Generalizing a Construction of Non-Strong Fuzzy Metrics from Metrics and Studying Their Induced Topology

Abstract

The problem of obtaining new examples of fuzzy metrics is of interest, as this type of fuzzy measurement has been proven to be useful in engineering applications. In this context, different works have addressed the problem of deriving fuzzy metrics from classical ones. This paper is devoted to generalizing a construction of non-strong fuzzy metrics from metrics already provided in the literature, both for continuous Archimedean t-norms and for the minimum t-norm. Moreover, we explore the conditions under which one adapts this generalized method to obtain fuzzy metrics in the sense of George and Veeramani. In addition, we investigate the connection between the topology associated with the fuzzy metric constructed via these procedures and that determined by the metric. Several examples are provided to support and illustrate our findings.

1. Introduction

Fuzzy metric spaces, as considered in this paper, are based on the concept introduced by Kramosil and Michalek in [1]. Since its introduction, this concept has undergone various modifications. For instance, George and Veeramani proposed strengthening some of the axioms for defining fuzzy metric spaces, as defined by Kramosil and Michalek, in order to obtain a Hausdorff topology induced by them. However, the same arguments via which George and Veeramani introduced the aforementioned Hausdorff topology remain valid for the notion due to Kramosil and Michalek. Furthermore, these arguments were applied even more broadly to the more general notion of fuzzy metric used throughout this paper (see Definition 2), which represents a minor refinement of the concept proposed by Kramosil and Michalek. Different authors have studied the topology induced by fuzzy metrics, specifically in the sense of George and Veeramani. However, most of the topological results obtained for fuzzy metric spaces in that sense are also recovered by the other approaches. Gregori and Romaguera [2] proved a notable result in this direction, and demonstrated that the class of metrizable topological spaces coincides with that of fuzzy metrizable ones—that is, topological spaces for which there exists a fuzzy metric space inducing the same topology. So, from a topological point of view, classical metrics and fuzzy metrics are equivalent. Nonetheless, they exhibit significant differences when pure metric issues are considered. For example, within the approach of George and Veeramani, there are fuzzy metric spaces that fail to admit a completion (see [3]). Additionally, fixed point theory in fuzzy metric spaces has shown substantial differences compared to its classical counterpart (see, for instance, [4,5]). Indeed, fixed point theory in fuzzy metric spaces remains an active field of research today (see, for instance, [6,7,8,9,10,11]).

The most substantial difference between classical metrics and fuzzy metrics lies in the fact that the latter include a parameter t in their definition. This feature has made fuzzy metrics interesting both from a theoretical point of view and in terms of their applicability. On the one hand, the parameter t has made it possible to introduce several concepts in fuzzy metric spaces that have no counterpart in the classical setting, and hence, this opens new lines of research in this field. Among these are the notions of strong fuzzy metric [12], p-convergence [13], and standard Cauchy sequence [14]. On the other hand, the t-parameter has provided fuzzy metrics with greater adaptability for addressing various real-world problems in computer science compared to classical metrics. Indeed, fuzzy metrics have been successfully applied in multiple aspects such as image processing [15,16,17,18], perceptual color difference [19], and clustering [20]. Nevertheless, in such applications, the examples of fuzzy metrics most commonly used are those introduced in the early stages of the development of fuzzy metric space theory. This issue was addressed in [21], where two novel approaches to deriving fuzzy metrics from classical metrics were proposed, allowing the creation of several new examples of fuzzy metrics. Moreover, to establish a duality relationship between classical and fuzzy metrics, ref. [22] proposed a method for deriving a non-strong fuzzy metric from a classical one. It should be noted that, prior to this, examples of non-strong fuzzy metric spaces were limited. Indeed, in [23], the open question of finding non-strong fuzzy metrics defined for t-norms other than the minimum was posed. This question was answered in [24] by providing two examples, which were ad hoc constructions developed specifically for this purpose.

The aim of this paper is to generalize the construction of (non-strong) fuzzy metrics from metrics provided in [22] following the ideas underlying the methods introduced in [21]. In this direction, we include a calibration function to manage the parameter t. Furthermore, we adapt the construction in [22] to generate fuzzy metrics under the minimum t-norm. It is worth noting that we identified an error in the proof of the theorem that provides a method for constructing fuzzy metrics from classical metrics for the minimum t-norm, as established in [21]. This error is detailed and corrected in the present work. Concerning the generalizations of the construction in [22], we first establish the necessary conditions for obtaining non-strong fuzzy (pseudo-)metrics for continuous Archimedean t-norms, as well as the additional requirements needed to obtain George and Veeramani fuzzy metrics. We also prove that our new method preserves topologies; that is, the topology associated with the obtained fuzzy metric is identical to the topology induced by the classical metric used in its construction. Furthermore, we show that, when the minimum t-norm is considered, a unique fuzzy pseudo-metric can be obtained. These new findings establish a novel method for obtaining new examples of non-strong fuzzy metrics, which provide a broader set of measurement tools for addressing the aforementioned engineering problems and improving upon previous approaches. To date, examples of non-strong fuzzy metrics have not yet been applied in any practical context. Moreover, compared with the results established in [22], a calibration function is included to manage the t parameter, providing flexibility for the constructed fuzzy metrics to be adapted to different contexts.

The subsequent sections of this paper are arranged as follows. Section 2 recalls the basic concepts on fuzzy metrics and t-norms necessary for the study. Section 3 presents the main results obtained, and Section 4 provides the conclusions.

2. Preliminaries

We first recall the concept of a continuous t-norm, which is essential for defining the fuzzy metric spaces considered in this work.

Definition 1.

A triangular norm (or t-norm) is a binary operation $\ast :[0,1]\times [0,1]\to [0,1]$ satisfying the following properties, for all $a,b,c,d\in [0,1]$:

(i) 

$a\ast b=b\ast a$;

(ii) 

$(a\ast b)\ast c=a\ast (b\ast c)$;

(iii) 

if $a\le c$ and $b\le d$, then $a\ast b\le c\ast d$;

(iv) 

$a\ast 1=a$.

A t-norm ∗ is said to be continuous if it is continuous as a function from $[0,1]\times [0,1]$ to $[0,1]$, with respect to the usual topology.

We continue by reviewing the notion of a fuzzy metric space that will be used throughout the paper. We remark that this definition is a modest modification of the reformulation given by Grabiec in [25] of the notion of a fuzzy metric initially proposed by Kramosil and Michalek in [1]. The main reasons for adopting this modification were detailed in [22].

Definition 2.

A fuzzy metric space is an ordered triple $(X,M,\ast )$ such that X is a (non-empty) set, ∗ is a continuous t-norm, and $M:X\times X\times ]0,\infty [\to [0,1]$ is a mapping satisfying, for all $x,y,z\in X$ and $t,s\in ]0,\infty [$, the following axioms:

(KM1) 

$M(x,y,t)=1$ for all $t\in ]0,\infty [$ if and only if $x=y;$

(KM2) 

$M(x,y,t)=M(y,x,t);$

(KM3) 

$M(x,y,t)\ast M(y,z,s)\le M(x,z,t+s);$

(KM4) 

The mapping $M_{xy}:]0,\infty [\to [0,1]$ is left-continuous, where $M_{xy}\left(t\right)=M(x,y,t)$ for each $t\in ]0,\infty [$.

As mentioned above, George and Veeramani strengthened some of the preceding axioms to define another notion of fuzzy metric, referred to as a $GV$-fuzzy metric, which can be formulated as follows.

Definition 3.

Let $(X,M,\ast )$ be a fuzzy metric space. $(X,M,\ast )$ is called a $GV$-fuzzy metric space if the mapping M satisfies, for all $x,y\in X$ and $t\in ]0,\infty [$, the following axioms:

(GV0) 

$M(x,y,t)>0$;

(GV1) 

$M(x,y,t_0)=1$ for some $t_0\in ]0,\infty [$ implies $x=y;$

(GV2) 

The mapping $M_{xy}:]0,\infty [\to [0,1]$ is continuous.

As is customary, we refer to $(M,\ast )$, or simply M when no ambiguity arises, as a ($GV$-)fuzzy metric on X. Analogously to the classical case, we say that $(X,M,\ast )$ constitutes a fuzzy pseudo-metric space if it meets all the conditions stated in Definition 2 except for axiom (KM1), which is replaced by the following weaker one:

(KM1’) 

$M(x,x,t)=1$ for all $t\in ]0,\infty [$.

Similarly, by substituting axiom (GV1) in the Definition 3 with (KM1’), we obtain the notion of a GV-fuzzy pseudo-metric space.

In [26], George and Veeramani demonstrated that every $GV$-fuzzy metric induces a Hausdorff topology ${\mathcal{T}}_M$ on X that has as a base the family of open balls $\{B_M(x,r,t):x\in X,r\in ]0,1[,t\in ]0,\infty \left[\right\}$, where $B_M(x,r,t)=\{y\in X:M(x,y,t)>1-r\}$ for each $x\in X$, $r\in ]0,1[$ and $t\in ]0,\infty [$. By applying reasoning analogous to that in [26], this result extends to fuzzy metrics as defined in Definition 2. Furthermore, a topology can be constructed from a ($GV$-)fuzzy pseudo-metric by means of the same family of open balls, which, in general, is not Hausdorff, as in the classical framework.

Based on Definitions 2 and 3, it becomes evident that $GV$-fuzzy metrics represent a specific subclass of fuzzy metrics. Consequently, in this work, we will formulate all definitions and results within the broader framework of fuzzy metrics, specifying explicitly whenever a concept or property applies exclusively in the George and Veeramani setting.

A noteworthy subclass of fuzzy (pseudo-)metrics, known as strong fuzzy (pseudo-)metrics, was introduced in [12] (see also [23]). They are defined as follows:

Definition 4.

Let $(X,M,\ast )$ be a fuzzy pseudo-metric space. We will say that $(X,M,\ast )$, or simply M, is strong if M additionally satisfies, for all $x,y,z\in X$ and $t\in ]0,\infty [$, the following axiom:

(S) 

$M(x,z,t)\ge M(x,y,t)\ast M(y,z,t)$.

Most known fuzzy metrics in the literature are of the strong type. In fact, obtaining examples of non-strong fuzzy metrics for specific t-norms other than the minimum was proposed as an open question in [23]. This question was addressed in [24], where two examples of non-strong fuzzy metrics were provided: one using the product t-norm and another using the Lukasiewicz t-norm. Furthermore, Ref. [24] encouraged the search for additional examples involving continuous t-norms lying above the product and below the minimum. Although these investigations were conducted in the George and Veeramani framework, the problem also becomes relevant within the more general notion of fuzzy metric. In this context, a method for constructing (non-strong) fuzzy metrics from classical metrics was introduced in [22], and will be presented later. This method relies on the notion of an additive generator of a t-norm, which is recalled below. For a more comprehensive treatment of t-norms and additive generators, we refer the reader to [27].

Definition 5.

Let ∗ be a continuous t-norm. A function $f_{\ast }:[0,1]\to [0,\infty ]$, which is continuous and strictly decreasing, is called an additive generator of ∗ if it satisfies $f_{\ast }\left(1\right)=0$ and for all $a,b\in [0,1]$, the following holds:

$$a\ast b=f_{\ast }^{(-1)}(f_{\ast }\left(a\right)+f_{\ast }\left(b\right)),$$

(1)

where $f_{\ast }^{(-1)}:[0,\infty ]\to [0,1]$ denotes the pseudo-inverse of $f_{\ast }$, defined as

$$f_{\ast }^{(-1)}\left(u\right)=\left\{\begin{array}{cc}{f}_{\ast }^{-1}\left(u\right),\hfill & ifu\in [0,f_{\ast }\left(0\right)[\hfill \\ 0,\hfill & ifu\in [f_{\ast }\left(0\right),\infty [\hfill \end{array}\right..$$

(2)

The following theorem characterizes continuous t-norms that admit an additive generator, known as Archimedean t-norms. These are precisely the t-norms that satisfy the property $a\ast a<a$ for each $a\in ]0,1[$. It is worth mentioning that the product and Lukasiewicz t-norms are Archimedean, whereas the minimum t-norm is not.

Theorem 1.

A function $\ast :{[0,1]}^2\to [0,1]$ is a continuous Archimedean t-norm if and only if there exists an additive generator $f_{\ast }$ of ∗.

At this point, we are now set to present the method outlined earlier for constructing (non-strong) fuzzy metrics provided in [22] (Theorem 3.1).

Theorem 2.

Let $(X,d)$ be a pseudo-metric space and let ∗ be a continuous t-norm with additive generator $f_{\ast }$. Then, $(M,\ast )$ is a fuzzy pseudo-metric on X, where M is the mapping defined on $X\times X\times ]0,\infty [$ as follows:

$$M(x,y,t)=f_{\ast }^{(-1)}\left(max\left\{d\right(x,y)-t,0\}\right),$$

(3)

for all $x,y\in X$ and for all $t\in ]0,\infty [$. Furthermore, $(M,\ast )$ is a fuzzy metric on X if and only if d is a metric on X.

In [22], it was pointed out that the fuzzy metric M defined in the preceding theorem is, in general, not strong. Moreover, Ref. [22] (Theorem 3.7) established a condition under which this fuzzy metric fails to be strong.

3. The Results

The goal of this section is to generalize the construction of (non-strong) fuzzy metrics presented in Theorem 2, following the ideas underlying the methods introduced in [21]. Moreover, we will study the connection between the topology generated by the resulting fuzzy pseudo-metric and that determined by the classical pseudo-metric from which it originates.

In [21], two new approaches for constructing strong (GV-)fuzzy metrics from classical metrics were proposed. One method relies on the use of an additive generator when dealing with continuous Archimedean t-norms, while the other employs a real function $g:[0,\infty ]\to [0,1]$ in the case of the minimum t-norm, which is continuous but not Archimedean. Additionally, an auxiliary function $\phi :]0,\infty [\to ]0,\infty [$ is employed, which must satisfy certain conditions.

We begin by recalling the first method presented in [21] (Theorem 3.1).

Theorem 3.

Let $(X,d)$ be a pseudo-metric space, let $\phi :]0,\infty [\to ]0,\infty [$ be an increasing and left-continuous function, and let ∗ be a continuous Archimedean t-norm. If $f_{\ast }$ is an additive generator of ∗, then $(X,M,\ast )$ is a strong fuzzy pseudo-metric space, where the mapping $M:X\times X\times ]0,\infty [$ is given, for each $x,y\in X$ and $t\in ]0,\infty [$, by

$$M(x,y,t)=f_{\ast }^{(-1)}\left({\displaystyle \frac{d(x,y)}{\phi \left(t\right)}}\right).$$

(4)

Moreover, M is a strong fuzzy metric if and only if d is a metric.

To adapt the previous result to the construction presented in Theorem 2, it will be necessary to add an additional condition on the function $\phi$, namely, superadditivity. Recall that a function $\phi :]0,\infty [\to ]0,\infty [$ is said to be superadditive if it satisfies $\phi (t+s)\ge \phi \left(t\right)+\phi \left(s\right)$ for each $t,s\in ]0,\infty [$ (see [28]). The following lemma, proved in [29], establishes an essential property of such functions for later use.

Lemma 1.

Let $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ be an increasing superadditive function. Then, $\underset{t\to 0^{+}}{lim}\phi \left(t\right)=0$ (where, as usual, $\underset{t\to 0^{+}}{lim}$ denotes the one-sided limit as t approaches 0 from the right).

We are now able to state and prove below the promised adaptation of Theorem 3 to the construction provided in Theorem 2.

Theorem 4.

Let $(X,d)$ be a pseudo-metric space, let $\phi :]0,\infty [\to ]0,\infty [$ be an increasing, superadditive and left-continuous function, and let ∗ be a continuous Archimedean t-norm. If $f_{\ast }$ is an additive generator of ∗, then $(X,M,\ast )$ is a fuzzy pseudo-metric space, where the mapping $M:X\times X\times ]0,\infty [$ is given, for each $x,y\in X$ and $t\in ]0,\infty [$, by

$$M(x,y,t)=f_{\ast }^{(-1)}\left(max\left\{d\right(x,y)-\phi (t),0\}\right).$$

(5)

Moreover, M is a fuzzy metric if and only if d is a metric.

Proof. 

Let $(X,d)$ be a pseudo-metric space, let $\phi :]0,\infty [\to ]0,\infty [$ be an increasing, superadditive and left-continuous function, and let ∗ be a continuous Archimedean t-norm. Suppose that $f_{\ast }$ is an additive generator of ∗. Define $M:X\times X\times ]0,\infty [$ by $M(x,y,t)=f_{\ast }^{(-1)}\left(max\left\{d\right(x,y)-\phi (t),0\}\right)$, for each $x,y\in X$ and $t\in ]0,\infty [$. We will see that M satisfies (KM1’), (KM2), (KM3), and (KM4).

First of all, for any $x\in X$, we have

$$M(x,x,t)=f_{\ast }^{(-1)}\left(max\left\{d\right(x,x)-\phi (t),0\}\right)=f_{\ast }^{(-1)}\left(max\{0-\phi (t),0\}\right)=f_{\ast }^{(-1)}\left(0\right)=1,$$

for all $t\in ]0,\infty [$, since d is a pseudo-metric on X. Therefore, axiom (KM1’) is satisfied.

Secondly, from the definition of M and the fact that d is a pseudo-metric, it follows immediately that (KM2) holds. Moreover, using again the definition of M, and noting that $\phi$ is left-continuous and $f_{\ast }$ is continuous, we conclude that (KM4) is also satisfied.

We now focus on verifying that M satisfies axiom (KM3). To this end, consider $x,y,z\in X$ and $t,s\in ]0,\infty [$. We claim that $max\left\{d\right(x,z)-\phi (t+s),0\}\le max\left\{d\right(x,y)-\phi (t),0\}+max\left\{d\right(y,z)-\phi (s),0\}.$ Indeed, if $d(x,z)\le \phi (t+s)$, then

$$max\left\{d\right(x,z)-\phi (t+s),0\}=0\le max\left\{d\right(x,y)-\phi (t),0\}+max\left\{d\right(y,z)-\phi (s),0\}.$$

Contrarily, if $d(x,z)>\phi (t+s)$, then, using the triangle inequality for the pseudo-metric d and the superadditivity of $\phi$, we obtain

$$max\left\{d\right(x,z)-\phi (t+s),0\}=d(x,z)-\phi (t+s)\le d(x,y)+d(y,z)-\phi \left(t\right)-\phi \left(s\right).$$

Moreover,

$$d(x,y)+d(y,z)-\phi \left(t\right)-\phi \left(s\right)\le max\left\{d\right(x,y)-\phi (t),0\}+max\left\{d\right(y,z)-\phi (s),0\}.$$

So, $max\left\{d\right(x,z)-\phi (t+s),0\}\le max\left\{d\right(x,y)-\phi (t),0\}+max\left\{d\right(y,z)-\phi (s),0\}$.

Now, we distinguish between two cases:

• Suppose that $max\{max\{d(x,y)-\phi \left(t\right),0\},max\{d(y,z)-\phi \left(s\right),0\left\}\right\}\ge f_{\ast }\left(0\right)$. Then, since $f_{\ast }^{(-1)}$ is decreasing and $f_{\ast }^{(-1)}\left(u\right)=0$, for all $u\in \left[0\right),\infty ]$, we get

  $$min\left\{M\right(x,y,t),M(y,z,s\left)\right\}=$$

  $$min\{f_{\ast }^{(-1)}\left(max\left\{d\right(x,y)-\phi (t),0\}\right),f_{\ast }^{(-1)}\left(max\left\{d\right(y,z)-\phi (s),0\}\right)\}=$$

  $$f_{\ast }^{(-1)}\left(f\left(0\right)\right)=0.$$
  Moreover, taking into account that $a\ast b\le min\{a,b\}$ for all $a,b\in [0,1]$, we conclude that $M(x,y,t)\ast M(y,z,s)=0$ and so $M(x,z,t+s)\ge M(x,y,t)\ast M(y,z,s)$ in this case.
• Contrarily, suppose that $max\{max\{d(x,y)-\phi \left(t\right),0\},max\{d(y,z)-\phi \left(s\right),0\left\}\right\}<f_{\ast }\left(0\right)$. Then, we have that $max\left\{d\right(x,y)-\phi (t),0\},max\left\{d\right(y,z)-\phi (s),0\}\in [0,f_{\ast }(0\left)\right[$. Taking into account that $f_{\ast }^{(-1)}$ is decreasing, by applying formula (2), we get

  $$M(x,z,t+s)=f_{\ast }^{(-1)}(max\{d(x,z)-\phi (t+s),0\})\ge$$

  $$f_{\ast }^{(-1)}\left(max\left\{d\right(x,y)-\phi (t),0\}+max\left\{d\right(y,z)-\phi (s),0\}\right)=$$

  $$f_{\ast }^{(-1)}\left(f_{\ast }(f_{\ast }^{(-1)}\left(max\left\{d\right(x,y)-\phi (t),0\}\right)\right)+f_{\ast }(f_{\ast }^{(-1)}\left(max\left\{d\right(y,z)-\phi (s),0\}\right)=$$

  $$f_{\ast }^{(-1)}\left(f_{\ast }\left(M\right(x,y,t\left)\right)+f_{\ast }\left(M\right(y,z,s\left)\right)\right)=M(x,y,t)\ast M(y,z,s).$$

Thus, in both cases, axiom (KM3) is satisfied.

What remains is to demonstrate that M constitutes a fuzzy metric precisely when d is a metric.

To prove the direct implication, assume that M is a fuzzy metric and let $x,y\in X$ such that $d(x,y)=0$. Then, for all $t\in ]0,\infty [$, we have $M(x,y,t)=f_{\ast }^{(-1)}(max\{0-\phi \left(t\right),0\})=f_{\ast }^{(-1)}\left(0\right)=1$, which leads us to conclude that $x=y$ because M is a fuzzy metric. Thus, d is a metric.

In the opposite direction, suppose that d is a metric and take $x,y\in X$ such that $M(x,y,t)=1$ for all $t\in ]0,\infty [$. Then, $f_{\ast }^{(-1)}(max\{d(x,y)-\phi \left(t\right),0\})=1$ for all $t\in ]0,\infty [$ and, since $f_{\ast }^{(-1)}:[0,0)]\to [0,1]$ is strictly decreasing and satisfies $f_{\ast }^{(-1)}\left(0\right)=1$, we conclude that $max\left\{d\right(x,y)-\phi (t),0\}=0$ for all $t\in ]0,\infty [$. Therefore, $d(x,y)\le \phi \left(t\right)$ for all $t\in ]0,\infty [$ and, by Lemma 1, taking the limit as t tends to 0, we get $d(x,y)\le \underset{t\to 0}{lim}\phi \left(t\right)=0$. Thus, since d is a metric, we conclude $x=y$. Hence, M is a fuzzy metric.  □

Above, it was stated that the superadditivity of $\phi$ is required to prove the preceding theorem. The following example justifies the necessity of this requirement.

Example 1.

Consider the (pseudo-)metric space $(\mathbb{R},d_u)$, where $d_u$ denotes the usual metric on $\mathbb{R}$, i.e., $d_u(x,y)=|x-y|$ for each $x,y\in \mathbb{R}$. Define $\phi :]0,\infty [\to ]0,\infty [$ by $\phi \left(t\right)=\sqrt{t}$ for all $t\in ]0,\infty [$, which is increasing and (left-)continuous, but is not superadditive. Indeed, $\phi (2+2)=\sqrt{2+2}=2<2\sqrt{2}=\sqrt{2}+\sqrt{2}=\phi \left(2\right)+\phi \left(2\right)$. Consider ${\ast }_P$ the product t-norm, which is continuous and Archimedean, and let $f_{{\ast }_P}$ be the additive generator of ${\ast }_P$ given by $f_{{\ast }_P}\left(a\right)=-log\left(a\right)$ for all $a\in [0,1]$. Then, all the requirements imposed in Theorem 4 are satisfied except for the superadditive of φ.

Now, $f_{{\ast }_P}^{(-1)}\left(u\right)=e^{-u}$ for all $u\in [0,\infty ]$ and so, applying expression (5), we get

$$M(x,y,t)=e^{-max\{d_u(x,y)-\sqrt{t},0\}},foreachx,y\in X,t\in ]0,\infty [.$$

Taking $x=0$, $y=1$, $z=2$, and $t=s=1$, we have $M(x,z,t+s)=e^{-max\{2-\sqrt{2},0\}}=e^{-2+\sqrt{2}}$ and $M(x,y,t)=M(y,z,s)=e^{-max\{1-\sqrt{1},0\}}=e^0=1$. Therefore,

$$M(x,z,t+s)=e^{-2+\sqrt{2}}<1=M(x,y,t){\ast }_{P}M(y,z,s),$$

and so M is not a fuzzy (pseudo-)metric on $\mathbb{R}$.

It should be noted that the construction of fuzzy (pseudo-)metrics provided in Theorem 4 generalizes that presented in Theorem 2. Indeed, by taking $\phi \left(t\right)=t$ in Theorem 4, we recover the construction given in that theorem. To demonstrate that Theorem 4 truly generalizes Theorem 2, it is enough to consider the function $\phi :]0,\infty [\to ]0,\infty [$ given by $\phi \left(t\right)=t^2$, which is increasing, continuous, and superadditive. Moreover, increasing convex functions, i.e., those increasing functions $\phi :]0,\infty [\to ]0,\infty [$ satisfying for each $x,y\in ]0,\infty [$ the property $\phi (\lambda x+(1-\lambda \left)y\right)\le \lambda \phi \left(x\right)+(1-\lambda )\phi \left(y\right)$ for all $\lambda \in ]0,1[$, are superadditve as long as $\underset{t\to 0}{lim}\phi \left(t\right)=0$. Therefore, a large class of functions, extensively studied in the literature, can be used in the construction provided in Theorem 4, only considering those that are increasing and left-continuous. For instance, $\phi :]0,\infty [\to ]0,\infty [$ given by $\phi \left(t\right)=k{t}^n$, for each $k,n\in \mathbb{R}$, with $k>0$ and $n\ge 1$, satisfies the requirements imposed in Theorem 4. To illustrate our new technique, we provide the next two corollaries, which generalize [22] (Corollaries 3.2 and 3.3). First, recall that the product t-norm ${\ast }_P$ and the Lukasievicz t-norm ${\ast }_L$ are continuous and Archimedean, and they have as an additive generator $f_{{\ast }_P}\left(a\right)=-log\left(a\right)$ and $f_{{\ast }_L}\left(a\right)=1-a$, for each $a\in [0,1]$, respectively.

Corollary 1.

Let $(X,d)$ be a pseudo-metric space and let $k,n\in \mathbb{R}$, with $k>0$ and $n\ge 1$. Then, $(M,{\ast }_P)$ is a fuzzy pseudo-metric on X, where M is the fuzzy set defined on $X\times X\times ]0,\infty [$ as follows:

$$M(x,y,t)=\left\{\begin{array}{cc}{\mathsf{e}}^{k{t}^n-d(x,y)},\hfill & ift\le {\left({\displaystyle \frac{d(x,y)}{k}}\right)}^{1/n}\hfill \\ 1,\hfill & ift>{\left({\displaystyle \frac{d(x,y)}{k}}\right)}^{1/n}\hfill \end{array}\right.$$

(6)

for all $x,y\in X$ and for all $t\in ]0,\infty [$. Furthermore, $(M_{d,f_{{\ast }_P}},{\ast }_P)$ is a fuzzy metric on X if and only if d is a metric on X.

Corollary 2.

Let $(X,d)$ be a pseudo-metric space and let $k,n\in \mathbb{R}$, with $k>0$ and $n\ge 1$. Then, $(M,{\ast }_L)$ is a fuzzy pseudo-metric on X, where M is the fuzzy set defined on $X\times X\times ]0,\infty [$ as follows:

$$M(x,y,t)=\left\{\begin{array}{cc}0,\hfill & ift\le {\left({\displaystyle \frac{d(x,y)-1}{k}}\right)}^{1/n}\hfill \\ 1+k{t}^n-d(x,y)\hfill & if{\left({\displaystyle \frac{d(x,y)-1}{k}}\right)}^{1/n}<t\le {\left({\displaystyle \frac{d(x,y)}{k}}\right)}^{1/n}\hfill \\ 1,\hfill & ift>{\left({\displaystyle \frac{d(x,y)}{k}}\right)}^{1/n}\hfill \end{array}\right.$$

(7)

for all $x,y\in X$ and for all $t\in ]0,\infty [$. Moreover, M is a fuzzy metric if and only if d is a metric.

Observe in the preceding corollaries that the parameters k and n are arbitrary. This fact provides flexibility in the applicability of the proposed fuzzy metrics to real-world problems, as they can be adjusted to achieve greater accuracy in the models.

At this point, we ask whether imposing any additional condition on $\phi$ allows us to obtain strong fuzzy (pseudo-)metrics via Theorem 4, regardless of the pseudo-metric space under consideration. As the next example shows, the answer to this question is negative.

Example 2.

Consider again the (pseudo-)metric space $(\mathbb{R},d_u)$. Let $\phi :]0,\infty [\to ]0,\infty [$ be an increasing, left continuous and superadditive function. Let ∗ be a continuous Archimedean t-norm and let $f_{\ast }$ be an additive generator of ∗. Fix $t_0\in ]0,\infty [$ and consider $x=0$, $y=\phi \left(t_0\right)$, and $z=2\phi \left(t_0\right)$. Then, $max\{d_u(x,z)-\phi \left(t_0\right),0\}=\phi \left(t_0\right)>0=max\{d_u(x,y)-\phi \left(t_0\right),0\}+max\{d_u(y,z)-\phi \left(t_0\right),0\}$. Therefore, $M(x,z,t_0)=f_{\ast }^{(-1)}\left(\phi \left(t_0\right)\right)<1=M(x,y,t_0)\ast M(y,z,t_0),$ and so M is not strong.

Based on the preceding example, regardless of whether the function $\phi :]0,\infty [\to ]0,\infty [$ considered is increasing, left continuous, or superadditive, and whether the continuous Archimedean t-norm is used, we can find a pseudo-metric space in which the construction provided by Theorem 4 fails to be strong, regardless of the additive generator of ∗ considered.

We now focus on adapting the second method presented in [21] (Theorem 4.1) to the construction of Theorem 2. It should be noted that this result contains an error in the proof, where it was claimed that (KM4) is satisfied by M. Specifically, it was asserted that the left continuity of g and $\phi$, together with the condition $\phi \left(t\right)>0$ for all $t\in ]0,\infty [$, implies that the mapping $M_{x,y}:]0,\infty [\to [0,1]$ is a left-continuous function. This assertion is incorrect, as demonstrated by the following example. The example is a slight modification of item (ii) in [21] (Example 9), where a computational error occurred.

Example 3.

Let $(X,d)$ be a metric space and let $\phi :(0,\infty )\to (0,\infty )$ given by $\phi \left(t\right)=t$ for all $t\in (0,\infty )$. It is clear that φ satisfies all conditions required in [21] (Theorem 4.1). Consider the function $g:[0,\infty ]\to [0,1]$ defined as follows:

$$g\left(a\right)=\left\{\begin{array}{cc}{\mathsf{e}}^{-a},\hfill & ifa\in [0,1]\hfill \\ {\mathsf{e}}^{-2a},\hfill & ifa\in ]1,\infty ]\hfill \end{array}\right..$$

(8)

Obviously, g is decreasing and $g\left(0\right)=1$. Moreover, g is left-continuous.

Let $x,y\in X$, with $x\ne y$, and $t\in ]0,\infty [$. Note that ${\displaystyle \frac{d(x,y)}{t}}\in [0,1]$ if and only if $d(x,y)\le t$ and, ${\displaystyle \frac{d(x,y)}{t}}\in ]1,\infty ]$ if and only if $0<t<d(x,y)$. So, for each $x,y\in X$, with $x\ne y$, and $t\in ]0,\infty [$, we obtain the next expression for the mapping M:

$$M(x,y,t)=\left\{\begin{array}{cc}{\mathsf{e}}^{-2{\displaystyle \frac{d(x,y)}{t}}},\hfill & if0<t<d(x,y)\hfill \\ {\mathsf{e}}^{-{\displaystyle \frac{d(x,y)}{t}}},\hfill & ift\ge d(x,y)\hfill \end{array}\right..$$

(9)

Note that, for each $x,y\in X$, with $x\ne y$, the assignment $M_{x,y}:(0,\infty )\to [0,1]$, given by $M_{x,y}\left(t\right)=M(x,y,t)$, is not left-continuous at $t=d(x,y)$. Thus, M does not satisfy axiom (KM4).

So, below, we provide a correction of [21] (Theorem 4.1). Moreover, we have relaxed the condition imposed in such a theorem to obtain a fuzzy metric.

Theorem 5.

Let $(X,d)$ be a pseudo-metric space, let $\phi :]0,\infty [\to ]0,\infty [$ be an increasing, left-continuous and superadditive function, and let $g:[0,\infty ]\to [0,1]$ be a decreasing and right-continuous function, such that $g\left(0\right)=1$. Then, $(X,M,{\ast }_M)$ is a fuzzy pseudo-metric space, where the mapping $M:X\times X\times ]0,\infty [$ is given, for each $x,y\in X$ and $t\in ]0,\infty [$, by

$$M(x,y,t)=g\left({\displaystyle \frac{d(x,y)}{\phi \left(t\right)}}\right).$$

(10)

If, in addition, $g^{-1}\left(1\right)\subseteq [0,1[$, then M is a fuzzy metric if and only if d is a metric.

Proof. 

Let $(X,d)$ be a pseudo-metric space, let $\phi :]0,\infty [\to ]0,\infty [$ be an increasing, left-continuous and superadditive function, and let $g:[0,\infty ]\to [0,1]$ be a decreasing and right-continuous function, such that $g\left(0\right)=1$. Define, $M(x,y,t)=g\left({\displaystyle \frac{d(x,y)}{\phi \left(t\right)}}\right)$, for each $x,y\in X$ and $t\in ]0,\infty [$.

We only prove that M satisfies (KM4) to show that $(X,M,{\ast }_M)$ is a fuzzy pseudo-metric space, since all the remaining conclusions follow from the arguments used in the proof of Theorem 4.1 in [21].

Obviously, if $x=y$, the assignment $M_{x,y}:]0,\infty [\to [0,1]$ is (left-)continuous since $M_{x,y}\left(t\right)=1$ for all $t\in ]0,\infty [$. So, fix $x,y\in X$, with $x\ne y$, and consider the function $M_{x,y}:]0,\infty [\to [0,1]$. Let $t_0\in ]0,\infty [$. We will see that, for each $\epsilon >0$, we can find $\delta >0$ such that, for all $t\in ]t_0-\delta ,t_0]$ we have $|M_{x,y}\left(t_0\right)-M_{x,y}\left(t\right)|<\epsilon$.

Fix $\epsilon >0$. On the one hand, g is right-continuous and decreasing, so there exists ${\delta }_1>0$ such that, for every $a\in [a_0,a_0+{\delta }_1[$, we have $g\left(a_0\right)-g\left(a\right)<\epsilon$, where $a_0={\displaystyle \frac{d(x,y)}{\phi \left(t_0\right)}}$. On the other hand, for ${\displaystyle \frac{\phi \left(t_0\right)}{2}}>0$, we can find ${\delta }_2>0$ such that $\phi \left(t_0\right)-\phi \left(t\right)<{\displaystyle \frac{\phi \left(t_0\right)}{2}}$, for every $t\in ]t_0-{\delta }_2,t_0]$, due to $\phi$ increasing and being left-continuous. Then, $\phi \left(t\right)>{\displaystyle \frac{\phi \left(t_0\right)}{2}}$ for each every $t\in ]t_0-{\delta }_2,t_0]$. Again, since $\phi$ is left-continuous and increasing, we have that, for ${\delta }_1·{\displaystyle \frac{\phi {\left(t_0\right)}^2}{2d(x,y)}}>0$, there exists ${\delta }_3>0$ such that $\phi \left(t_0\right)-\phi \left(t\right)<{\delta }_1·{\displaystyle \frac{\phi {\left(t_0\right)}^2}{2d(x,y)}}$, for every $t\in ]t_0-{\delta }_3,t_0]$.

Let $\delta =min\{{\delta }_2,{\delta }_3\}$. By the exposed above, we have $\phi \left(t\right)>{\displaystyle \frac{\phi \left(t_0\right)}{2}}$ and $\phi \left(t_0\right)-\phi \left(t\right)<{\delta }_1·{\displaystyle \frac{\phi {\left(t_0\right)}^2}{2d(x,y)}}$, for each $t\in ]t_0-\delta ,t_0]$. Therefore, for each $t\in ]t_0-\delta ,t_0]$, we get

$${\delta }_1>{\displaystyle \frac{\phi \left(t_0\right)-\phi \left(t\right)}{\frac{\phi {\left(t_0\right)}^2}{2d(x,y)}}}=d(x,y)·{\displaystyle \frac{\phi \left(t_0\right)-\phi \left(t\right)}{\phi \left(t_0\right)·\frac{\phi \left(t_0\right)}{2}}}>d(x,y)·{\displaystyle \frac{\phi \left(t_0\right)-\phi \left(t\right)}{\phi \left(t\right)·\phi \left(t_0\right)}}=$$

$$d(x,y)\left({\displaystyle \frac{1}{\phi \left(t\right)}}-{\displaystyle \frac{1}{\phi \left(t_0\right)}}\right)={\displaystyle \frac{d(x,y)}{\phi \left(t\right)}}-{\displaystyle \frac{d(x,y)}{\phi \left(t_0\right)}}.$$

Taking into account the preceding inequality and due to $\phi$ increasing, we conclude

$${\displaystyle \frac{d(x,y)}{\phi \left(t_0\right)}}\le {\displaystyle \frac{d(x,y)}{\phi \left(t\right)}}<{\displaystyle \frac{d(x,y)}{\phi \left(t_0\right)}}+{\delta }_1.$$

Thus, ${\displaystyle \frac{d(x,y)}{\phi \left(t\right)}}\in [a_0,a_0+{\delta }_1[$ for each $t\in ]t_0-\delta ,t_0]$, where $a_0={\displaystyle \frac{d(x,y)}{\phi \left(t_0\right)}}$. Therefore, the above arguments ensure that

$$|M_{x,y}\left(t_0\right)-M_{x,y}\left(t\right)|=g\left({\displaystyle \frac{d(x,y)}{\phi \left(t_0\right)}}\right)-g\left({\displaystyle \frac{d(x,y)}{\phi \left(t\right)}}\right)<\epsilon .$$

Hence, the function $M_{x,y}$ is left-continuous at $t_0$, and since $t_0$ is arbitrary, we conclude that $M_{x,y}$ is left-continuous.

It remains to demonstrate that, if $g^{-1}\left(1\right)\subseteq [0,1[$, then M is a fuzzy metric if and only if d is a metric. So, suppose that $g^{-1}\left(1\right)\subseteq [0,1[$.

For the direct implication, suppose that M is a fuzzy metric and let $x,y\in X$ such that $d(x,y)=0$. Then, by our assumption on g, we have

$$M(x,y,t)=g\left({\displaystyle \frac{0}{\phi \left(t\right)}}\right)=g\left(0\right)=1,\mathrm{for}\mathrm{every}t\in ]0,\infty [.$$

Then, $x=y$ since M is a fuzzy metric on X.

Conversely, suppose that d is a metric and assume $M(x,y,t)=1$ for all $t\in ]0,\infty [$. Then, by formula (10), we have

$$1=M(x,y,1)=g\left({\displaystyle \frac{d(x,y)}{\phi \left(t\right)}}\right),\mathrm{for}\mathrm{every}t\in ]0,\infty [.$$

By our assumption on g, we get ${\displaystyle \frac{d(x,y)}{\phi \left(t\right)}}\in [0,1[$ for every $t\in ]0,\infty [$ or, equivalently, $d(x,y)<\phi \left(t\right)$ for every $t\in ]0,\infty [$. Taking the limit as t tends to 0, we conclude by Lemma 1 that $d(x,y)=0$. Hence, $x=y$ due to d being a metric on X.  □

We now present the announced adaptation. As before, additional conditions are required, this time on the function g.

Theorem 6.

Let $(X,d)$ be a pseudo-metric space, let $\phi :]0,\infty [\to ]0,\infty [$ be an increasing, superadditive and left-continuous function, and let $g:[0,\infty ]\to [0,1]$ be a decreasing and right-continuous function such that $g\left(0\right)=1$ and satisfying, for each $u\in ]0,\infty [$, the property $g\left(u\right)\ge g\left(v\right)$ for all $v\in \left[{\displaystyle \frac{u}{2}},u\right[$. Then, $(X,M,{\ast }_M)$ is a fuzzy pseudo-metric space, where the mapping $M:X\times X\times ]0,\infty [$ is given, for each $x,y\in X$ and $t\in ]0,\infty [$, by

$$M(x,y,t)=g\left(max\left\{d\right(x,y)-\phi (t),0\}\right).$$

(11)

Moreover, if, in addition, $g^{-1}\left(1\right)=\left\{0\right\}$, then M is a fuzzy metric if and only if d is a metric.

Proof. 

Let $(X,d)$ be a pseudo-metric space, let $\phi :]0,\infty [\to ]0,\infty [$ be an increasing, superadditive and left-continuous function, and let $g:[0,\infty ]\to [0,1]$ be a decreasing and right-continuous function, such that $g\left(0\right)=1$ and satisfying, for each $u\in ]0,\infty [$, the property $g\left(u\right)\ge g\left(v\right)$ for all $v\in \left[{\displaystyle \frac{u}{2}},u\right[$. Define $M:X\times X\times ]0,\infty [$ by $M(x,y,t)=g\left(max\left\{d\right(x,y)-\phi (t),0\}\right)$, for each $x,y\in X$ and $t\in ]0,\infty [$.

It is easy to show that M satisfies conditions (KM1’) and (KM2) using arguments similar to those employed in the proof of Theorem 4. Moreover, the same reasoning used in the proof of Theorem 5 remains valid to show that M satisfies (KM4). So, we will only see that M satisfies (KM3).

Let $x,y,z\in X$ and $t,s\in ]0,\infty [$. If $max\left\{d\right(x,z)-\phi (t+s),0\}=0$, then using the assumption $g\left(0\right)=1$, we get

$$M(x,z,t+s)=g(max\{d(x,z)-\phi (t+s),0\})=g\left(0\right)=1\ge M(x,y,t){\ast }_{M}M(y,z,s).$$

Now assume that $max\left\{d\right(x,z)-\phi (t+s),0\}=d(x,z)-\phi (t+s)$, i.e., $d(x,z)\ge \phi (t+s)$. Since d is a pseudo-metric and $\phi$ is superadditive, it follows that either $d(x,y)\ge \phi \left(t\right)$ or $d(y,z)\ge \phi \left(s\right)$. Indeed, if $d(x,y)<\phi \left(t\right)$ and $d(y,z)<\phi \left(s\right)$, then $\phi \left(t\right)+\phi \left(s\right)>d(x,y)+d(y,z)\ge d(x,z)\ge \phi (t+s)$, which contradicts the superadditivity of $\phi$. We distinguish two cases:

• Suppose that $d(x,y)-\phi \left(t\right)\ge d(y,z)-\phi \left(s\right)$; then, $d(x,y)\ge \phi \left(t\right)$ and so

  $$M(x,y,t){\ast }_{M}M(y,z,s)=$$

  $$g(max\{d(x,y)-\phi \left(t\right),0\}){\ast }_{M}g(max\{d(y,z)-\phi \left(s\right),0\})=g(d(x,y)-\phi \left(t\right)).$$
  On the other hand, our assumption implies $d(x,y)-\phi \left(t\right)-d(y,z)\ge -\phi \left(s\right)$ and, since d is a pseudo-metric and $\phi$ is superadditive, we obtain

  $$d(x,z)-\phi (t+s)\le d(x,y)+d(y,z)-\phi \left(t\right)-\phi \left(s\right)\le$$

  $$d(x,y)+d(y,z)-\phi \left(t\right)+d(x,y)-\phi \left(t\right)-d(y,z)=2\left(d\right(x,y)-\phi (t\left)\right).$$
  Set $u=d(x,z)-\phi (t+s)$ and $v=d(x,y)-\phi \left(t\right)$. Then, the previous inequality ensures $v\in \left[{\displaystyle \frac{u}{2}},\infty \right[$. So, if $v\ge u$, since g is decreasing, we get $g\left(u\right)\ge g\left(v\right)$. Contrarily, if $v\in \left[{\displaystyle \frac{u}{2}},u\right[$, the (additional) condition imposed on g guarantees again that $g\left(u\right)\ge g\left(v\right)$. Therefore,

  $$M(x,z,t+s)=g(d(x,z)-\phi (t+s))\ge g(d(x,y)-\phi \left(t\right))=M(x,y,t){\ast }_{M}M(y,z,s),$$

  and hence M satisfies (KM3) in this case.
• The remaining case $d(x,y)-\phi \left(t\right)<d(y,z)-\phi \left(s\right)$ follows by similar arguments.

To show that M is a fuzzy metric if and only if d is a metric, one can use the same arguments used for this purpose in Theorem 4, but employing that $g\left(0\right)=1$ for the direct implication, and the condition $g^{-1}\left(1\right)=\left\{0\right\}$ for the converse.  □

A natural question that arises from the preceding theorem is whether the additional condition on g is in fact necessary. The next example shows that this requirement cannot be omitted.

Example 4.

Consider again the (pseudo-)metric space $(\mathbb{R},d_u)$ and let $\phi :]0,\infty [\to ]0,\infty [$ given by $\phi \left(t\right)=t$, for all $t\in ]0,\infty [$, which is increasing, (left-)continuous, and superadditive. Define $g:[0,\infty ]\to [0,1]$ given by $g\left(u\right)={\displaystyle \frac{1}{1+u}}$, for all $u\in [0,\infty [$ and $g(\infty )=0$. Obviously, g is decreasing, (right-)continuous and $g\left(0\right)=1$. Nonetheless, g does not satisfy the condition: for each $u\in ]0,\infty [$, g satisfies the property $g\left(u\right)\ge g\left(v\right)$ for all $v\in \left[{\displaystyle \frac{u}{2}},u\right[$. Indeed, given $u=1$ for $v={\displaystyle \frac{1}{2}}\in \left[{\displaystyle \frac{1}{2}},1\right[$, we have that

$$g\left(v\right)=g\left({\displaystyle \frac{1}{2}}\right)={\displaystyle \frac{2}{3}}>{\displaystyle \frac{1}{2}}=g\left(1\right)=g\left(u\right).$$

Observe that M constructed following expression (11), for d and φ under consideration, is not a fuzzy metric on $\mathbb{R}$ for the minimum t-norm. Indeed, $M(x,y,t)={\displaystyle \frac{1}{1+max\left\{d\right(x,z)-t\}}}$, and setting $x=0$, $y=2$, $z=4$, and $t=s=1$ we get

$$M(x,z,t+s)={\displaystyle \frac{1}{3}}<{\displaystyle \frac{1}{2}}=M(x,y,t){\ast }_{M}M(y,z,s).$$

After showing that the aforementioned condition is necessary to obtain the conclusion of Theorem 6, we now prove the following result, which characterizes the functions that satisfy it.

Proposition 1.

Let $g:[0,\infty [\to [0,1]$ be a function, such that $g\left(0\right)=1$. Then, g is decreasing and satisfies, for each $u\in ]0,\infty [$, the property $g\left(u\right)\ge g\left(v\right)$ for all $v\in \left[{\displaystyle \frac{u}{2}},u\right[$ if and only if g is constant on $]0,\infty [$.

Proof. 

For the direct implication, let $g:[0,\infty [\to [0,1]$ be a decreasing function which, for each $u\in ]0,\infty [$, satisfies the property $g\left(u\right)\ge g\left(v\right)$ for all $v\in \left[{\displaystyle \frac{u}{2}},u\right[$. We will see that, for each $u,v\in ]0,\infty [$, with $u\ne v$, we have that $g\left(u\right)=g\left(v\right)$. So, let $u_0,v_0\in ]0,\infty [$, with $u_0\ne v_0$. Assume without loss of generality that $u_0>v_0$. Since g is decreasing, we conclude that $g\left(v_0\right)\ge g\left(u_0\right)$.

We will now prove by induction on n that, for each $n\in \mathbb{N}$, $g\left(u_0\right)\ge g\left(w\right)$ is satisfied for all $w\in \left[{\displaystyle \frac{u_0}{2^n}},u_0\right[$. The base case $n=1$ is given by our hypothesis on g, since it satisfies the property $g\left(u_0\right)\ge g\left(w\right)$, for all $w\in \left[{\displaystyle \frac{u_0}{2}},u_0\right[$. Suppose the statement is true for some arbitrary n, i.e., $g\left(u_0\right)\ge g\left(w\right)$ for all $w\in \left[{\displaystyle \frac{u_0}{2^n}},u_0\right[$, and we will prove it for $n+1$. Let $w\in \left[{\displaystyle \frac{u_0}{2^{n+1}}},u_0\right[$. We distinguish two cases:

• If $w\in \left[{\displaystyle \frac{u_0}{2^n}},u_0\right[$, by the hypothesis induction we obtain that $g\left(u_0\right)\ge g\left(w\right)$.
• Contrarily, if $w\in \left[{\displaystyle \frac{u_0}{2^{n+1}}},{\displaystyle \frac{u_0}{2^n}}\right[$, then, by our assumption on g, applied to $u={\displaystyle \frac{u_0}{2^n}}$, we get $g\left({\displaystyle \frac{u_0}{2^n}}\right)\ge g\left(w\right)$ for all $w\in \left[{\displaystyle \frac{u_0}{2^{n+1}}},{\displaystyle \frac{u_0}{2^n}}\right[$. Moreover, by the induction hypothesis, we have $g\left(u_0\right)\ge g\left({\displaystyle \frac{u_0}{2^n}}\right)$ and, combining it with the preceding inequality, we obtain $g\left(u_0\right)\ge g\left({\displaystyle \frac{u_0}{2^n}}\right)\ge g\left(w\right)$.

Therefore, by induction on n, we have that, for each $n\in \mathbb{N}$, $g\left(u_0\right)\ge g\left(w\right)$ is satisfied for all $w\in \left[{\displaystyle \frac{u_0}{2^n}},u_0\right[$.

Now, since $u_0>v_0$, there exists $n_0\in \mathbb{N}$ such that ${\displaystyle \frac{u_0}{2^{n_0}}}<v_0$. Then, $v_0\in \left[{\displaystyle \frac{u_0}{2^{n_0}}},u_0\right[$ and so $g\left(u_0\right)\ge g\left(v_0\right)$. Hence, $g\left(u_0\right)=g\left(v_0\right)$, since the above also proved the inequality $g\left(u_0\right)\le g\left(v_0\right)$.

Taking into account that $u_0,v_0\in ]0,\infty [$, with $u_0\ne v_0$, were arbitrary elements, we conclude that g is constant on $]0,\infty [$.

For the converse, assume g is constant on $]0,\infty [$. Since $g\left(0\right)=1$, it is obvious that g is decreasing since it is assumed to be constant on $]0,\infty [$. Additionally, this assumption provides that, for each $u\in ]0,\infty [$, we have that $g\left(u\right)=g\left(v\right)$ for each $v\in \left[{\displaystyle \frac{u}{2}},u\right[$, and so, we obtain the converse implication.  □

Remark 1.

After proving the preceding result, and taking into account Example 4, we conclude that, in order to adapt the method presented in [21] (Theorem 4.1) to the construction in Theorem 2, we can only use right-continuous and decreasing functions $g:[0,\infty ]\to [0,1]$ that are constant on $]0,\infty [$ such that $g\left(0\right)=1$ and $g(\infty )\le g\left(u\right)$ for each $u\in ]0,\infty [$. Therefore, g must be a constant function $g:[0,\infty ]\to [0,1]$, with $g\left(u\right)=1$ for all $u\in [0,\infty ]$. So, the adaptation of the method presented in [21] (Theorem 4.1) to the construction in Theorem 2 only provides the trivial fuzzy pseudo-metric given by $M(x,y,t)=1$ for all $x,y\in X$ and $t\in ]0,\infty [$, which lacks interest.

We continue now by discussing the possibility of obtaining a version of Theorem 4 within the context of George and Veeramani. Note that Theorem 3.2 in [21] established a construction of $GV$-fuzzy (pseudo-)metrics using the same ideas as in expression (3) by imposing additional conditions. We claim that, in general, the construction provided by Theorem 4 can be formulated to yield fuzzy metrics in the sense of George and Veeramani, but only $GV$-fuzzy pseudo-metrics. This claim will be justified later on. Before doing so, we state the $GV$ version of Theorem 3. It should be noted that [21] (Theorem 3.2) requires an additional condition on the t-norm. Specifically, the t-norm must be strict, i.e., those t-norms ∗ that are continuous and satisfy $a\ast b>0$ for each $a,b\in ]0,1]$. The next result on strict t-norms was essential to prove the aforementioned theorem.

Proposition 2

(See [27]). Let ∗ be a continuous Archimedean t-norm and let $f_{\ast }:[0,1]\to [0,\infty ]$ be a continuous additive generator of ∗. Then, ∗ is strict if and only if $f_{\ast }\left(0\right)=\infty$.

Now, we are able to establish the $GV$ version of Theorem 3.

Theorem 7.

Let $(X,d)$ be a pseudo-metric space, let $\phi :]0,\infty [\to ]0,\infty [$ be an increasing, superadditive and continuous function, and let ∗ be a strict Archimedean t-norm. If $f_{\ast }$ is an additive generator of ∗, then $(X,M,\ast )$ is a $GV$-fuzzy pseudo-metric space, where the mapping $M:X\times X\times ]0,\infty [$ is defined as in (5).

Proof. 

Let $(X,d)$ be a pseudo-metric space, let $\phi :]0,\infty [\to ]0,\infty [$ be an increasing, superadditive and continuous function, and let ∗ be a strict Archimedean t-norm. Suppose that $f_{\ast }$ is an additive generator of ∗. Define M as in (5). By Theorem 4, we have that $(X,M,\ast )$ is a fuzzy pseudo-metric space. Therefore, M satisfies axioms (KM1’), (KM2), and (KM3). It remains to show that M also satisfies axioms (GV0) and (GV2).

On the one hand, let $x,y\in X$ and $t\in ]0,\infty [$. Then, taking into account that ∗ is Archimedean and strict, we have that $f_{\ast }^{(-1)}\left(u\right)>0$, for each $u\in [0,\infty [$. Therefore, since $max\left\{d\right(x,y)-\phi (t),0\}\in [0,\infty [$, we get

$$M(x,y,t)=f_{\ast }^{(-1)}\left(max\left\{d\right(x,y)-\phi (t),0\}\right)>0,$$

and so (GV0) is satisfied.

On the other hand, (GV2) follows from the continuity of $\phi$ and $f_{\ast }^{(-1)}$.

Hence, $(X,M,\ast )$ is a $GV$-fuzzy pseudo-metric space.  □

In contrast to Theorem 4, the preceding result does not, in general, yield a $GV$-fuzzy metric when a metric is considered. It suffices to take a metric space $(X,d)$ in which there exist $x,y\in X$ with $x\ne y$. Then, choose a function $\phi :]0,\infty [\to ]0,\infty [$ such that we can find $t_0\in ]0,\infty [$ satisfying $\phi \left(t_0\right)>d(x,y)$. Then, $max\{d(x,y)-\phi \left(t_0\right),0\}=0$ and so $M(x,y,t_0)=f_{\ast }^{(-1)}\left(0\right)=1$, which means that (GV1) does not hold.

To conclude this section, we will establish the relationship between the topology induced by the fuzzy pseudo-metric constructed in Theorem 4 and that induced by the classical pseudo-metric from which it is derived. It should be noted that [29] established such a relationship when considering constructions of [21] (Theorems 3.1 and 4.1). On the one hand, it was proved in [29] that both topologies coincide for the construction of [21] (Theorem 3.1) whereas they can be different when the method of [21] (Theorem 4.1) is under consideration. Additionally, two additional conditions on g were imposed in order to preserve the topology in the construction of [21] (Theorem 4.1). Taking into account that, in this paper, we have corrected a mistake in [21] (Theorem 4.1), we provide below a new version of [29] (Theorem 5), in which continuity on g at 0 is deleted in the statement. Note that left-continuity on g must be changed by right-continuity to prove [21] (Theorem 4.1), as shown above. Then, the right-continuity on g implies continuity at 0.

Theorem 8.

Let $(X,d)$ be a pseudo-metric space, let $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ be an increasing left-continuous superadditive function, and let $g:[0,\infty ]\to [0,1]$ be a decreasing right-continuous function such that $g\left(0\right)=1$. If there exists $a\in {\mathbb{R}}^{+}$ such that $g\left(a\right)<1$, then the topology induced by d on X coincides with the topology induced by M on X, where M is defined as in expression (4).

The next result describes how the topology induced by the fuzzy pseudo-metric in Theorem 4 relates to that of the classical pseudo-metric from which it is constructed.

Theorem 9.

Let $(X,d)$ be a pseudo-metric space, let $\phi :]0,\infty [\to ]0,\infty [$ be an increasing, superadditive and left-continuous function, and let ∗ be a continuous Archimedean t-norm. If $f_{\ast }$ is an additive generator of ∗, then the topology $\mathcal{T}\left(d\right)$ induced by d on X coincides with the topology ${\mathcal{T}}_M$ induced by M on X, where M is defined as in expression (5).

Proof. 

Let $(X,d)$ be a pseudo-metric space, let $\phi :]0,\infty [\to ]0,\infty [$ be an increasing, superadditive and left-continuous function, and let ∗ be a continuous Archimedean t-norm. Suppose that $f_{\ast }$ is an additive generator of ∗. Then, by Theorem 4, we have that $(X,M,\ast )$ is a fuzzy pseudo-metric space. We will see that ${\mathcal{T}}_M=\mathcal{T}\left(d\right)$.

Firstly, we show that ${\mathcal{T}}_M\subseteq \mathcal{T}\left(d\right)$. With this aim, let $A\in {\mathcal{T}}_M$. Then, for each $x\in A$, there exists $r_0\in ]0,1[$ and $t_0\in ]0,\infty [$ such that $B_M(x,r_0,t_0)\subseteq A$. Fix $x\in A$. We claim that there exists ${\epsilon }_0>0$ such that $B_d(x;{\epsilon }_0)\subseteq B_M(x,r_0,t_0)\subseteq A$. Indeed, let ${\epsilon }_0=\phi \left(t_0\right)$ and consider $y\in B_d(x;{\epsilon }_0)$. Then, $d(x,y)<\phi \left(t_0\right)$ and we get

$$M(x,y,t_0)=f_{\ast }^{(-1)}\left(max\{d(x,y)-\phi \left(t_0\right),0\}\right)=f_{\ast }^{(-1)}\left(0\right)=1>1-r_0.$$

Thus, $y\in B_M(x,r_0,t_0)$ and so ${\mathcal{T}}_M\subseteq \mathcal{T}\left(d\right)$.

Secondly, we show that $\mathcal{T}\left(d\right)\subseteq {\mathcal{T}}_M$. We proceed similarly by considering $A\in \mathcal{T}\left(d\right)$. Let $x\in A$; then, there exists ${\epsilon }_0>0$ such that $B_d(x;{\epsilon }_0)\subseteq A$. By continuity of $f_{\ast }$, we can find $r_0\in ]0,1[$ such that $0<f_{\ast }(1-r)<{\epsilon }_0$. Moreover, by Lemma 1, we can find $t_0>0$ such that $\phi \left(t_0\right)<{\epsilon }_0-f_{\ast }(1-r)$. We claim that $B_M(x,r_0,t_0)\subseteq B_d(x;{\epsilon }_0)$. Indeed, let $y\in B_M(x,r_0,t_0)$; then,

$$1-r<M(x,y,t_0)=f_{\ast }^{(-1)}\left(max\{d(x,y)-\phi \left(t_0\right),0\}\right).$$

Observe that, in such a case, $max\{d(x,y)-\phi \left(t_0\right),0\}<f_{\ast }\left(0\right)$ since, in the contrary case, $M(x,y,t_0)=0$. Therefore,

$$1-r<M(x,y,t_0)=f_{\ast }^{-1}\left(max\{d(x,y)-\phi \left(t_0\right),0\}\right)$$

and so

$$f_{\ast }(1-r)>max\{d(x,y)-\phi \left(t_0\right),0\}\ge d(x,y)-\phi \left(t_0\right).$$

Thus, $d(x,y)<f_{\ast }(1-r)+\phi \left(t_0\right)<{\epsilon }_0$. Hence, $y\in B_d(x;{\epsilon }_0)$ and so $\mathcal{T}\left(d\right)\subseteq {\mathcal{T}}_M$.  □

4. Conclusions and Future Work

In this work, we have generalized the construction of (non-strong) fuzzy pseudo-metrics from classical pseudo-metrics introduced in [22] (Theorem 3.1) (see Theorem 2), based on the methods developed in [21] (Theorems 3.1 and 4.1). Our approach incorporates a real-valued function to handle the parameter t, and we have tried to extend Theorem 2 to obtain fuzzy metrics for the minimum t-norm.

On the one hand, we have established in Theorem 4 the necessary conditions for constructing (non-strong) fuzzy (pseudo-)metrics for continuous Archimedean t-norms via their additive generators. We have also proved in Theorem 9 that the proposed construction preserves the topology of the original metric space, and we have identified in Theorem 7 the additional requirements needed to obtain $GV$-fuzzy metrics. On the other hand, we have shown (see Remark 1) that, for the minimum t-norm, the method yields a unique fuzzy pseudo-metric which is the trivial one.

A further contribution of this paper is the identification of an error in the proof of [21] (Theorem 4.1), related to the construction of fuzzy metrics from classical metrics under the minimum t-norm. This issue has been fully detailed here and corrected in Theorem 5.

Future work may focus on two directions. On the one hand, a theoretical study can be conducted to extend the present work by generalizing the way in which the distance and the t parameter are combined before applying the additive generator. On the other hand, several new fuzzy metrics obtained through the constructions presented in this paper could be employed in image processing, perceptual color difference, or clustering, in order to compare their performance with previously reported approaches in the literature that make use of fuzzy metrics.