Fuzzy Metrics in Terms of Fuzzy Relations

: In this paper, we study the concept of fuzzy metrics from the perspective of fuzzy relations. Speciﬁcally, we analyze the commonly used deﬁnitions of fuzzy metrics. We begin by noting that crisp metrics can be uniquely characterized by linear order relations. Further, we explore the criteria that crisp relations must satisfy in order to determine a crisp metric. Subsequently, we extend these conditions to obtain a fuzzy metric and investigate the additional axioms involved. Additionally, we introduce the deﬁnition of an extensional fuzzy metric or E - d -metric, which is a fuzziﬁcation of the expression d ( x , y ) = t . Thus, we examine fuzzy metrics from both the linear order and from the equivalence relation perspectives, where one argument is a value d ( x , y ) and the other is a number within the range [ 0, + ∞ ) .


Introduction
Since L.A. Zadeh introduced fuzzy sets in 1965 [1], researchers have been actively exploring ways to integrate traditional mathematical concepts and theories into the fuzzy sets context. Among the pioneering successes in this endeavor were the development of fuzzy topologies by C.L. Chang [2], the introduction of fuzzy algebraic structures by A. Rosenfeld [3], fuzzy category theory by A. Šostak [4], etc. Fuzzy sets have also been widely used for practical applications that involve uncertainty, vagueness, and imprecision. They have already proven their efficiency in natural language processing, decision-making, pattern recognition, and optimization problems. As the potential applications of fuzzy metrics in real-world problem solving became evident, the idea of establishing a fuzzy counterpart to a metric space gained traction. Several researchers took on this challenge, and notable contributions to the field of fuzzy metrics were made by I. Kramosil and J. Michalek [5], A. George and P. Veeramani [6], Z. Deng [7], and O. Kaleva and S. Seikkala [8]. It is worth mentioning that each of these researchers used different initial prerequisites in their approach, which adds diversity to the developments in the field. These advancements in fuzzy metrics open up new avenues for studying and addressing complex real-world issues through the flexible and adaptable nature of fuzzy sets and metrics. As research in this area continues to progress, we can anticipate even more valuable applications and insights into a wide range of problems. Currently, there is growing interest in exploring the topological properties of fuzzy metrics, as this line of study holds promise not only for theoretical constructions but also for fixed-point theorems and various practical applications. Regarding the investigation of the topological properties of classical fuzzy metrics, extensive references can be found in [9][10][11][12][13][14][15][16][17][18][19][20][21]. Although fuzzy metrics have demonstrated successful applications in image processing problems [22][23][24], their full potential remains untapped. These metrics hold significant promise, particularly in addressing segmentation, spectralization, and compression problems. Furthermore, fuzzy metrics have showcased their capability in tackling optimization problems [25]. As research in this area continues to progress, it is likely that we will witness even more innovative applications and fruitful outcomes from the study of fuzzy metrics and their properties. Continuing the exploration of fuzzy metrics from the perspective of fuzzy relations opens up exciting avenues for enhancing our theoretical understanding of their properties. By delving into the fuzzy relations aspect, we can establish insightful connections between various structures and gain deeper insights into the fundamental nature of fuzzy metrics. Fuzzy relations provide a powerful framework to analyze the relationships and interactions within fuzzy metrics, shedding light on the underlying mathematical intricacies. This approach not only offers a fresh lens to examine existing fuzzy metrics but also allows us to uncover hidden patterns and unveil novel properties. Moreover, investigating fuzzy metrics from the fuzzy relations point of view paves the way for constructing new examples. By leveraging the inherent flexibility of fuzzy relations, we can create innovative fuzzy metric spaces that possess specific properties tailored to tackle real-world problems. The synergy between fuzzy metrics and fuzzy relations also holds promise for cross-disciplinary applications. It enables us to leverage a wealth of knowledge from different fields and merge their insights to address complex challenges more comprehensively. In conclusion, venturing into the study of fuzzy metrics through the lens of fuzzy relations not only enriches our theoretical understanding of these structures but also opens up a vast landscape of possibilities for practical applications.
In the current literature, the concept of a fuzzy metric is predominantly based on the axioms introduced in [6,9], which are essentially a reformulation of the original axioms defined in [5]. In [5], the idea of defining a fuzzy metric stems from the assumption that the evaluated value d(x, y) of a crisp metric d to be fuzzified or approximated is smaller than a predetermined real number t. In other words, the statement d(x, y) < t is fuzzified. This paper aims to justify this fact. It is crucial to take into account this fact when working with applications.
Thus, the main idea of the paper is to show how the notion of a fuzzy metric arises from the crisp order relation R, by demonstrating that every metric d can be determined by an order relation R. Subsequently, we investigate the criteria that crisp relations must satisfy in order to establish a crisp metric. Furthermore, we fuzzify the axioms of R to obtain a fuzzy metric and examine the conditions that a fuzzy relation R must satisfy in order to be considered a fuzzy metric. Finally, we introduce a different approach to the fuzzy metric concept, where we extend a crisp metric d on a set X by means of a fuzzy equivalence relation E on the set IR + . We call it an E-d metric or an extensional fuzzy metric.
The paper is structured as follows. In Section 2, we provide a recap of the key results and concepts that are used in the paper. Specifically, we introduce and discuss triangular norms and fuzzy relations. Section 3 is devoted to the examination of classical metrics and their representation by means of linear order relations. The primary objective of the paper is addressed in Section 4. Here, we analyze the existing definition of fuzzy metrics and propose a method for its construction, employing fuzzy order relations. Extensional fuzzy metrics are explored in Section 5. Finally, in Section 6, we conclude the paper.

Triangular Norms
We start with the definition of a t-norm, which plays a crucial role in the definition of transitivity for fuzzy relations: • T(a, 1) = a (a boundary condition).
Some of the commonly used t-norms are mentioned below: otherwise , the Hamacher t-norm.
A t-norm T is called Archimedean if and only if, for all pairs (a, b) ∈ (0, 1) 2 , there is n ∈ N such that T n (a) < b, where T n (a) is defined by induction: T 1 (a) = T(a, a), T 2 (a) = T(a, T 1 (a)), . . . , T n (a) = T(a, T n−1 (a)).
Product, Łukasiewicz, and Hamacher t-norms are Archimedean, while minimum t-norm is not.

Fuzzy Relations
We continue with an overview of basic definitions and results of fuzzy relations. Definitions of a fuzzy order relation and a fuzzy equivalence relation were first introduced by L.A. Zadeh in 1971 [27] under the names of a fuzzy ordering and a fuzzy similarity relation. In our paper, we use results of a fuzzy order defined with respect to a fuzzy equivalence relation studied in [28,29].

Definition 2 ([27]). A fuzzy binary relation R on a set S is a mapping R
Definition 3 (see, e.g., [28]). A fuzzy binary relation E on a set S is called a fuzzy equivalence relation with respect to a t-norm T (or a T-equivalence) if and only if the following three axioms are fulfilled for all a, b, c ∈ S: 1.

Definition 4 ([29]).
A fuzzy binary relation L on a set S is called a fuzzy order relation with respect to a t-norm T and a T-equivalence E (or T-E-order) if and only if the following three axioms are fulfilled for all a, b, c ∈ S: T(L(a, b), L(b, c)) ≤ L(a, c) T-transitivity; 3.
T(L(a, b), L(b, a)) ≤ E(a, b) T-E-antisymmetry. A fuzzy order relation L is called strongly linear if and only if for all a, b ∈ S: The following theorem states that strongly linear fuzzy order relations are uniquely characterized as fuzzifications of crisp linear orders. Preliminary, let us recall the definition of compactability: 29]). Let be a crisp order on a set S, and let E be a fuzzy equivalence relation on S. E is called compatible with if and only if the following implication holds for all a, b, c ∈ S: Theorem 1 ([29]). Let L be a binary fuzzy relation on S, and let E be a T-equivalence on S. Then, the following two statements are equivalent: L is a strongly linear T-E-order on S;

2.
There is a linear order with which the relation E is compatible, such that L can be represented as follows:

Crisp Metrics
The concepts of a metric and a metric space, first introduced by M. Fréchet in 1906 [30], now belong to the most fundamental concepts of modern mathematics. For convenience of presentation, we recall them in the next definition: Definition 6. Metric space is an ordered pair (X, d), where X is a set and d is a metric on X, i.e., a function d : X × X → [0, ∞), satisfying the following axioms for all points x, y, z ∈ X: As the next theorem shows, metric spaces (X, d) are fully characterized by pairs

Theorem 2.
A metric d on a set X is uniquely determined by the following function:

Proof. Let us prove that two metrics d 1 and d 2 differ if and only if
Now we investigate how to define R d : X × X × [0, ∞) → {0, 1} in order to reflect axioms from Definition 6: 1.
If R d (x, y, t) = 1 if and only if d(x, y) < t, then, if t = 0, d(x, y) < 0 cannot be fulfilled for any x, y ∈ X and R(x, y, 0) = 0. However, we still want to invent an axiom for R d that is equivalent to the axiom d(x, y) = 0 if and only if x = y. The axiom is: Let us prove that this axiom is equivalent to the axiom d(x, y) = 0 ⇐⇒ x = y: If d(x, y) = 0, then obviously R d (x, y, t) = 1 for all t > 0, and from (1), it follows that x = y. If x = y, then from (1), R d (x, y, t) = 1 for all t > 0, which means d(x, y) < t for all t > 0, and then d(x, y) = 0. Let us prove the opposite. If R d (x, y, t) = 1 for all t > 0, then d(x, y) < t for all t > 0, which implies d(x, y) = 0 and, finally, x = y. The opposite direction is also fulfilled.
It is obvious that condition (2) is equivalent to axiom (2) from Definition 6: Inequality (3) comes from the assertion: Thus, axioms (1)-(3) from Definition 6 are equivalent to the following axioms for function R d : The question is whether a metric d on a set X is uniquely determined by a function R : X × X × (0, ∞) → {0, 1}, satisfying for all x, y ∈ X and t ∈ [0, ∞) the three abovementioned conditions.
It is clear that function R, which satisfies the three above-mentioned conditions, is non-decreasing with respect to the third argument: for all t, s > 0. That means that, for the fixed x, y, the value R(x, y, t) = 0 when t is less than or equal to /less than some λ and R(x, y, t) = 1 otherwise. Then, we can define a metric d : X × X → [0, ∞) as d(x, y) = inf{t : R(x, y, t) = 1}. The only thing to take into account is that two functions could define the same metric (if R differs for fixed x, y only in one point); thus, we ask function R to be left-semicontinuous to be in accordance with the condition R(x, y, t) = 1 =⇒ d(x, y) < t. Note that the metric d : X × X → [0, ∞) can also be defined as d(x, y) = sup{t : R(x, y, t) = 0}, which is equal to d(x, y) = max{t : R(x, y, t) = 0}, since R is left-semicontinuous.
Thus, we have the following theorem: Theorem 3. A metric d on a set X is uniquely determined by a function R : X × X × (0, ∞) → {0, 1}, which is left-semicontinuous with respect to the third argument and for which the following conditions are fulfilled for all x, y ∈ X and t, s ∈ (0, ∞): R(x, y, t) = R(y, x, t); 3.

Remark 1.
In the previous theorem, it was sufficient to define the domain of R as X × X × (0, ∞) (not including 0 in the interval (0, ∞)). Intuitively, it could be explained by the fact that d(x, y) cannot be less than 0. On the other hand, this does not prevent us from defining d(x, x). If we still want to work with domain X × X × [0, ∞) for R, we should define R(x, x, 0) for all x ∈ X, since otherwise it could be both 0 and 1. If, in the previous proof, we want to define d(x, y) as sup{t : R(x, y, t) = 0}, we should add the following condition for R: R(x, x, 0) = 0.

Remark 2. The function R
Based on this fact, we will call a function R : X × X × [0, ∞) → {0, 1} as a parametric relation.
From the above theorems, we obtain the following principal result: we obtain a parametric relation satisfying properties (1) According to the definition of nonexpansive, continuous, and uniformly continuous functions in terms of metric spaces (X, d), it is possible to define these functions in terms of spaces (X, R), where R satisfies properties (1)-(3) of Theorem 3 and is left-semicontinuous. In the next propositions, we suppose that (X 1 , R d 1 ) and (X 2 , R d 2 ) are spaces isomorphic to (X 1 , d 1 ) and (X 2 , d 2 ) in the sense of Corollary 1, (d 1 (x, y) = inf{t : R d 1 (x, y, t) = 1} and d 2 (x, y) = inf{t : R d 1 (x, y, t) = 1}): is nonexpansive if and only if, for every pair of points x and y in X 1 , it holds that:

Proposition 2. A function
is continuous if and only if, for every x ∈ X 1 and every ε > 0, there exists δ > 0 such that, for every point y in X 1 , it holds that: is uniformly continuous if and only if, for every ε > 0, there exists δ > 0 such that, for every pair of points x and y in X 1 , it holds that: The proof of the previous three propositions relies on the direct application of nonexpansive, continuous, and uniformly continuous functions in terms of metric spaces (X, d) and Theorem 3. It is possible to study categorical aspects of metric spaces in terms of metrics defined by relations, but we left the study of this topic for the future.
We continue in this paper to explain the definition of commonly used fuzzy metrics by extending the definition of a metric space in terms of relation R taking values in unit interval [0, 1].

Fuzzy Metrics
Now we can use the last theorem from the previous section to define a fuzzy metric expanding the set {0, 1} to the interval [0, 1] and using arbitrary t-norm T instead of the minimum t-norm that was used in the previous section:
M(x, y, t) = 1 for all t > 0 if and only if x = y; 2.
In the case of Definition 7, the nondecreasing condition is fulfilled in the case of any tnorm T, and the condition lim t→∞ M(x, y, t) = 1 is skipped by other authors since it comes from the statistical metric spaces and does not play any role in the context of fuzzy sets.

Example 1.
These examples fulfill fuzzy metrics axioms (0)-(2) and (4) and axiom (3) for the corresponding t-norm and for any crisp metric d that is used for the construction: 1.
Axiom (3) is fulfilled for any t-norm T.
Defining fuzzy metrics in this way, we should clearly understand that the value M(x, y, t) shows the degree to which d(x, y) < t for a metric d, which is explained by the roots of this definition proposed in the previous section.
The conditions (0) and (1) are quite strong especially when they are used together. Condition (0) shows that M(x, y, 0) = 0 since, for any metric and for all x, y ∈ X, condition d(x, y) < 0 is not fulfilled, i.e., d(x, y) ≥ 0 for any x, y ∈ X. In the fuzzy sense, this leads to the assumption that if d(x, y) ≥ t, then M(x, y, t) should be always 0. The condition (1) leads to the assumption that, if d(x, y) < t, then M(x, y, t) = 1, but it is not clear why it is fulfilled only in the case x = y. Both assumptions together lead us to the crisp case explained in the previous section.
M(x, y, t) = 1 if and only if x = y; 2.
In this definition, the authors do not allow function M to take the value 0 and allow it to take the value 1 only when x = y: M(x, x, t) = 1.
These requirements are quite strong. Additionally, in using this definition, it is impossible to construct a crisp metric d from the function M even if we use the definition of fuzzy linear order R, where R(d(x, y), t) = M(x, y, t). This means that it is not clear which metric d the fuzzy metric M fuzzifies.
To overcome the problem of revealing the metric d that is fuzzified by M, we propose two approaches. The first approach is to define the fuzzy metric as the function M : Let us come back to our initial idea of defining a metric through an order, but this time in a fuzzy sense. We first introduce a definition of a compatible fuzzy relation with an order ≤. This property can be interpreted as follows: if we have a three-element chain a < b < c, then the degree that a < c is greater then the degree of a < b and of b < c.
The next theorem shows that it is enough for a fuzzy relation R : X × X → [0, 1], defined as R(d(x, y), t) for a metric d, to be compatible with ≤ (where ≤ is a linear order on S = [0, ∞)) to fulfill the axioms from Definition 7. Thus, we do not need to require T-transitivity of the fuzzy relation R.
, is left-semicontinuous with respect to the second argument, and satisfies conditions R(a, t) = 1 ∀t > 0 ⇐⇒ a = 0 is a fuzzy metric.
M(x, y, t) = R(d(x, y), t) = 1 for all t > 0 if and only if d(x, y) = 0, but that it is fulfilled if and only if x = y;

Extensional Fuzzy Metrics
In this section, we invite the reader to trace the development of the ideas of the previous sections. Whereas in the previous section we fuzzified the statement d(x, y) < t, where d is a metric and t ∈ [0, ∞), here we explain the idea of fuzzification of the statement d(x, y) = t.
Consider a metric space (X, d) and a T-equivalence relation E. We define a fuzzy metric as an extension of the given metric d with respect to a T-equivalence relation E on the set [0, ∞) (codomain of the metric d). In the definition of an extensional fuzzy metric, we use a strongly linear T-E-order on [0, ∞), defined as: Thus, whereas in the previous section we used a fuzzy order relation, here we rely on a fuzzy equivalence relation. This approach has been developed in [32]; we outline here the main ideas to illustrate the approach and the logical development of the ideas of the previous section.
We propose to define a fuzzy metric as the degree to which the observed distance d(x, y) between points x and y is equal to the real number t, or equal in a certain fuzzy sense determined by fuzzy equivalence E. That is, we define a fuzzy metric (called the E-d-metric) as a mapping M Ed : X × X × [0, ∞) → [0, 1] as follows: Definition 10. Let d be a crisp metric on a set X, t ∈ [0, ∞) and E be a fuzzy T-equivalence. Let a mapping M Ed : X × X × [0, ∞) → [0, 1] be defined as: The fuzzy set M Ed is called an extensional fuzzy metric determined by metric d and fuzzy equivalence E or E-d-metric if the following condition is satisfied: T(E(d(x, y), t), E(d(y, z), s)) ≤ R E (d(x, z), t + s).
Condition (8) shows that d(x, y) = t and that d(y, z) = s implies d(x, z) ≤ t + s in a certain fuzzy sense. In other words, it is a fuzzy version of the triangular inequality.
If we have a crisp fuzzy equivalence relation: and corresponding T-E-order, then condition (8) holds for any t-norm T and metric d; it actually follows from the triangular inequality of the metric d. We finish this section by noting the fact that the inequality (8) is quite natural and is fulfilled for Archimedean t-norms automatically. Theorem 6. Let T be a continuous Archimedean t-norm, and let T-equivalence be defined by: where g is an additive generator of t-norm T. Then, the condition T(E(d(x, y), t), E(d(y, z), s)) ≤ R E (d(x, z), t + s) is fulfilled for any metric d. • M E L d (x, y, t) = E L (d(x, y), t) = max(1 − |d(x, y) − t|, 0) in the case of T, which is the Łukasiewicz t-norm; • M E P d (x, y, t) = E P (d(x, y), t) = e −|d(x,y)−t| in the case of T, which is the product t-norm; 1+|d(x,y)−t| in the case of T, which is the Hamacher t-norm.

Conclusions
In this paper, we explained the definitions of fuzzy metrics used in the literature and analyzed which notions they fuzzified. This explanation is important for finding possible applications, since in applying fuzzy constructions, we should clearly understand the essence of the construction. Thus, we draw the reader's attention to the fact that the classical fuzzy metric definition directly arises from the fuzzification of the expression d(x, y) < t. To explain this, first we studied which properties fulfill the crisp relation R : X × X × [0, ∞) → {0, 1} to uniquely define a crisp metric d : X × X → [0, ∞). Then, we fuzzified these conditions in order to obtain a fuzzy metric. Namely, we allowed relation R to be fuzzy or to take values in the interval [0, 1], and we used the t-norm as a generalized conjunction instead of a crisp conjunction. Then, we analyzed the obtained conditions. Since relation R in the crisp case determines a metric d and is defined as R(x, y, t) = 1 if and only if d(x, y) < t, we invite the reader to understand that, in Kramosil-Michalek, Grabisch, and George-Veeramani fuzzy metric cases, the expression d(x, y) < t is fuzzified. We also studied which conditions a fuzzy relation should fulfill in order to determine a fuzzy metric. For completeness, we recalled and revised the definition of a fuzzy metric that fuzzifies the expression d(x, y) = t, as introduced in [32].