^{*}}}(x,y,t+s)\ge T(E^{d,f_{T^{*}}}(x,z,t),E^{d,f_{T^{*}}}(z,y,s)),$$

for all $x,y,z\in X$ and $t,s\ge 0$.

$\left(2\right)\Rightarrow \left(3\right)$. Let d be a modular quasi-pseudo-metric on X. Observe that, in particular, d is a function that satisfies the modular triangular inequality on X. Therefore, $E^{d,f_{T^{*}}}$ is a modular T-transitive relation on X. Consequently, it is enough to show that $E^{d,f_{T^{*}}}(x,x,t)=1$ for all $x\in X$ and for all $t>0$.

Based on the fact that d is a modular quasi-pseudo-metric on X, we know that $d(x,x,t)=0$ for all $x\in X$ and for all $t>0$. Therefore, assertion (a) in Lemma 2 ensures that $E^{d,f_{T^{*}}}$ is a modular fuzzy pre-order for T on X.

$\left(2\right)\Rightarrow \left(4\right)$. Let d be a modular relaxed pseudo-metric on X. Observe that, in particular, d is a function that satisfies the modular triangular inequality on X. Therefore, $E^{d,f_{T^{*}}}$ is a modular T-transitive relation on X.

Moreover, since d is a modular relaxed pseudo-metric on X, we know that $d(x,y,t)=d(y,x,t)$ for all $x,y\in X$ and for all $t>0$. Hence, it can be deduced from assertion (b) in Lemma 2 that $E^{d,f_{T^{*}}}$ is a modular T-relaxed indistinguishability relation on X.

$\left(3\right)\Rightarrow \left(5\right)$. Let d be a modular pseudo-metric on X. Notice that, in particular, d is a modular quasi-pseudo-metric on X. Therefore, $E^{d,f_{T^{*}}}$ is a modular T-fuzzy pre-order on X.

In addition, based on the fact that d is a modular pseudo-metric, we know that d is symmetric. Using assertion (b) in Lemma 2, we find that $E^{d,f_{T^{*}}}$ is also symmetric. Consequently, $E^{d,f_{T^{*}}}$ is a modular indistinguishability relation.

$\left(3\right)\Rightarrow \left(6\right)$. Let d be a modular quasi-metric on X. Notice that, in particular, d is a modular quasi-pseudo-metric on X. Therefore, $E^{d,f_{T^{*}}}$ is a modular T-fuzzy pre-order on X.

Additionally, since d satisfies the distance asymmetric point separation condition, $E^{d,f_{T^{*}}}$ satisfies the asymmetric point separation condition by assertion (d) in Lemma 2. Consequently, we find that $E^{d,f_{T^{*}}}$ is a modular T-fuzzy partial order on X.

$\left(4\right)\Rightarrow \left(5\right)$. Let d be a modular pseudo-metric on X. Notice that, in particular, d is a modular relaxed pseudo-metric on X. Therefore, $E^{d,f_{T^{*}}}$ is a modular T-relaxed indistinguishability relation on X.

In addition, based on the fact that $d(x,x,t)=1$ for all $x\in X$ and for all $t>0$, it can be deduced form assertion (a) in Lemma 2 that $E^{d,f_{T^{*}}}$ is a modular T-indistinguishability relation on X.

$\left(5\right)\Rightarrow \left(7\right)$. Let d be a modular metric on X. Notice that, in particular, d is a modular pseudo-metric on X. Therefore, $E^{d,f_{T^{*}}}$ is a modular T-indistinguishability relation on X. In addition, assertion (c) in Lemma 2 ensures that $E^{d,f_{T^{*}}}$ is a modular T equality on X.

$\left(6\right)\Rightarrow \left(7\right)$. Let d be a modular metric on X. Notice that, in particular, d is a modular quasi-metric on X. Therefore, $E^{d,f_{T^{*}}}$ is a modular T-fuzzy partial order on X.

Moreover, assertion (b) in Lemma 2 helps us to prove that $E^{d,f_{T^{*}}}$ is symmetric and, consequently, a modular T equality on X.

$\left(7\right)\Rightarrow \left(1\right)$. We will prove that if $E^{d,f_{T^{*}}}$ is a modular T equality on X whenever d is a modular metric on X, then $T^{*}(a,b)\ge T(a,b)$ for any $a,b\in [0,1]$.

Notice that whenever $\max \{a,b\}=1$, $T^{*}(a,b)=T(a,b)$. Similarly, if $\min \{a,b\}=0$, we also have $T^{*}(a,b)=T(a,b)$. Hence, we only need to consider the case of $a,b\in ]0,1[$. Let $X=\{x,y,z\}$, where $x,y,z$ are all different from each other, and let $a,b\in ]0,1[$. The function expressed as $d:X\times X\times ]0,\infty [\to [0,\infty ]$ is defined, for all $t>0$, as follows:

$$d(u,v,t)=d(v,u,t)=\left\{\begin{array}{c}0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if}u=v\hfill \\ f_{T^{*}}\left(a\right)+f_{T^{*}}\left(b\right), \mathrm{if}u=x\mathrm{and}v=y\hfill \\ f_{T^{*}}\left(a\right),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if}u=x\mathrm{and}v=z\hfill \\ f_{T^{*}}\left(b\right),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if}u=y\mathrm{and}v=z\hfill \end{array}\right..$$

It can be easily shown that d is a modular metric on X. As a consequence, $E^{d,f_{T^{*}}}$ is a modular T equality on X. Then, given $t>0$, we obtain the desired result:

$$\begin{array}{cc}\hfill T(a,b)& =T(f_{T^{*}}^{(-1)}\left(f_{T^{*}}\left(a\right)\right),f_{T^{*}}^{(-1)}\left(f_{T^{*}}\left(b\right)\right))=T(E^{d,f_{T^{*}}}(x,z,t),E^{d,f_{T^{*}}}(z,y,s))\hfill \\ \hfill & \le E^{d,f_{T^{*}}}(x,y,t+s)=f_{T^{*}}^{(-1)}(f_{T^{*}}\left(a\right)+f_{T^{*}}\left(b\right))=T^{*}(a,b).\hfill \end{array}$$

To sum up, we have proven that $\left(1\right)\Rightarrow \left(2\right)\Rightarrow \left(3\right)\Rightarrow \left(5\right)\Rightarrow \left(7\right)\Rightarrow \left(1\right)$, as well as that $\left(3\right)\Rightarrow \left(6\right)\Rightarrow \left(7\right)\Rightarrow \left(1\right)\Rightarrow \left(3\right)$ and that $\left(2\right)\Rightarrow \left(4\right)\Rightarrow \left(5\right)\Rightarrow \left(7\right)\Rightarrow \left(1\right)\Rightarrow \left(2\right)$. Therefore, we can conclude that all the statements are equivalent. □

Notice that Theorem 6 improves Theorem 3 not only by extending the number of equivalences considering more cases of modular T-transitive relations but also by introducing a new equivalence based on the dominance of triangular norms (statement (1)), which was not considered in Theorem 3 and Proposition 7 in [8].

4. Aggregation of Modular $\mathit{T}$-Transitive Relations and Modular Generalized Metrics

In this section, our goal is to prove that functions that aggregate each one of the modular generalized metrics studied in Section 3 can be obtained by means of a specific construction of functions that merge some type of modular T-transitive fuzzy relations involving additive generators and their pseudo-inverses. In particular, we want to provide a theorem in the spirit of Theorem 4 for each one of the previously analyzed pairs (T-transitive fuzzy relation and modular generalized metric). Throughout this section, all t-norms will be considered to be continuous and Archimedean because the proofs of the presented results depend on Theorem 6.

4.1. Aggregation of Modular Fuzzy Pre-Orders and Modular Quasi-Pseudo-Metrics

The aforementioned construction for the case in which we consider those functions that merge collections of modular indistinguishability relations and those that aggregate collections of modular pseudo-metrics was treated in [30] and introduced as Theorem 4 here. Moreover, in [29] (Theorems 2 and 8), the sets of those functions that merge collections of modular indistinguishability relations that aggregate modular fuzzy pre-orders and that aggregate fuzzy pre-orders were shown to be the same. In particular, the following result was proven.

Theorem 7.

Let $n\in \mathbb{N}$, let $\mathcal{T}={\left\{T_i\right\}}_{i=1}^n$ be a collection of continuous Archimedean t-norms, and let ${\left\{f_{T_i}\right\}}_{i=1}^n$ be a collection of additive generators of $\mathcal{T}$. If T is a continuous Archimedean t-norm and $F:{[0,1]}^n\to [0,1]$ is a function, then the following assertions are equivalent:

(1) 

F aggregates $\mathcal{T}$-fuzzy pre-orders into a T-fuzzy pre-order.

(2) 

F aggregates modular $\mathcal{T}$-fuzzy pre-orders into a modular T-fuzzy pre-order.

(3) 

F aggregates modular $\mathcal{T}$-indistinguishability relations into a modular T-indistinguishability relation.

Taking all this into account, the next result can be directly deduced.

Theorem 8.

Let $n\in \mathbb{N}$, let $\mathcal{T}={\left\{T_i\right\}}_{i=1}^n$ be a collection of continuous Archimedean t-norms, and let ${\left\{f_{T_i}\right\}}_{i=1}^n$ be a collection of additive generators of $\mathcal{T}$. If T is a continuous Archimedean t-norm and $F:{[0,1]}^n\to [0,1]$ is a function, then the following assertions are equivalent:

(1) 

F aggregates $\mathcal{T}$-fuzzy pre-orders into a T-fuzzy pre-order.

(2) 

F aggregates modular $\mathcal{T}$-fuzzy pre-orders into a modular T-fuzzy pre-order.

(3) 

F aggregates modular $\mathcal{T}$-indistinguishability relations into a modular T-indistinguishability relation.

(4) 

The function expressed as $G:{[0,\infty ]}^n\to [0,f_T\left(0\right)]$, where $G=f_T\circ F\circ (f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)})$, satisfies

(a) 

$G\left(0_n\right)=0$.

(b) 

$G\left(c\right)\le G\left(a\right)+G\left(b\right)$ for all $a,b,c\in {[0,\infty ]}^n$ such that $c_i\le a_i+b_i$ for all $i\in \{1,\dots ,n\}$.

(5) 

The function expressed as $G:{[0,\infty ]}^n\to [0,f_T\left(0\right)]$, where $G=f_T\circ F\circ (f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)})$, aggregates every collection of modular pseudo-metrics (${\left\{d_i\right\}}_{i=1}^n$) into a $f_T\left(0\right)$-bounded modular pseudo-metric.

(6) 

The function expressed as $G:{[0,\infty ]}^n\to [0,f_T\left(0\right)]$, where $G=f_T\circ F\circ (f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)})$, aggregates every collection of modular quasi-pseudo-metrics (${\left\{d_i\right\}}_{i=1}^n$) into a $f_T\left(0\right)$-bounded modular quasi-pseudo-metric.

(7) 

The function expressed as $H:{\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]\to [0,f_T\left(0\right)]$, where $H=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, aggregates every collection of ${\left(f_{T_i}\left(0\right)\right)}_{i=1}^n$-bounded modular pseudo-metrics (${\left\{d_i\right\}}_{i=1}^n$) into a $f_T\left(0\right)$-bounded modular pseudo-metric.

(8) 

The function expressed as $H:{\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]\to [0,f_T\left(0\right)]$, where $H=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, aggregates every collection of ${\left(f_{T_i}\left(0\right)\right)}_{i=1}^n$-bounded modular quasi-pseudo-metrics (${\left\{d_i\right\}}_{i=1}^n$) into a $f_T\left(0\right)$-bounded modular quasi-pseudo-metric.

(9) 

The function expressed as $H:{\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]\to [0,f_T\left(0\right)]$, where $H=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, satisfies

(a) 

$H\left(0_n\right)=0$.

(b) 

$H\left(c\right)\le H\left(a\right)+H\left(b\right)$ for all $a,b,c\in {\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]$ such that $c_i\le a_i+b_i$ for all $i\in \{1,\dots ,n\}$.

Proof. 

The equivalence of $\left(1\right)\iff \left(2\right)\iff \left(3\right)$ is guaranteed by Theorem 7 (Theorems 2 and 8 in [29]). The equivalence of $\left(3\right)\iff \left(4\right)\iff \left(5\right)\iff \left(7\right)\iff \left(9\right)$ follows from Theorem 4.

Next, we show the equivalence of $\left(4\right)\iff \left(6\right)$. It is evident that $\left(6\right)\Rightarrow \left(5\right)$. Next, we show that $\left(5\right)\Rightarrow \left(6\right)$. We know that the equivalence of $\left(5\right)\iff \left(4\right)$ is true. As a consequence, we show that $\left(4\right)\Rightarrow \left(6\right)$. Now, assume that ${\left\{d_i\right\}}_{i=1}^n$ is a collection of modular quasi-pseudo-metrics on a non-empty set (X). Then, $G(d_1,\dots ,d_n)(x,x,t)=G(d_1(x,x,,t),\dots ,d_n(x,x,,t))=G(0,\dots ,0)=0$ for all $x\in X$ and for all $t>0$. Moreover, we find that

$$G(d_1,\dots ,d_n)(x,z,t+s)\le G(d_1,\dots ,d_n)(x,y,t)+G(d_1,\dots ,d_n)(y,z,t)$$

for all $x,y,z\in X$ and for all $t,s>0$, since $d_i(x,z,t+s)\le d_i(x,y,t)+d_i(y,z,s)$ for all $x,y,z\in X$, for all $t,s>0$ and for all $i\in \{1,\dots ,n\}$. Finally, we have $G\left(a\right)\le f_T\left(0\right)$ for all $a\in {[0,\infty ]}^n$. So $G(d_1,\dots ,d_n)(x,y,t)\le f_T\left(0\right)$ for all $x,y\in X$ and for all $t>0$. Therefore, we conclude that $G(d_1,\dots ,d_n)$ is a $f_T\left(0\right)$-bounded modular quasi-pseudo-metric on X.

Finally, we show the equivalence of $\left(7\right)\iff \left(8\right)$. It is clear that $\left(8\right)\Rightarrow \left(7\right)$. In addition, we know that the equivalence of $\left(9\right)\iff \left(7\right)$ holds. The same arguments applied to prove the implication of $\left(4\right)\Rightarrow \left(6\right)$ remain valid to show that $\left(9\right)\Rightarrow \left(8\right)$. □

In the particular case in which T is a strict continuous Archimedean t-norm, we have $f_T\left(0\right)=\infty$; thus, the following corollary can be deduced.

Corollary 1.

Let $n\in \mathbb{N}$, let $\mathcal{T}={\left\{T_i\right\}}_{i=1}^n$ be a collection of strict continuous Archimedean t-norms, and let ${\left\{f_{T_i}\right\}}_{i=1}^n$ be a collection of additive generators of $\mathcal{T}$. If T is a strict continuous Archimedean t-norm and $F:{[0,1]}^n\to [0,1]$ is a function, then the following assertions are equivalent:

(1) 

F aggregates $\mathcal{T}$-fuzzy pre-orders into a T-fuzzy pre-order.

(2) 

F aggregates modular $\mathcal{T}$-fuzzy pre-orders into a modular T-fuzzy pre-order.

(3) 

F aggregates modular $\mathcal{T}$-indistinguishability relations into a modular T-indistinguishability relation.

(4) 

The function expressed as $G:{[0,\infty ]}^n\to [0,\infty ]$, where $G=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, satisfies the following:

(a) 

$G\left(0_n\right)=0$.

(b) 

$G\left(c\right)\le G\left(a\right)+G\left(b\right)$ for all $a,b,c\in {[0,\infty ]}^n$ such that $c_i\le a_i+b_i$ for all $i\in \{1,\dots ,n\}$.

(5) 

The function expressed as $G:{[0,\infty ]}^n\to [0,\infty )]$, where $G=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, aggregates every collection of modular pseudo-metrics (${\left\{d_i\right\}}_{i=1}^n$) into a modular pseudo-metric.

(6) 

The function expressed as $G:{[0,\infty ]}^n\to [0,\infty ]$, where $G=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, aggregates every collection of modular quasi-pseudo-metrics (${\left\{d_i\right\}}_{i=1}^n$) into a modular quasi-pseudo-metric.

4.2. Aggregation of Modular Equalities and Modular Metrics

Taking inspiration from [30], we next explore the possibility of obtaining a result in the spirit of Theorem 4 for the particular case of modular equalities and modular metrics. First, we need to prove the following auxiliary result.

Lemma 3.

Let T be a continuous Archimedean t-norm, and let $f_T$ be a continuous additive generator of T. For all $a,b\in [0,\infty ]$, we have $f_T^{(-1)}(a+b)=f_T^{(-1)}(f_T\circ f_T^{(-1)}\left(a\right)+f_T\circ f_T^{(-1)}\left(b\right))$.

Proof. 

Given that $f_T$ is a continuous additive generator, notice that

$$f_T^{(-1)}(a+b)=f_T^{-1}\left(\min \{f_T\left(0\right),a+b\}\right)$$

and

$$\begin{array}{cc}\hfill f_T^{(-1)}(f_T\circ f_T^{(-1)}\left(a\right)+f_T\circ f_T^{(-1)}\left(b\right))& =f_T^{(-1)}(\min \{f_T\left(0\right),a\}+\min \{f_T\left(0\right),b\})\hfill \\ \hfill & =f_T^{-1}\left(\min \{f_T\left(0\right),\min \{f_T\left(0\right),a\}+\min \{f_T\left(0\right),b\}\}\right)\hfill \end{array}.$$

Let us consider two possible scenarios:

(i)

$\max \{a,b\}\ge f_T\left(0\right)$: In such a case, $a+b\ge f_T\left(0\right)$ and $\min \{f_T\left(0\right),a\}+\min \{f_T\left(0\right),b\}\ge f_T\left(0\right)$. Hence,

$$f_T^{(-1)}(a+b)=0=f_T^{(-1)}(f_T\circ f_T^{(-1)}\left(a\right)+f_T\circ f_T^{(-1)}\left(b\right)).$$

(ii)

$\max \{a,b\}<f_T\left(0\right)$: In such a case, $\min \{f_T\left(0\right),a\}+\min \{f_T\left(0\right),b\}=a+b$. Thus,

$$f_T^{(-1)}(a+b)=f_T^{(-1)}(f_T\circ f_T^{(-1)}\left(a\right)+f_T\circ f_T^{(-1)}\left(b\right)).$$

 □

In [29] (Theorem 19), the following characterization of those functions that merge collections of modular fuzzy equalities was proven.

Theorem 9.

Let $n\in \mathbb{N}$, let $\mathcal{T}={\left\{T_i\right\}}_{i=1}^n$ be a collection of continuous Archimedean t-norms, and let ${\left\{f_{T_i}\right\}}_{i=1}^n$ be a collection of additive generators of $\mathcal{T}$. If T is a continuous Archimedean t-norm and $F:{[0,1]}^n\to [0,1]$ is a function, then the following assertions are equivalent:

(1) 

F aggregates $\mathcal{T}$-fuzzy partial orders into a T-fuzzy partial order.

(2) 

F aggregates modular $\mathcal{T}$-fuzzy partial orders into a modular T-fuzzy partial order.

(3) 

F aggregates modular $\mathcal{T}$ equalities into a modular T equality.

(4) 

$F\left(1_n\right)=1$: If $F\left(a\right)=1$, then there exists $i\in \{1,\dots ,n\}$ such that $a_i=1$ and $T\left(F\right(a),F(b\left)\right)\le F\left(c\right)$ whenever $a,b,c\in {[0,1]}^n$ such that $T_i(a_i,b_i)\le c_i$ for all $i\in \{1,\dots ,n\}$.

Moreover, in [29] (Proposition 21), the following useful implication was proven.

Proposition 1.

Let $n\in \mathbb{N}$, let $\mathcal{T}={\left\{T_i\right\}}_{i=1}^n$ be a collection of continuous Archimedean t-norms, and let ${\left\{f_{T_i}\right\}}_{i=1}^n$ be a collection of additive generators of $\mathcal{T}$. If T is a continuous Archimedean t-norm and $F:{[0,1]}^n\to [0,1]$ is a function, then among the following, $\left(1\right)\Rightarrow \left(2\right)$:

(1) 

F aggregates $\mathcal{T}$-fuzzy partial orders into a T-fuzzy partial order.

(2) 

Let $a,b\in {[0,1]}^n$. If $F\left(a\right)=F\left(b\right)=1$, then there exists $i\in \{1,\dots ,n\}$ such that $a_i=b_i=1$.

In [25] (Theorem 9), the following characterization of those functions that aggregate collections of modular metrics was obtained.

Theorem 10.

Let $n\in \mathbb{N}$, and let $G:{[0,\infty ]}^n\to [0,\infty ]$ be a function. The statements below are equivalent:

(1) 

G aggregates every collection of modular metrics (${\left\{d_i\right\}}_{i=1}^n$) into a modular metric.

(2) 

G aggregates every collection of modular quasi-metrics (${\left\{d_i\right\}}_{i=1}^n$) into a modular quasi-metric.

(3) 

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

Moreover, the next useful result was proven in [26] (Lemma 1). Let us recall that a function expressed as $G:{[0,+\infty ]}^n\to [0,+\infty ]$ is sub-additive, provided that $G(a+b)\le G\left(a\right)+G\left(b\right)$ for all $a,b\in {[0,\infty ]}^n$.

Lemma 4.

Let $n\in \mathbb{N}$, and let $G:{[0,\infty ]}^n\to [0,\infty ]$ be a sub-additive 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 $G\left(a\right)=0$.

(2) 

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

In view of the previous results, we can obtain the desired theorem.

Theorem 11.

Let $n\in \mathbb{N}$, let $\mathcal{T}={\left\{T_i\right\}}_{i=1}^n$ be a collection of continuous Archimedean t-norms, and let ${\left\{f_{T_i}\right\}}_{i=1}^n$ be a collection of additive generators of $\mathcal{T}$. If T is a continuous Archimedean t-norm and $F:{[0,1]}^n\to [0,1]$ is a function, then the following assertions are equivalent:

(1) 

F aggregates $\mathcal{T}$-fuzzy partial orders into a T-fuzzy partial order.

(2) 

F aggregates modular $\mathcal{T}$-fuzzy partial orders into a modular T-fuzzy partial order.

(3) 

F aggregates modular $\mathcal{T}$ equalities into a modular T equality.

(4) 

The function expressed as $G:{[0,\infty ]}^n\to [0,f_T\left(0\right)]$, where $G=f_T\circ F\circ (f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)})$, satisfies the following:

(a) 

$G\left(0_n\right)=0$.

(b) 

If $G\left(b\right)=0$ and $b\in {[0,\infty ]}^n$, then $b_i=0$ for some $i\in \{1,\dots ,n\}$.

(c) 

$G\left(c\right)\le G\left(a\right)+G\left(b\right)$ for all $a,b,c\in {[0,\infty ]}^n$ such that $c_i\le a_i+b_i$ for all $i\in \{1,\dots ,n\}$.

(5) 

The function expressed as $G:{[0,\infty ]}^n\to [0,f_T\left(0\right)]$, where $G=f_T\circ F\circ (f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)})$, aggregates every collection of modular metrics (${\left\{d_i\right\}}_{i=1}^n$) into a $f_T\left(0\right)$-bounded modular metric.

(6) 

The function expressed as $G:{[0,\infty ]}^n\to [0,f_T\left(0\right)]$, where $G=f_T\circ F\circ (f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)})$, aggregates every collection of modular quasi-metrics (${\left\{d_i\right\}}_{i=1}^n$) into a $f_T\left(0\right)$-bounded modular quasi-metric.

(7) 

The function expressed as $H:{\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]\to [0,f_T\left(0\right)]$, where $H=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, aggregates every collection of ${\left(f_{T_i}\left(0\right)\right)}_{i=1}^n$-bounded metrics (${\left\{d_i\right\}}_{i=1}^n$) into a $f_T\left(0\right)$-bounded metric.

(8) 

The function expressed as $H:{\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]\to [0,f_T\left(0\right)]$, where $H=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, aggregates every collection of ${\left(f_{T_i}\left(0\right)\right)}_{i=1}^n$-bounded quasi-metrics ${\left\{d_i\right\}}_{i=1}^n$ into a $f_T\left(0\right)$-bounded quasi-metric.

(9) 

The function expressed as $H:{\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]\to [0,f_T\left(0\right)]$, where $H=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, satisfies the following:

(a) 

$H\left(0_n\right)=0$.

(b) 

If $H\left(b\right)=0$ and $b\in {\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]$, then $b_i=0$ for some $i\in \{1,\dots ,n\}$.

(c) 

$H\left(c\right)\le H\left(a\right)+H\left(b\right)$ for all $a,b,c\in {\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]$ such that $c_i\le a_i+b_i$ for all $i\in \{1,\dots ,n\}$.

Proof. 

The equivalence of $\left(1\right)\iff \left(2\right)\iff \left(3\right)$ is warranted by Theorem 9.

$\left(3\right)\Rightarrow \left(4\right)$. Let us start by proving that $G\left(0_n\right)=0$. Provided that $\mathcal{T}$ is a family of continuous Archimedean t-norms and that ${\left\{f_{T_i}\right\}}_{i=1}^n$ is a collection of additive generators of $\mathcal{T}$, then ${\left\{f_{T_i}\right\}}_{i=1}^n$ is a collection of continuous functions and $f_{T_i}^{(-1)}\left(x\right)=f_{T_i}^{-1}\left(\min \{f_{T_i}\left(0\right),x\}\right)$ for all $x\in [0,1]$. As a consequence, $f_{T_i}^{(-1)}\left(0\right)=1$ for all $i\in \{1,\dots ,n\}$ and

$$(f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)})\left(0_n\right)=(f_{T_1}^{(-1)}\left(0\right)\times \dots \times f_{T_n}^{(-1)}\left(0\right))=1_n.$$

According to assertion (4) in Theorem 9, we find that $F\left(1_n\right)=1$. Hence, we obtain $G\left(0_n\right)=f_T\circ F\left(1_n\right)=f_T\left(1\right)=0$.

Next, assume that there exists $b\in {[0,\infty ]}^n$ such that $G\left(b\right)=0$. Then, the following equality is satisfied:

$$f_T^{(-1)}\circ f_T\circ F\circ (f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)})\left(b\right)=f_T^{(-1)}\circ G\left(b\right)=f_T^{(-1)}\left(0\right)=1.$$

Given that $f_T^{(-1)}\circ f_T=id$ and according to assertion (4) in Theorem 9, $F\left(1_n\right)=1$ implies the existence of $i\in \{1,\dots ,n\}$ such that $a_i=1$, and we know that there exists $i\in \{1,\dots ,n\}$ such that $f_{T_i}^{(-1)}\left(b_i\right)=1$. Hence, there exists $i\in \{1,\dots ,n\}$ such that $b_i=0$.

To conclude, consider $a,b,c\in {[0,\infty ]}^n$ such that $a_i+b_i\ge c_i$ for all $i\in \{1,\dots ,n\}$. Recall that any additive generator and its pseudo-inverse are decreasing functions. Then, for all $i\in \{1,\dots ,n\}$, we find that $f_{T_i}^{(-1)}(a_i+b_i)\le f_{T_i}^{(-1)}\left(c_i\right)$. Moreover, Lemma 3 and the definition of an additive generator warrant

$$\begin{array}{c}{T}_i\left(f_{T_i}^{(-1)}\left(a\right),f_{T_i}^{(-1)}\left(b\right)\right)\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =f_{T_i}^{(-1)}\left(f_{T_i}\circ f_{T_i}^{(-1)}\left(a\right)+f_{T_i}\circ f_{T_i}^{(-1)}\left(b\right)\right)=f_{T_i}^{(-1)}(a_i+b_i)\le f_{T_i}^{(-1)}\left(c_i\right).\end{array}$$

Therefore, based on assertion (4) in Theorem 9, it can be deduced that

$$\begin{array}{c}F\left(f_{T_1}^{(-1)}\left(c_1\right)\times \dots \times f_{T_n}^{(-1)}\left(c_n\right)\right)\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \ge T\left(F(f_{T_1}^{(-1)}\left(a_1\right)\times \dots \times f_{T_n}^{(-1)}\left(a_n\right)),F(f_{T_1}^{(-1)}\left(b_1\right)\times \dots \times f_{T_n}^{(-1)}\left(b_n\right))\right).\end{array}$$

Using the definition of an additive generator once more and recalling that $f_T$ is decreasing, we finally get

$$\begin{array}{cc}\hfill G\left(c\right)& =f_T\circ F\left(f_{T_1}^{(-1)}\left(c_1\right)\times \dots \times f_{T_n}^{(-1)}\left(c_n\right)\right)\hfill \\ \hfill & \le f_T\circ F\left(f_{T_1}^{(-1)}\left(a_1\right)\times \dots \times f_{T_n}^{(-1)}\left(a_n\right)\right)+f_T\circ F\left(f_{T_1}^{(-1)}\left(b_1\right)\times \dots \times f_{T_n}^{(-1)}\left(b_n\right)\right)\hfill \\ \hfill & =G\left(a\right)+G\left(b\right)\hfill \end{array}.$$

$\left(4\right)\Rightarrow \left(5\right)$. According to Theorem 10, the function expressed as $G(d_1,\dots ,d_n):X\times X\times ]0,\infty [\to [0,\infty ]$ is a modular metric. It only remains to prove that $G(d_1,\dots ,d_n)$ is $f_T\left(0\right)$-bounded. We find that $G\left(a\right)\le f_T\left(0\right)$ for all $a\in {[0,\infty ]}^n$. So $G(d_1,\dots ,d_n)(x,y,t)\le f_T\left(0\right)$ for all $x,y\in X$ and for all $t>0$. Therefore, we conclude that $G(d_1,\dots ,d_n)$ is a $f_T\left(0\right)$-bounded modular metric on X.

$\left(5\right)\Rightarrow \left(7\right)$. It is obvious.

$\left(7\right)\Rightarrow \left(9\right)$. Let us consider $a,b,c\in {\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]$ such that $a_i+b_i\ge c_i$ for all $i\in \{1,\dots ,n\}$ and the non-empty set expressed as $X=\{x,y,z\}$, with $x,y,z$ all different from each other. Consider a collection of functions (${\left\{d_i\right\}}_{i=1}^n$) with $d_i:X\times X\times ]0,\infty [\to [0,\infty ]$ satisfying the following conditions:

$$d_i(x,x,t)=d_i(y,y,t)=d_i(z,z,t)=0\mathrm{for} \mathrm{all} t>0,$$

$$d_i(x,y,t)=d_i(y,x,t)=\left\{\begin{array}{c}{f}_{T_i}\left(0\right),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{if}0<t\le 1\hfill \\ \min \{f_{T_i}\left(0\right),a_i+b_i\},\hspace{1em}\mathrm{if}1<t\le 2 ,\hfill \\ c_i,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if}2<t\hfill \end{array}\right.$$

$$d_i(x,z,t)=d_i(z,x,t)=\left\{\begin{array}{c}{f}_{T_i}\left(0\right),\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{if}0<t\le 1\hfill \\ a_i,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if}1<t\le 2 ,\hfill \\ \min \{c_i,a_i\},\hspace{1em} \mathrm{if}2<t\hfill \end{array}\right.$$

$$d_i(y,z,t)=d_i(z,y,t)=\left\{\begin{array}{c}{f}_{T_i}\left(0\right),\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{if}0<t\le 1\hfill \\ b_i,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if}1<t\le 2 .\hfill \\ \min \{c_i,b_i\},\hspace{1em} \mathrm{if}2<t\hfill \end{array}\right.$$

Let us prove that ${\left\{d_i\right\}}_{i=1}^n$ is a collection of ${\left(f_{T_i}\left(0\right)\right)}_{i=1}^n$-bounded modular metrics. Notice that for all $u,v\in X$ with $u\ne v$, there exists $t>0$ such that $d_i(u,v,t)\ne 0$ (the condition of separation of points is satisfied). Additionally, by definition, each $d_i$ is symmetric and fulfills $d_i(u,u,t)=0$ for all $u\in X$ and for all $t>0$. Hence, it is enough to prove that each $d_i$ satisfies the modular triangular inequality. Observe that, with a fixed $u,v\in X$, the function expressed as $d_i(u,v,·):]0,\infty [\to [0,\infty ]$ is decreasing for all $i\in \{1,\dots ,n\}$. Then, for all $u,v\in X$ and for all $t,s>0$, we get

$$d_i(u,u,t+s)=0\le d_i(u,v,t)+d_i(v,u,s),$$

$$d_i(u,v,t+s)\le d_i(u,v,t)=d_i(u,v,t)+d_i(v,v,s)$$

and

$$d_i(u,v,t+s)\le d_i(u,v,s)=d_i(u,u,t)+d_i(u,v,s).$$

Therefore, we can focus on the case where $u,v,w\in X$ are all different from each other. Let us consider all possible scenarios:

(i)

$\min \{t,s\}\le 1$: In such a case, $d_i(u,v,t)+d_i(v,w,s)\ge f_{T_i}\left(0\right)\ge d_i(u,w,t+s)$.

(ii)

$1<\min \{t,s\}$ and $\max \{t,s\}\le 2$: In such a case, $d_i(u,v,t)+d_i(v,w,s)\ge a_i+b_i\ge d_i(u,w,t+s)$.

(iii)

$\max \{t,s\}>2$: In such a case, $d_i(u,v,t)+d_i(v,w,s)\ge \min \{c_i,a_i\}+\min \{c_i,b_i\}\ge c_i\ge d_i(u,w,t+s)$.

Now that we have proven that ${\left\{d_i\right\}}_{i=1}^n$ is a collection of ${\left(f_{T_i}\left(0\right)\right)}_{i=1}^n$-bounded modular metrics, let us begin by proving condition $\left(a\right)$. By hypothesis, we know that $H(d_1,\dots ,d_n)$ is a $f_T\left(0\right)$-bounded modular metric. Then, $0=H(d_1,\dots ,d_n)(x,x,1)=H\left(0_n\right)$.

Next, we show that condition $\left(c\right)$ is fulfilled. Indeed, assume that $a,b,c\in {\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]$ such that $c_i\le a_i+b_i$ for all $i\in \{1,\dots ,n\}$. Then, we find that $d_i(x,y,4)=c_i,d_i(x,z,2)=a_i,d_i(z,y,2)=b_i$ for all $i\in \{1,\dots ,n\}$. Then,

$$\begin{array}{cc}\hfill H\left(c\right)& =H(d_1(x,y,4),\dots ,d_n(x,y,4))\hfill \\ \hfill & \le H(d_1(x,z,2),\dots ,d_n(x,z,2))+H(d_1(z,y,2),\dots ,d_n(z,y,2))=H\left(a\right)+H\left(b\right).\hfill \end{array}$$

Finally, we prove that condition (b) holds. Assume that there exists $b\in {[0,\infty ]}^n$ such that $H\left(b\right)=0$ and $b_i\ne 0$ for all $i\in \{1,\dots ,n\}$. Now, consider the collection of ${\left(f_{T_i}\left(0\right)\right)}_{i=1}^n$-bounded modular metrics (${\left\{d_i\right\}}_{i=1}^n$) on $\{x,y\}$ ($x\ne y$), given as follows: $d_i(x,y,t)=b_i$ and $d_i(x,x,t)=d_i(y,y,t)=0$ for all $t>0$ and for all $i\in \{1,\dots ,n\}$. Then, $H(d_1,\dots ,d_n)(x,y,t)=H\left(b\right)=0$ for all $t>0$. Since $H(d_1,\dots ,d_n)$ is a $f_T\left(0\right)$-bounded modular metric, we conclude that $x=y$, which contradicts the fact that $x\ne y$.

$\left(9\right)\Rightarrow \left(3\right)$. Assume that ${\left\{E_i\right\}}_{i=1}^n$ is a family of modular $\mathcal{T}$ equalities on a non-empty set (X). Let us start by proving that $F(E_1,\dots ,E_n)$ is reflexive. Based on the reflexivity of $E_i$, we know that $f_T\left(E_i(x,x,t)\right)=0$ for all $x\in X$ and for all $t>0$. Then,

$$\begin{array}{cc}\hfill 0=H\left(0_n\right)& =f_T\circ F\circ \left(f_{T_1}^{-1}\left(f_{T_1}\left(E_1(x,x,t)\right)\right)\times \dots \times f_{T_n}^{-1}(f_{T_n}\left(E_n(x,x,t)\right)\right)\hfill \\ \hfill & =f_T\circ F(E_1(x,x,t),\dots ,E_n(x,x,t))\hfill \end{array}.$$

Since $f_T$ is strictly decreasing, we find that $F(E_1(x,x,t),\dots ,E_n(x,x,t))=1$ for all $x\in X$ and for all $t>0$.

Next, assume that there exist $x,y\in X$ such that $F(E_1,\dots ,E_n)(x,y,t)=1$ for all $t>0$. We know that $f_{T_i}\left(E_i(x,y,t)\right)\in [0,f_{T_i}\left(0\right)]$ for all $i\in \{1,\dots ,n\}$. Thus, we obtain

$$\begin{array}{cc}\hfill & H(f_{T_1}\left(E_1(x,y,t)\right),\dots ,f_{T_n}\left(E_n(x,y,t)\right))\hfill \\ \hfill & =f_T\circ F(f_{T_1}^{-1}\left(f_{T_1}\left(E_1(x,y,t)\right)\right),\dots ,f_{T_n}^{-1}\left(f_{T_n}\left(E_n(x,y,t)\right)\right))\hfill \\ \hfill & =f_T\circ F\left(E_1(x,y,t)\right),\dots ,E_n(x,y,t)))=f_T\left(1\right)=0\hfill \end{array}$$

for all $t>0$. It follows that there exists $i_0\in \{1,\dots ,n\}$ such that $f_{T_{i_0}}\left(E_{i_0}(x,y,t)\right)=0$ for all $t>0$. So $E_{i_0}(x,y,t)=0$ for all $t>0$. Consequently, $x=y$.

Finally, we prove that $F(E_1,\dots ,E_n)$ satisfies the T-transitivity condition. According to assertion (7) in Theorem 5, $f_{T_i}\left(E_i(x,y,t)\right)+f_{T_i}(E_i(y,z,s)\ge f_{T_i}\left(E_i(x,z,t+s)\right)$ for all $i\in \{1,\dots ,n\}$, for all $x,y,z\in X$ and for all $t,s>0$, since $d_{E,f_{T_I}}(x,y,t)=f_{T_i}\left(E(x,y,t)\right)$ is a modular metric on X for all $i\in \{1,\dots ,n\}$. According to condition $\left(c\right)$, we also have

$$\begin{array}{cc}\hfill & H(f_{T_1}\left(E_1(x,y,t)\right),\dots ,f_{T_n}\left(E_n(x,y,t)\right))+H(f_{T_1}\left(E_1(y,z,s)\right),\dots ,f_{T_n}\left(E_n(y,z,s)\right))\hfill \\ \hfill & \ge H(f_{T_1}\left(E_1(x,z,t+s)\right),\dots ,f_{T_n}(E_n(x,z,t+s))).\hfill \end{array}$$

Since $f_T^{(-1)}$ is a decreasing function, we get

$$\begin{array}{cc}\hfill & f_T^{(-1)}\left(H(f_{T_1}\left(E_1(x,y,t)\right),\dots ,f_{T_n}\left(E_n(x,y,t)\right))+H(f_{T_1}\left(E_1(y,z,s)\right),\dots ,f_{T_n}\left(E_n(y,z,s)\right))\right)\hfill \\ \hfill & \le f_T^{(-1)}\left(H(f_{T_1}\left(E_1(x,z,t+s)\right),\dots ,f_{T_n}(E_n(x,z,t+s)))\right).\hfill \end{array}$$

Furthermore

$$f_T^{(-1)}\left(H(f_{T_1}\left(E_1(x,z,t+s)\right),\dots ,f_{T_n}\left(E_n(x,z,t+s)\right)\right)=F(E_1,\dots ,E_n)(x,z,t+s)$$

and

$$\begin{array}{cc}\hfill & f_T^{(-1)}\left(H(f_{T_1}\left(E_1(x,y,t)\right),\dots ,f_{T_n}\left(E_n(x,y,t)\right))+H(f_{T_1}\left(E_1(y,z,s)\right),\dots ,f_{T_n}(E_n(y,z,s)))\right)\hfill \\ \hfill & =T(F(E_1,\dots ,E_n)(x,y,t),F(E_1,\dots ,E_n)(y,z,s)).\hfill \end{array}$$

Therefore, we can conclude that

$$T(F(E_1,\dots ,E_n)(x,y,t),F(E_1,\dots ,E_n)(y,z,s))\le F(E_1,\dots ,E_n)(x,z,t+s).$$

Therefore, $F(E_1,\dots ,E_n)$ is a T equality on X.

$\left(6\right)\Rightarrow \left(5\right)$. It is evident.

$\left(5\right)\Rightarrow \left(6\right)$. We know that the equivalence of $\left(5\right)\iff \left(4\right)$ is true. Now, assume that ${\left\{d_i\right\}}_{i=1}^n$ is a collection of modular quasi-metrics on a non-empty set (X). Then, according to Theorem 10, the function expressed as $G(d_1,\dots ,d_n):X\times X\times ]0,\infty [\to [0,\infty ]$ is a modular quasi-metric. Finally, we find that $G\left(a\right)\le f_T\left(0\right)$ for all $a\in {[0,\infty ]}^n$. So $G(d_1,\dots ,d_n)(x,y,t)\le f_T\left(0\right)$ for all $x,y\in X$ and for all $t>0$. Therefore, we conclude that $G(d_1,\dots ,d_n)$ is a $f_T\left(0\right)$-bounded modular quasi-metric on X.

$\left(8\right)\Rightarrow \left(7\right)$. It is evident.

$\left(7\right)\Rightarrow \left(8\right)$. Arguments simiar to those applied to the proof of the implication of $\left(5\right)\Rightarrow \left(6\right)$ remain valid in this case.

 □

Recall that $f_T\left(0\right)=\infty$ in the particular case in which T is a strict continuous Archimedean t-norm. Taking this property into account, the following corollary can be deduced.

Corollary 2.

Let $n\in \mathbb{N}$, let $\mathcal{T}={\left\{T_i\right\}}_{i=1}^n$ be a collection of strict continuous Archimedean t-norms, and let ${\left\{f_{T_i}\right\}}_{i=1}^n$ be a collection of additive generators of $\mathcal{T}$. If T is a strict continuous Archimedean t-norm and $F:{[0,1]}^n\to [0,1]$ is a function, then the following assertions are equivalent:

(1) 

F aggregates $\mathcal{T}$-fuzzy partial orders into a T-fuzzy partial order.

(2) 

F aggregates modular $\mathcal{T}$-fuzzy partial orders into a modular T-fuzzy partial order.

(3) 

F aggregates modular $\mathcal{T}$ equalities into a modular T equality.

(4) 

The function expressed as $G:{[0,\infty ]}^n\to [0,\infty ]$, where $G=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, satisfies the following:

(a) 

$G\left(0_n\right)=0$.

(b) 

If $G\left(b\right)=0$ and $b\in {[0,\infty ]}^n$, then $b_i=0$ for some $i\in \{1,\dots ,n\}$.

(c) 

$G\left(c\right)\le G\left(a\right)+G\left(b\right)$ for all $a,b,c\in {[0,\infty ]}^n$ such that $c_i\le a_i+b_i$ for all $i\in \{1,\dots ,n\}$.

(5) 

The function expressed as $G:{[0,\infty ]}^n\to [0,\infty ]$, where $G=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, aggregates every collection of modular metrics (${\left\{d_i\right\}}_{i=1}^n$) into a modular metric.

(6) 

The function expressed as $G:{[0,\infty ]}^n\to [0,\infty ]$, where $G=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, aggregates every collection of modular quasi-metrics (${\left\{d_i\right\}}_{i=1}^n$) into a modular quasi-metric.

4.3. Aggregation of Modular Relaxed Indistinguishability Relations and Modular Relaxed Pseudo-Metrics

In this subsection, we will approach the aggregation problem for modular relaxed indistinguishability relations and modular relaxed pseudo-metrics. In order to do so, we recall the following characterization introduced in [27] (Theorem 5).

Theorem 12.

Let $n\in \mathbb{N}$, let $\mathcal{T}={\left\{T_i\right\}}_{i=1}^n$ be a collection of t-norms and let $F:{[0,1]}^n\to [0,1]$ be a function; then, the following assertions are equivalent:

(1) 

F aggregates modular $\mathcal{T}$-relaxed indistinguishability relations into a modular T-relaxed indistinguishability relation.

(2) 

$T\left(F\right(a),F(b\left)\right)\le F\left(c\right)$ whenever $a,b,c\in {[0,1]}^n$ such that $T_i(a_i,b_i)\le c_i$ for all $i\in \{1,\dots ,n\}$.

Moreover, the result presented below, which can be found in ([28] Theorem 3), will be useful later on.

Theorem 13.

Let $n\in \mathbb{N}$. If $G:{[0,\infty ]}^n\to [0,\infty ]$ is a function, then the following assertions are equivalent:

(1) 

G aggregates every collection of modular relaxed pseudo-metrics (${\left\{d_i\right\}}_{i=1}^n$) into a modular relaxed pseudo-metric.

(2) 

$G\left(c\right)\le G\left(a\right)+G\left(b\right)$ for all $a,b,c\in {[0,\infty ]}^n$, with $c_i\le a_i+b_i$ for all $i\in \{1,\dots ,n\}$.

Next, we present the characterization in the case at hand.

Theorem 14.

Let $n\in \mathbb{N}$, let $\mathcal{T}={\left\{T_i\right\}}_{i=1}^n$ be a collection of continuous Archimedean t-norms, and let ${\left\{f_{T_i}\right\}}_{i=1}^n$ be a collection of additive generators of $\mathcal{T}$. If T is a continuous Archimedean t-norm and $F:{[0,1]}^n\to [0,1]$ is a function, then the following assertions are equivalent:

(1) 

F aggregates modular $\mathcal{T}$-relaxed indistinguishability relations into a modular T-relaxed indistinguishability relation.

(2) 

The function expressed as $G:{[0,\infty ]}^n\to [0,f_T\left(0\right)]$, where $G=f_T\circ F\circ (f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)})$, satisfies $G\left(c\right)\le G\left(a\right)+G\left(b\right)$ for all $a,b,c\in {[0,\infty ]}^n$ such that $c_i\le a_i+b_i$ for all $i\in \{1,\dots ,n\}$.

(3) 

The function expressed as $G:{[0,\infty ]}^n\to [0,f_T\left(0\right)]$, where $G=f_T\circ F\circ (f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)})$, aggregates every collection of modular relaxed pseudo-metrics (${\left\{d_i\right\}}_{i=1}^n$) into a $f_T\left(0\right)$-bounded modular relaxed pseudo-metric.

(4) 

The function expressed as $H:{\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]\to [0,f_T\left(0\right)]$, where $H=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, aggregates every collection of ${\left(f_{T_i}\left(0\right)\right)}_{i=1}^n$-bounded modular relaxed pseudo-metrics (${\left\{d_i\right\}}_{i=1}^n$) into a $f_T\left(0\right)$-bounded modular relaxed pseudo-metric.

(5) 

The function expressed as $H:{\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]\to [0,f_T\left(0\right)]$, where $H=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, satisfies $H\left(c\right)\le H\left(a\right)+H\left(b\right)$ for all $a,b,c\in {\prod }_{i=1}^n[0,f_{T_i}\left(0\right)]$ such that $c_i\le a_i+b_i$ for all $i\in \{1,\dots ,n\}$.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$. The arguments applied in the proof of the implication of $\left(3\right)\Rightarrow \left(4\right)$ in Theorem 11 are valid in this case but now making use of Theorem 12.

$\left(2\right)\Rightarrow \left(3\right)$. The same reasoning applied to the proof of the implication of $\left(4\right)\Rightarrow \left(5\right)$ in Theorem 11 remains valid in this case but now making use of Theorem 12.

$\left(3\right)\Rightarrow \left(4\right)$. It is obvious.

$\left(4\right)\Rightarrow \left(5\right)$. Provided that the collection of ${\left(f_{T_i}\left(0\right)\right)}_{i=1}^n$-bounded modular metrics given in the proof of the implication of $\left(7\right)\Rightarrow \left(9\right)$ in Theorem 11 is, in particular, a collection of modular relaxed pseudo-metrics, reasoning to analogous to that exposed in the proof of the aforementioned implication can be used to prove the desired statement.

$\left(5\right)\Rightarrow \left(1\right)$. This follows from the same argument applied to the proof of the implication of $\left(9\right)\Rightarrow \left(3\right)$ in Theorem 11 but now making use of assertion (2) in Theorem 5. □

Once more, based on the fact that $f_T\left(0\right)=\infty$ whenever T is a strict continuous Archimedean t-norm, the following corollary can be deduced.

Corollary 3.

Let $n\in \mathbb{N}$, let $\mathcal{T}={\left\{T_i\right\}}_{i=1}^n$ be a collection of strict continuous Archimedean t-norms, and let ${\left\{f_{T_i}\right\}}_{i=1}^n$ be a collection of additive generators of $\mathcal{T}$. If T is a strict continuous Archimedean t-norm and $F:{[0,1]}^n\to [0,1]$ is a function, then the following assertions are equivalent:

(1) 

F aggregates modular $\mathcal{T}$-relaxed indistinguishability relations into a modular T-relaxed indistinguishability relation.

(2) 

The function expressed as $G:{[0,\infty ]}^n\to [0,\infty ]$, where $G=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, satisfies $G\left(c\right)\le G\left(a\right)+G\left(b\right)$ for all $a,b,c\in {[0,\infty ]}^n$ such that $c_i\le a_i+b_i$ for all $i\in \{1,\dots ,n\}$.

(3) 

The function expressed as $G:{[0,\infty ]}^n\to [0,\infty ]$, where $G=f_T\circ F\circ (f_{T_1}^{-1}\times \dots \times f_{T_n}^{-1})$, aggregates every collection of modular relaxed pseudo-metrics (${\left\{d_i\right\}}_{i=1}^n$) into a modular relaxed pseudo-metric.

4.4. Relationship Between the Aggregation Problems

With the purpose of providing a quick guide to the results found so far, in Table 3, we present a summary of the different types of the (modular) generalized metrics that can be aggregated by the G function, depending on the family of (modular) T-transitive relations that can be merged by the F function. Observe that Table 3 also works to summarize the different types of (modular) T-transitive relations that can be aggregated by F in terms of each one of the families of (modular) generalized metrics that can be merged by G, since the fact that $G=f_T\circ F\circ \left(f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)}\right)$ implies that $F=f_T^{(-1)}\circ G\circ (f_{T_1}\times \dots \times f_{T_n})$.

Table 3. Summary of the generalized metrics that can be aggregated by the H function, depending on the family of T-transitive relations that can be merged by F. More precisely, each row indicates which families of (modular) generalized metrics can be aggregated by H when considering that F can merge the type of (modular) T-transitive relation from that row. Each column indicates which families of (modular) T-transitive relations can be aggregated by F when considering that H can merge the type of (modular) generalized metric from that column. Whenever the possibility exists, it is represented by “Y”, and we use “N” if it is not possible. Note that relaxed indistinguishability relations are abbreviated as “relaxed indist.”.

		Generalized Metrics Aggregated by G = fT ∘ F ∘ (${\mathit{f}}_{{\mathit{T}}_1}^{\mathbf{(}-1\mathbf{)}}\times \dots \times {\mathit{f}}_{{\mathit{T}}_{\mathit{n}}}^{\mathbf{(}-1\mathbf{)}}$)
Fuzzy Relation Aggregated by $\mathit{F}$		Pseudo- Metrics	Metrics	Quasi- Pseudo- Metrics	Quasi- Metrics	Relaxed Pseudo- Metrics	Modular Pseudo- Metrics	Modular Metrics	Modular Quasi- Pseudo- Metrics	Modular Quasi- Metrics	Modular Relaxed Pseudo- Metrics
Non-modular	indistinguishabilities	Y	N	N	N	Y	N	N	N	N	N
	fuzzy equalities	Y	Y	N	N	Y	N	N	N	N	N
	fuzzy pre-orders	Y	N	Y	N	Y	Y	N	Y	N	Y
	fuzzy partial-orders	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y
	relaxed indist.	N	N	N	N	Y	N	N	N	N	N
Modular	indistinguishabilities	Y	N	Y	N	Y	Y	N	Y	N	Y
	fuzzy equalities	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y
	fuzzy pre-orders	Y	N	Y	N	Y	Y	N	Y	N	Y
	fuzzy partial-orders	Y	Y	Y	Y	Y	Y	Y	Y	Y	Y
	relaxed indist.	N	N	N	N	Y	N	N	N	N	Y

5. Some Illustrative Instances

Next, we will present some instances of how functions that merge collections of modular generalized metrics can be obtained from functions that merge collections of modular transitive fuzzy relations and vice versa. For simplicity reasons, we will consider that all the t-norms belonging to collection $\mathcal{T}$ are the same for our examples. It must be pointed out that this is not mandatory and that all of them could be different from each other (as long as they are continuous and Archimedean t-norms).

In the following examples, we will consider, on the one hand, the product t-norm ($T_P$), which is strictly continuous and Archimedean. On the other hand, we will use the Łukasiewicz t-norm ($T_L$), which is nilpotent, continuous and Archimedean. Recall that for all $a,b\in [0,1]$, we have $T_P(a,b)=a·b$ and $T_L(a,b)=\max \{a+b-1,0\}$. Additionally, according to [32], $f_{T_P}\left(a\right)=-\ln \left(a\right)$ and $f_{T_L}\left(a\right)=1-a$ for all $a\in [0,1]$. Moreover, remember that we have

$$f_{T_P}^{(-1)}=\exp \{-x\} \mathrm{for} \mathrm{all} a\in [0,\infty ]\mathrm{and}{f}_{T_L}^{(-1)}\left(x\right)=\left\{\begin{array}{cc}1-x,\hfill & \mathrm{if}0\le x\le 1\hfill \\ 0,\hfill & \mathrm{if}1<x\hfill \end{array}\right..$$

Let us begin by showing, in Example 1, how to get a function that aggregates modular metrics into a modular metric by means of a function that merges modular fuzzy equalities into a modular fuzzy equality.

Example 1.

Let $x\in {[0,\infty ]}^n$, and let $F:{[0,1]}^n\to [0,1]$ be the function given by $F\left(a\right)=\min \{a_1,\dots ,a_n\}$ for all $a\in {[0,1]}^n$. It is not hard to check that F aggregates modular $\mathcal{T}$ equalities into a modular T equality in all cases considered below. Next, we provide the expression of $G=f_T\circ F\circ (f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)})$ for the following different scenarios:

• $T=T_L$ and $T_i=T_L$ for all $i\in \{1,\dots ,n\}$. In such a case, the function expressed as $G:{[0,\infty ]}^n\to [0,1]$ obtained from the previously detailed construction is defined as follows:

  $$G\left(x\right)=f_{T_L}\circ \underset{i\in \{1,\dots ,n\}}{\min }\left\{f_{T_L}^{(-1)}\left(x_i\right)\right\}.$$
  Since $f_{T_L}^{(-1)}$ is a decreasing function, we obtain

  $$\underset{i\in \{1,\dots ,n\}}{\min }\left\{f_{T_L}^{(-1)}\left(x_i\right)\right\}=f_{T_L}^{(-1)}\left(\underset{i\in \{1,\dots ,n\}}{\max }\left\{x_i\right\}\right).$$
  Therefore, the G function is expressed as follows:

  $$G\left(x\right)=\left\{\begin{array}{cc}{\max }_{i\in \{1,\dots ,n\}}\left\{x_i\right\},\hfill & if{\max }_{i\in \{1,\dots ,n\}}\left\{x_i\right\}\le 1\hfill \\ 1,\hfill & otherwise\hfill \end{array}\right..$$
• $T=T_P$ and $T_i=T_P$ for all $i\in \{1,\dots ,n\}$. Using analogous reasoning and taking into account that $T_P$ is strict (and thus $f_{T_P}\circ f_{T_P}^{(-1)}=id.$), we find that the function ($G:{[0,\infty ]}^n\to [0,\infty ]$) obtained from the aforementioned construction is defined as follows:

  $$G\left(x\right)=f_{T_P}\circ f_{T_P}^{(-1)}\left(\underset{i\in \{1,\dots ,n\}}{\max }\left\{x_i\right\}\right)=\underset{i\in \{1,\dots ,n\}}{\max }\left\{x_i\right\}.$$
  Observe that $G\left(x\right)={\max }_{i\in \{1,\dots ,n\}}\left\{x_i\right\}$ whenever all the involved t-norms are the same strict, continuous Archimedean t-norm.
• $T=T_L$ and $T_i=T_P$ for all $i\in \{1,\dots ,n\}$. Using analogous reasoning, again, we find that the function expressed as $G=f_{T_L}\circ F\circ (f_{T_P}^{-1}\times \dots \times f_{T_P}^{-1}):{[0,\infty ]}^n\to [0,1]$ is given as follows:

  $$G\left(x\right)=f_{T_L}\circ f_{T_P}^{(-1)}\left(\underset{i\in \{1,\dots ,n\}}{\max }\left\{x_i\right\}\right)=1-\exp \{-\underset{i\in \{1,\dots ,n\}}{\max }\left\{x_i\right\}\}.$$

It is worth noting that, under the conditions of Theorem 11, if $G=f_T\circ F\circ (f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)})$, then $F=f_T^{(-1)}\circ G\circ (f_{T_1}\times \dots \times f_{T_n})$. Now, we are ready for Example 2, in which we get a function that aggregates modular fuzzy equalities into a modular fuzzy equality by means of a function that merges modular metrics into a modular metric.

Example 2.

Let $\mathcal{T}={\left\{T_i\right\}}_{i=1}^n$ be a collection of continuous Archimedean t-norms, and let ${\left\{f_{T_i}\right\}}_{i=1}^n$ be a collection of additive generators of $\mathcal{T}$. Consider a continuous Archimedean t-norm (T), and let $f_T$ be an additive generator of T. Let $w_i\in [0,\infty [$ for all $i\in \{1,\dots ,n\}$. Consider the function expressed as $G:{[0,\infty ]}^n\to [0,f_T\left(0\right)]$ and defined by

$$G\left(x\right)=\min \left\{{\left(\sum _{i=1}^n{\left(w_i{x}_i\right)}^p\right)}^{{\displaystyle \frac{1}{p}}},f_T\left(0\right)\right\}$$

for all $x\in {[0,\infty ]}^n$, with $p\in [1,\infty [$. It is not hard to prove that G fulfills all conditions in assertion (3) in Theorem 10. So it aggregates every collection of modular metrics (${\left\{d_i\right\}}_{i=1}^n$) into a modular metric. Indeed, by construction, it aggregates every collection of modular metrics (${\left\{d_i\right\}}_{i=1}^n$) into a $f_T\left(0\right)$-bounded modular metric.

Next we show what the function expressed as $F:{[0,1]}^n\to [0,1]$ aggregating modular $\mathcal{T}$ equalities into a modular T equality could give us the G function through the construction expressed as $G=f_T\circ F\circ (f_{T_1}^{(-1)}\times \dots \times f_{T_n}^{(-1)})$ provided in Theorem 11 when the considered t-norms are those contemplated in the exposed cases below.

Assume that $a\in {[0,1]}^n$. We provide the expression of $F=f_T^{(-1)}\circ G\circ (f_{T_1}\times \dots \times f_{T_n})$ when the considered t-norms are the following:

• $T=T_L$ and $T_i=T_L$ for all $i\in \{1,\dots ,n\}$. Let us denote $\alpha \left(a\right)=G\circ (f_{T_L}\left(a_1\right)\times \dots \times f_{T_L}\left(a_n\right))$. Hence,

  $$\alpha \left(a\right)=\min \left\{{\left(\sum _{i=1}^n{\left(w_i(1-a_i)\right)}^p\right)}^{{\displaystyle \frac{1}{p}}},1\right\}$$

  and

  $$F\left(a\right)=f_{T_L}^{(-1)}\left(\alpha \left(a\right)\right)=1-\min \left\{{\left(\sum _{i=1}^n{\left(w_i(1-a_i)\right)}^p\right)}^{{\displaystyle \frac{1}{p}}},1\right\}.$$
  In the particular case where $w_i\in [0,1]$ for all $i\in \{1,\dots ,n\}$ and ${\sum }_{i=1}^n{w}_i=1$, it is direct to prove that $0\le \alpha \left(a\right)\le 1$ and, thus, that

  $$F\left(a\right)=1-\alpha \left(a\right)=1-{\left(\sum _{i=1}^n{\left(w_i(1-a_i)\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}.$$
• $T=T_P$ and $T_i=T_P$ for all $i\in \{1,\dots ,n\}$. Taking $\alpha \left(a\right)=G\circ (f_{T_P}\left(a_1\right)\times \dots \times f_{T_P}\left(a_n\right))$, we get

  $$\alpha \left(a\right)={\left(\sum _{i=1}^n{(-w_i\ln \left(a_i\right))}^p\right)}^{{\displaystyle \frac{1}{p}}}.$$
  Thus, we obtain the following:

  $$\begin{array}{cc}\hfill F\left(a\right)& =f_{T_P}^{(-1)}\left(\alpha \left(a\right)\right)=\exp \{-{\left(\sum _{i=1}^n{(-w_i\ln \left(a_i\right))}^p\right)}^{{\displaystyle \frac{1}{p}}}\}.\hfill \end{array}$$
• $T=T_P$ and $T_i=T_L$ for all $i\in \{1,\dots ,n\}$. We have already seen that whenever $T_i=T_L$, for all $i\in \{1,\dots ,n\}$, we get

  $$\alpha \left(a\right)={\left(\sum _{i=1}^n{\left(w_i(1-a_i)\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}.$$
  Then,

  $$F\left(a\right)=f_{T_P}^{(-1)}\left(\alpha \left(a\right)\right)=\exp \left\{-{\left(\sum _{i=1}^n{\left(w_i(1-a_i)\right)}^p\right)}^{{\displaystyle \frac{1}{p}}}\right\}.$$

Notice that the G and F functions obtained in Examples 1 and 2 are actually particular instances of functions aggregating any class of modular T-transitive relations and any type of modular generalized metrics (or even any of the possible combinations presented in Table 3).

6. Conclusions and Future Work

Many authors have studied the possibility of constructing generalized metrics from many types of transitive fuzzy relations via the use of an additive generator of the involved t-norm. Moreover, a method for generating many classes of transitive fuzzy relations from generalized metrics but now by means of the use of the pseudo-inverse of an additive generator of the t-norm has been developed. Taking advantage of this duality relationship, the possibility of constructing functions that merge collections of indistinguishability relations (equalities) by means of those that aggregate specific collections of pseudo-metrics (metrics) was explored in [14]. In particular, a specific method to construct them was provided by means of the use of additive generators of the involved t-norms and their associated pseudo-inverses.

Motivated, on the one hand, by the method exposed in [14] and, on the other hand, by the practical advantages offered by working within the modular framework, we have proven that such a method can be extended to this context. Hence, we have extend the aforementioned technique to cases in which the pairs formed by many types of modular T-transitive relations and their dual modular generalized metrics are considered—in other words, pairs that are obtained by relaxing the reciprocal conditions from the definitions of modular equality and modular metrics. More precisely, we have proven the following results:

• The known duality between modular equalities and modular metrics (and between modular indistinguishability relations and modular pseudo-metrics) still exists when reciprocal conditions are relaxed for both functions. More precisely, we have proven that this duality exists between modular fuzzy pre-orders and modular quasi-pseudo-metrics, between modular fuzzy partial orders and modular quasi-metrics, and between modular relaxed indistinguishability relations and modular relaxed pseudo-metrics.
• Functions that merge the aforementioned types of modular generalized metrics can be used to obtain functions that merge the dual type of modular T-transitive relations by means of a specific construction based on the use of additive generators of t-norms and their associated pseudo-inverses.
• Aggregation functions for some families of T-transitive relations can also be obtained from functions that aggregate generalized metrics different from their dual counterpart. The precise families of functions that can be used to do so are presented in Table 3.
• Some instances of how the construction method works are provided.

As for future work, taking advantage of the duality relationship, we plan to improve existing applications based on modular T-transitive relations or modular generalized metrics. In particular, we plan to explore whether modular generalized metrics that better suit certain situations in artificial intelligence or decision making can be obtained by taking advantage of the similarity interpretation of modular T-transitive relations and vice versa.