A New Approach to Fuzzy Metric Spaces and Their Similarity-Based Construction

Soylu, Gültekin; Güloğlu, Mutlu

doi:https://doi.org/10.1155/2016/2945210

Journal of Function Spaces

On this page

Abstract Introduction Preliminaries Conclusions References Copyright Related Articles

Research Article | Open Access

Volume 2016 | Article ID 2945210 | https://doi.org/10.1155/2016/2945210

A New Approach to Fuzzy Metric Spaces and Their Similarity-Based Construction

Gültekin Soylu¹and Mutlu Güloğlu¹

Academic Editor: Hugo Leiva

Received13 Jan 2016

Accepted23 Mar 2016

Published26 May 2016

Abstract

We introduce a space of functions which can be interpreted as a similarity-based approach to fuzzy metric spaces. The triangle inequality we propose is defined by means of a fuzzy ordering. We compare the introduced space with fuzzy metric spaces in the sense of Seikkala and Kaleva. Finally we complete the work discovering the corresponding classical as well as fuzzy topologies.

1. Introduction

The theory of metric spaces, first introduced by Fréchet in 1906 [1], stems from the idealization of the notion of distance. In a metric space the distance between two points of a set is represented by a single nonnegative real number. However, in many cases of physical or other scientific problems of interest, it seems to be too ideal to associate the distance between two points of a set with a single nonnegative real number. Most probably this fact inspired Menger in 1942 [2] to generalize the concept of a metric space in a statistical manner. He was modeling the distance by a probability of being less than a value. Wald [3] and later Schweizer and Sklar [4] made worthful contributions to the theory of statistical metric spaces. Motivated by these works in 1975 Kramosil and Michalek [5] introduced an equivalent notion, namely, the fuzzy metric space. They were modeling the case where the inexactness of the distance is due to incomplete information rather than randomness. A lot of work is done on this subject with small modifications especially by George and Veeramani [6] and by Gregori and Romaguera [7]. However, Kaleva and Seikkala [8] felt that (and the authors of this paper share this feeling) it is a more natural way to define the fuzzy metric space by setting the distance between two points to be a nonnegative fuzzy number. Following this latter approach, in this work we introduce the similarity-based fuzzy metric spaces. We try to model the case where the inexactness of the distance between two points is due to the indistinguishability of the measured values. Kaleva and Seikkala propose a triangular inequality on the fuzzy numbers which employs both the addition of fuzzy numbers and a crisp partial ordering between them. The fuzzy triangular inequality we propose employs the notion of fuzzy orderings rather than crisp orderings. In fact our definition of fuzzy triangular inequality is just a rendering of the crisp one to the semantics of -norm fuzzy logic. This seems to be advantageous in the theoretic point of view because this new concept is in harmony with formerly defined similarity-based fuzzy concepts such as fuzzy orderings and fuzzy limits. The smooth connection between those concepts is a result of the underlying -norm logic in common and is therefore promising for possible further works related to the introduced fuzzy metric. On the other hand from the practical point of view a fuzzy metric in our setting is easy to understand and very easy to construct: all you need to start with is a fuzzy set. Nevertheless the proposed approach is not completely different from its predecessors; the close relation between our proposal and the one of Kaleva and Seikkala is also presented in the work. At the end of the paper, for the sake of completeness of the work, we establish the corresponding classical and fuzzy topologies to a given similarity-based fuzzy metric.

2. Preliminaries

For simplicity in this paper the set of truth degrees is considered as the unit interval and is denoted by . The set of all real numbers is shown by and the symbol will denote a left-continuous -norm [9].

The support of a fuzzy subset of a set is defined as the crisp set, , and is denoted by .

The core of a fuzzy subset of a set is defined as the crisp set, , and is denoted by .

A fuzzy subset of a set is called to be normal if there exists with (i.e., is nonempty).

Let be a fuzzy subset of the space ; then we say that is fuzzy convex [10] if , for all , and all .

Suppose that is a topological space, is a point in , and is a function. We say that is upper semicontinuous at if for every there exists a neighborhood of such that for all .

A fuzzy number [11] is a fuzzy subset of the real line , that is, a mapping which is normal, fuzzy convex, and upper semicontinuous, and is bounded. The set of all fuzzy numbers will be denoted by .

The -level sets of a fuzzy number are denoted by and are defined by , .

A fuzzy subset of is fuzzy convex if and only if each of its -level sets is a convex set in . The upper semicontinuity condition guarantees that the -level set of a fuzzy number is a closed interval . One should note that the intervals and real numbers can be embedded in since every interval can be represented by a fuzzy number defined byand, similarly, every real number can be represented by a fuzzy number defined by

If for all then is called a nonnegative fuzzy number and the set of all nonnegative fuzzy numbers will be denoted by .

For a choice of three real numbers , and with a triangular fuzzy number is represented by and is defined by

A map is called a -similarity relation on iff the following three axioms are satisfied:(E.1), ,(E.2), ,(E.3), .A -similarity relation on is said to be separated (i.e., a -equality relation) iff the implication is satisfied for all .

Although fuzzy orderings related to an underlying -similarity relation were first defined by Höhle and Blanchard [12] and developed by researchers like Bělohlávek [13], we will use throughout this paper Bodenhofer’s contribution [14, 15] to this subject.

Definition 1. Let be a -similarity on a set ; a fuzzy relation is called an -partial ordering on if it satisfies the following properties: (EPO.1) , (EPO.2) , (EPO.3) , An -partial ordering on is called a strongly linear -ordering if it satisfies. (EPO.4)

Definition 2. Let be a crisp ordering on a set ; -similarity is called compatible with if and only if the following implication holds for all :

Definition 3. Let be a binary operation on A -similarity is called invariant under if it satisfies the following condition for all :

Example 4. Let be the real line and let the binary operation be the addition. The relations defined by and are -similarities invariant under +, where is the product -norm and the Lukasiewicz -norm, respectively.

In [14] also the following important theorem is proved.

Theorem 5. is a strongly linear -ordering if and only if there exists a linear ordering on that is compatible with such that can be represented as

Throughout the text, whenever a -similarity on compatible with a linear ordering on is given the corresponding -ordering will be assumed to be the one defined in (6).

3. Similarity-Based Fuzzy Metric

The concept of probabilistic metric spaces [4] was introduced to mathematics as a tool which deals with situations where the distance between two points is uncertain. The uncertainty was assumed to be caused by randomness. But later trying to deal with uncertainty caused by incomplete information in [8] the authors have introduced the following definition for a fuzzy metric space.

Let be a nonempty set and a mapping from into and let the mappings be symmetric, nondecreasing in both arguments and satisfy and Denote

The quadruple is called a fuzzy metric space and a fuzzy metric, if(F1),(F2) for all , and all ,(F3)for all ,(a) whenever , , and ,(b) whenever , , and .

In this definition axiom (F3) is a generalization of the triangular inequality and typical examples for the mappings and are -norms and -conorms, respectively. For the particular choices and (F3) is equivalent towhere is Zadeh’s Extension [16] of addition to fuzzy numbers and is a crisp partial ordering on fuzzy numbers [17]. An example for fuzzy metric spaces will be given in the sequel right after Theorem 10. We propose the following definition.

Definition 6. Let be a -equality on and an E-ordering on . A fuzzy set in is called a similarity-based fuzzy metric if it satisfies the following properties:(FM1), ,(FM2), ,(FM3), ,(FM4).The quadruple will be called a similarity-based fuzzy metric space.

This definition is a slight modification of the one in [18]. The axioms given here are the many valued counterparts of the well-known metric axioms. Among these axioms (FM1), (FM2), and (FM3) are direct transformations whereas (FM4) is the -norm logic counterpart of the following reformulation of the triangular inequality.

If the distances between , and , and , are , , and , respectively, then should be less than or equal to . Analogously (FM4) can be interpreted as follows: if it is true to a certain degree that the distances between , and , and , are , , and , respectively, then it should be true at least to the same degree that is less than or equal to . One should note that the -ordering plays an important role in this last axiom: the fact that in the two-valued case (FM4) will be equivalent to the classic triangular inequality is a consequence of the fact that in this case the -ordering becomes a crisp ordering.

In the sequel we will denote for , .

In the following theorem we establish a connection between fuzzy metric spaces and metric spaces. The proof will be in a similar fashion to the proof of Theorem3.2 in [8] in order to show the close relation between the two approaches.

Definition 7. A uniformity for a set is a nonempty family of subsets of which satisfies the following conditions: (i)Each member of contains the diagonal .(ii)If , then .(iii)If , then for some in , where denotes the composite of with itself.(iv)If , then .(v)If and , then .If is a uniformity, then the pair is called uniform space [19].

Theorem 8. A nonempty family of subsets of is a base for some uniformity for if and only if one has the following:(i)Each member of contains the diagonal.(ii)If , then contains a member of .(iii)If , then for some in .(iv)Intersection of two members of contains a member [19].

A space with a uniform topology is a Hausdorff space iff is the diagonal In this case is said to be Hausdorff [19].

Theorem 9. Let be a similarity-based fuzzy metric space, where is a -equality on compatible with , is the corresponding -ordering, and is a nonempty set for any choice of in Then the family of sets forms a basis for a Hausdorff uniformity on and the corresponding uniform topology is metrizable.

Proof. To see that the family of the sets forms a basis for a Hausdorff uniformity on , we will show that the sets satisfy the axioms for a basis for a Hausdorff uniformity [19, pages 174–180] as follows.
We keep in mind that is nonempty and therefore always exists.
(i) Let By (FM3) we obtain that which implies and thereby (ii) Let Since this means that and by (FM2) is symmetric we get . We conclude that is symmetric.
(iii) Let . On the other hand let . So we have and and therefore .
Let ; by (FM4) we obtain and hence . We conclude that .
(iv) Let . Let which implies and therefore and . Hence and We conclude that .
Finally for being Hausdorff we observe the following.
Let ; by (FM1) and (FM3) we have . Setting , we obtain . We deduce that the intersection of the members of the family is the diagonal of .
Now since for a nonnegative sequence converging to the family is a countable basis for the Hausdorff uniformity of discourse, the metrizability of the corresponding uniform topology follows [19, page 186].

The following theorem states the construction of a similarity-based fuzzy metric on a metric space, where is equipped with a -equality.

Theorem 10. Let be a metric space, a -equality on compatible with , invariant under , and the corresponding -ordering on . The fuzzy relation defined by is a similarity-based fuzzy metric on .

Proof. We will show that satisfies the properties (FM1)–(FM4).(FM1)Let and . Suppose that This would yield and since is a -equality. So (FM2)Let .(FM3)Let ; then by definition and since is a -equality we have and this implies Conversely, let which implies and , which in return by definition yields .(FM4)In order to see that satisfies (FM4) it is sufficient to investigate the case where since otherwise the RHS of the inequality would be equal to 1.The rest of the proof consists of some subcases.
(i) Suppose that , , and . Keeping in mind that is the corresponding ordering to the inequality in (FM4) becomesWe exploit the triangular inequality for the metric function , , and obtain the inequality, . Now by -compatibility of we obtain which yields the desired inequality:(ii) Suppose that ; in this case we haveSince is invariant under + and is a symmetric, transitive relation, we can write the following:(iii) We are left with the case where , and (or and which is a symmetric case). (a)We assume Consider Therefore we can write the following: (b)We assume . The proof is identical to (iii)(a).(c)We assume . Here again from -compatibility of we obtain the following inequality: (d)We assume . The proof is identical to (iii)(c).(e)Finally we assume .This time we make use of -compatibility of as follows: Hence,finishes the proof.

Remark 11. It is clear that is a nonempty set for any choice of in and we can conclude that is metrizable.

A result of the famous representation theorem of Valverde [20] is the following.

Let be the Lukasiewicz -norm and a family of fuzzy sets; then

is a -similarity relation.

Combining this result with the preceding theorem we obtain a method for constructing a similarity-based fuzzy metric just by starting with a “sample” fuzzy set. In the following example we use a triangular fuzzy number for this purpose although it is possible to choose from several other types of fuzzy numbers.

Example 12. Let be the Lukasiewicz -norm and let be a given triangular fuzzy number. We define the family to contain all horizontal shifts of the given membership function. That is,Now we construct a -similarity relation by . It is easy to verify that , where . It can also be observed that is a -equality compatible with and invariant under + so we can conclude that , is a similarity-based fuzzy metric.

The intersection of fuzzy metric spaces and similarity-based fuzzy metric spaces is nonempty as it is shown in the following theorem.

Theorem 13. Let be a metric space, a -equality on compatible with , invariant under , and the corresponding -ordering on . The fuzzy relation defined by is both a similarity-based fuzzy metric and a fuzzy metric on .

Proof. We will only show that the fuzzy relation is a fuzzy metric.
The first two properties (F1) and (F2) are easy to observe so let us prove that holds the third property.
We set and , where denotes the binary operation maximum.
(i) Let or . Since one of the conditions of (F3)(b) will not hold (F3)(b) drops. Let us check (F3)(a).
The conditions and of (F3)(a) naturally hold and since either or we obtain(ii) Let and . We will investigate 2 subcases; namely, (or ) and and :(a)Let (or which is the symmetric case). In this case and for (F3)(a) the assumptions imply . The desired inequality can easily be observed since On the other hand for (F3)(b) let and . If we have and therefore If we observe that if we have , , and , and therefore Meanwhile if we have ; on the other hand since we have . Combining these two facts we observe that (b)Let and . In this case if , considering that we have Since and since is compatible with and invariant under + we have and similar to this So we obtain the required inequality by the following: If the fact that and and implies so the desired inequality holds and the proof is completed.

Example 14. Consider the metric space with its standard metric. Let be the product -norm. It is well-known that is a -equality on ; moreover, it is easy to check that is invariant under + and compatible with the ordering on . Finally let be the corresponding -ordering. The fuzzy relation defined by is an example for both a similarity-based fuzzy metric and a fuzzy metric on , where the functions and are chosen to be product and maximum, respectively.

Now we will try to show a connection between fuzzy metric spaces and fuzzy topologies.

One of the possible formulations of the limit of a sequence in a metric space might be given as follows: where is a sequence in

A many-valued counterpart of this relation is given in [18, 21] for the definition of similarity-based fuzzy limits.

Denoting the set by ,

Later in the same works the authors prove that similarity-based fuzzy limits coincide with limit maps of a specific fuzzy topology on

At this point we aim to establish a connection between similarity-based fuzzy metrics and -topologies via similarity-based fuzzy limits; to do this let us first briefly recall some concepts in -topologies.

Let be a nonempty set and complete quasi-monodial lattice; a subfamily of satisfying the following properties is called -topology on and is given by Höhle and Šostak in [22]:(T1) and ,(T2),(T3), where is any index set.

In this definition, choosing and the connective as the -norm remains a special case which is named Chang’s [23] fuzzy topology in former literature but later on called the -topology on (where is not restricted to being the minimum). From now on by -topology we mean the case where and

Starting with a set and a -similarity on it is always possible to obtain a -topology on as shown below.

Let be a set and a similarity on and ; if

then is called -extensional.

The set of all -extensional maps denoted by is a -topology on [24].

In [18, 21] the authors have shown that coincides with the limit map [22] on , where is a -equality on invariant under + and compatible with and is a -similarity relation on defined by

In the following proposition we construct a similarity-based fuzzy limit by the mean of a similarity-based fuzzy metric.

Proposition 15. Let be a metric space, a -equality on compatible with , invariant under , and the corresponding -ordering on . For a sequence in the fuzzy relation defined byis a similarity-based fuzzy limit on

Corollary 16. Let be a metric space, a -equality on compatible with , invariant under , and the corresponding -ordering on ; then coincides with the limit map on

By this corollary can be considered as the fuzzy topology induced by the similarity-based fuzzy metric .

4. Conclusions and Future Works

We have developed a new version of fuzzy metrics by making use of similarity relations. We have shown the relation between our metrics and former ones and we also have proposed some easy constructions of them. Nevertheless the more important thing we observed was the connection of similarity-based fuzzy metrics with fuzzy topologies. This connection yields to the conclusion that the new fuzzy metric seems to fit well in a complete theory, namely, the similarity-based real line, that is the main motivation behind this work and is encouraging further works in this direction. Towards the similarity-based fuzzy real line some concepts already exist in the literature. Among these in this paper we have mentioned similarity-based orderings and similarity-based fuzzy limits; another example is the similarity-based positive class [25]. The next step might immediately be the absolute value and trying to complete the structure, it follows naturally that similarity-based continuity and differentiation will lie at the heart of other future works.

Competing Interests

Hereby the authors declare that there is no person with conflict of interest regarding the publication of this paper.

References

M. M. Fréchet, “Sur quelques points du calcul fonctionnel,” Rendiconti del Circolo Matematico di Palermo, vol. 22, no. 1, pp. 1–72, 1906.
View at: Publisher Site | Google Scholar
K. Menger, “Statistical metrics,” Proceedings of the National Academy of Sciences of the United States of America, vol. 28, pp. 535–537, 1942.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Wald, “On a statistical generalization of metric spaces,” Proceedings of the National Academy of Sciences of the United States of America, vol. 29, pp. 196–197, 1943.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, pp. 313–334, 1960.
View at: Publisher Site | Google Scholar | MathSciNet
I. Kramosil and J. Michalek, “Fuzzy metrics and statistical metric spaces,” Kybernetika, vol. 5, pp. 336–344, 1975.
View at: Google Scholar
A. George and P. Veeramani, “On some results in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 64, no. 3, pp. 395–399, 1994.
View at: Publisher Site | Google Scholar | MathSciNet
V. Gregori and S. Romaguera, “Some properties of fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 115, no. 3, pp. 485–489, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
O. Kaleva and S. Seikkala, “On fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 12, no. 3, pp. 215–229, 1984.
View at: Publisher Site | Google Scholar | MathSciNet
V. Novăk, Fuzzy Sets and Their Applications, Adam Hilger, Bristol, UK, 1989.
L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
D. Dubois and H. Prade, “Operations on fuzzy numbers,” International Journal of Systems Science, vol. 9, no. 6, pp. 613–626, 1978.
View at: Google Scholar | MathSciNet
U. Höhle and N. Blanchard, “Partial ordering in L-underdeterminate sets,” Information Sciences, vol. 35, no. 2, pp. 133–144, 1985.
View at: Publisher Site | Google Scholar | MathSciNet
R. Bělohlávek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic/Plenum Press, New York, NY, USA, 2002.
U. Bodenhofer, “A similarity-based generalization of fuzzy orderings preserving the classical axioms,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 8, no. 5, pp. 593–610, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
U. Bodenhofer, “Representations and constructions of similarity-based fuzzy orderings,” Fuzzy Sets and Systems, vol. 137, no. 1, pp. 113–136, 2003.
View at: Publisher Site | Google Scholar | MathSciNet
L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning-I,” Information Sciences, vol. 8, no. 3, pp. 199–249, 1975.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
M. Mizumoto and K. Tanaka, “Some properties of fuzzy numbers,” in Advances in Fuzzy Set Theory and Applications, pp. 153–164, North-Holland, New York, NY, USA, 1979.
View at: Google Scholar | MathSciNet
G. Soylu, On the vague real line [Ph.D. thesis], Akdeniz Üniversitesi, 2008.
J. L. Kelley, General Topology, Springer, New York, NY, USA, 1975.
View at: MathSciNet
L. Valverde, “On the structure of F-indistinguishability operators,” Fuzzy Sets and Systems, vol. 17, no. 3, pp. 313–328, 1985.
View at: Publisher Site | Google Scholar | MathSciNet
G. Soylu, “Similarity-based fuzzy limits,” Fuzzy Sets and Systems, vol. 159, no. 24, pp. 3380–3387, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
U. Höhle and A. P. Šostak, “Axiomatic foundations of fixed-basis fuzzy topology,” in Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, U. Höhle and S. E. Rodabaugh, Eds., vol. 3 of The Handbooks of Fuzzy Sets Series, pp. 123–272, Kluwer Academic Publishers, Boston, Mass, USA, 1999.
View at: Google Scholar
C. L. Chang, “Fuzzy topological spaces,” Journal of Mathematical Analysis and Applications, vol. 24, pp. 182–190, 1968.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
U. Höhle, “Topological aspects of non-convergent sequences: a comment on Burginfis concept of fuzzy limits,” Fuzzy Sets and Systems, vol. 149, no. 3, pp. 399–412, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
G. Soylu, “E-vague positive class,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 17, no. 1, pp. 137–146, 2009.
View at: Publisher Site | Google Scholar | MathSciNet

Copyright

Copyright © 2016 Gültekin Soylu and Mutlu Güloğlu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

1927

Downloads

1056

Citations