Answering a recent question posed by Gregori et al. [On a class of completable fuzzy metric spaces, Fuzzy Sets and Systems 161 (2010), 2193-2205] we present two examples of non-strong fuzzy metrics (in the sense of George and Veeramani). © 2010 Elsevier B.V. All rights reserved.This research was supported by the Ministry of Science and Innovation of Spain under Grants MTM2009-12872-C02-01 and MTM2009-12872-C02-02. J. Gutierrez Garcia also acknowledges financial support from the University of the Basque Country under Grant GIU07/27.Gutiérrez García, J.; Romaguera Bonilla, S. (2011). Examples of non-strong fuzzy metrics. Fuzzy Sets and Systems. 162(1):91-93. https://doi.org/10.1016/j.fss.2010.09.017S9193162

George

Gregori

J. Gutiérrez García

S. Romaguera

English

Answering a recent question posed by Gregori et al. [On a class of completable fuzzy metric spaces, Fuzzy Sets and Systems 161 (2010), 2193-2205] we present two examples of non-strong fuzzy metrics (in the sense of George and Veeramani). © 2010 Elsevier B.V. All rights reserved.This research was supported by the Ministry of Science and Innovation of Spain under Grants MTM2009-12872-C02-01 and MTM2009-12872-C02-02. J. Gutierrez Garcia also acknowledges financial support from the University of the Basque Country under Grant GIU07/27.Gutiérrez García, J.; Romaguera Bonilla, S. (2011). Examples of non-strong fuzzy metrics. Fuzzy Sets and Systems. 162(1):91-93. https://doi.org/10.1016/j.fss.2010.09.0179193162

Gutiérrez García, J.

Romaguera Bonilla, Salvador

RiuNet

 Document downloaded from:  This paper must be cited as:                     The final publication is available at   Copyright http://dx.doi.org/10.1016/j.fss.2010.09.017http://hdl.handle.net/10251/46839ElsevierGutiérrez García, J.; Romaguera Bonilla, S. (2011). Examples of non-strong fuzzy metrics.Fuzzy Sets and Systems. 162(1):91-93. doi:10.1016/j.fss.2010.09.017.Examples of non strong fuzzy metricsIJ. Gutie´rrez Garcı´aa,∗, S. RomaguerabaDepartamento de Matema´ticas, Universidad del Paı´s Vasco-Euskal Herriko Unibertsitatea,Apdo. 644, 48080, Bilbao, SpainbInstituto Universitario de Matema´tica Pura y Aplicada, Universidad Polite´cnica de Valencia, 46071, Valencia, SpainAbstractAnswering a recent question posed by Gregori, Morillas and Sapena (“On a class of completablefuzzy metric spaces”, Fuzzy Sets and Systems, 161 (2010), 2193–2205) we present two examplesof non strong fuzzy metrics (in the sense of George and Veeramani).Keywords: Fuzzy metric space, Strong fuzzy metric2000 MSC: 54A40, 54E35, 54E70The theory of fuzzy metric completion (in the sense of George and Veeramani [1]) is very dif-ferent from the classical theories of metric completion and probabilistic metric completion sincethere are fuzzy metric spaces which are not completable. The authors of [3] became interested instrong fuzzy metrics when looking for a class of completable fuzzy metrics. In [3] they providean example of a non-strong fuzzy metric for the minimum t-norm and state the open question tofind a non-strong fuzzy metric for another t-norm different from the minimum.In what follows we answer this question and present two examples of non-strong fuzzy met-rics for the product and the Łukasiewicz t-norm, respectively. Terms and undefined conceptsmay be found in [3].Definition 1. ([1]) A fuzzy metric on a (nonempty) set X is a pair (M, ∗) such that M is a fuzzyset on X × X × (0,+∞) and ∗ is a continuous t-norm satisfying the following conditions:(FM1) M(x, y, t) > 0 for all x, y ∈ X and all t > 0;(FM2) M(x, y, t) = 1 for t > 0 if and only if x = y;(FM3) M(x, y, t) = M(y, x, t) for all x, y ∈ X and all t > 0;(FM4) M(x, z, t + s) ≥ M(x, y, t) ∗ M(y, z, s) for all x, y, z ∈ X and t, s > 0;(FM5) M(x, y, ·) is continuous for each x, y ∈ X.IThis research was supported by the Ministry of Science and Innovation of Spain under grants MTM2009-12872-C02-01 and MTM2009-12872-C02-02. J. Gutie´rrez Garcı´a also acknowledges financial support from the University ofthe Basque Country under grant GIU07/27.∗Corresponding authorEmail addresses: javier.gutierrezgarcia@ehu.es (J. Gutie´rrez Garcı´a), sromague@mat.upv.es (S.Romaguera)URL: http://www.ehu.es/javiergutierrezgarcia/ (J. Gutie´rrez Garcı´a)Note that condition (FM4) implies that M(x, y, ·) is non-decreasing for each x, y ∈ X.A special type of fuzzy metrics has been recently considered in [3] and [2]:Definition 2. ([3]) Let (X, M, ∗) be a fuzzy metric space. M is said to be strong if it satisfies thefollowing additional axiom(FM4’) M(x, z, t) ≥ M(x, y, t) ∗ M(y, z, t) for all x, y, z ∈ X and t > 0.Clearly, if d is a metric on a set X, then the fuzzy metric (Md, ∗) is strong for every continuoust-norm ∗ such that ∗ ≤ ·, where Md is defined by Md(x, y, t) = t/(t + d(x, y)), for all x, y ∈ X andt > 0.In Section 3 of [3] (see also [2]) the authors observed that, however, the fuzzy metric (Md,∧)is strong if and only if d is an ultrametric. Then, they posed the natural question of finding a nonstrong fuzzy metric (M, ∗) where ∗ is not ∧.The following examples solve this question.Example 1. Let X = {x, y, z}, ∗ = · and M : X × X × (0,+∞)→ [0, 1] defined for each t > 0 asM(x, x, t) = M(y, y, t) = M(z, z, t) = 1,M(x, z, t) = M(z, x, t) = M(y, z, t) = M(z, y, t) =tt + 1,M(x, y, t) = M(y, x, t) =t2(t + 2)2.It is easy to check that M satisfies (FM1), (FM2), (FM3) and (FM5). With respect to (FM4), wehave thatM(x, y, t + s)−M(x, z, t)∗M(z, y, s) = (t + s)2(t + s + 2)2− tt + 1· ss + 1=(t − s)2(s + t + 1)(t + s + 2)2(t + 1)(s + 1)≥ 0Hence M(x, y, t + s) ≥ M(x, z, t) ∗ M(z, y, s). AlsoM(x, z, t + s) =t + st + s + 1≥ ss + 1= M(z, y, s) ≥ M(x, z, t) ∗ M(z, y, s)M(y, z, t + s) =t + st + s + 1≥ tt + 1= M(y, x, t) ≥ M(y, x, t) ∗ M(x, z, s).Consequently, (M, ·) is a fuzzy metric.However, for each t > 0 we have thatM(x, y, t) =t2(t + 2)2<t2(t + 1)2= M(x, z, t) ∗ M(z, y, t).It follows that (M, ·) is not strong.Example 2. Let X = {x, y, z}, ∗ = TL (the Łukasiewicz t-norm), and M : X×X×(0,+∞)→ [0, 1]defined for each t > 0 asM(x, x, t) = M(y, y, t) = M(z, z, t) = 1,M(x, z, t) = M(z, x, t) = M(y, z, t) = M(z, y, t) =2t + 12t + 2,M(x, y, t) = M(y, x, t) =tt + 2.2It is easy to check that M satisfies (FM1), (FM2), (FM3) and (FM5). With respect to (FM4), wehave thatM(x, y, t + s) − M(x, z, t) ∗ M(z, y, s) = t + st + s + 2−max{2t + 12t + 2+2s + 12s + 2− 1, 0}=2(t − s)2(t + s + 2)(2t + 2)(2s + 2)≥ 0.Hence M(x, y, t + s) ≥ M(x, z, t) ∗ M(z, y, s). AlsoM(x, z, t + s) =2(t + s) + 12(t + s) + 2≥ 2s + 12s + 2= M(z, y, s) ≥ M(x, z, t) ∗ M(z, y, s)M(y, z, t + s) =2(t + s) + 12(t + s) + 2≥ 2t + 12t + 2= M(y, x, t) ≥ M(y, x, t) ∗ M(x, z, s).Consequently, (M,TL) is a fuzzy metric.However, for each t > 0 we have thatM(x, y, t) =tt + 2<tt + 1=2t + 12t + 2+2t + 12t + 2− 1 = M(x, z, t) ∗ M(z, y, t).It follows that (M,TL) is not strong.Conclusion and perspectivesIn this short note we answer a question posed in [3] and present two examples of non strongfuzzy metrics (in the sense of [1]). Note that in our first example we have used the product t-norm. In this direction it would be interesting to find further examples for continuous t-normsthat are greater than the product but different from minimum.References[1] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994), 395–399.[2] A. Sapena, S. Morillas, On strong fuzzy metrics, Proceedings of the Workshop in Applied Topology WiAT09:“Applied Topology: Recent progress for Computer Science, Fuzzy Mathematics and Economics”, 135–141.[3] V. Gregori, S. Morillas, A. Sapena, On a class of completable fuzzy metric spaces, Fuzzy Sets and Systems 161(2010), 2193–2205.3

Examples of non-strong fuzzy metrics

Answering a recent question posed by Gregori et al. [On a class of completable fuzzy metric spaces, Fuzzy Sets and Systems 161 (2010), 2193-2205] we present two examples of non-strong fuzzy metrics (in the sense of George and Veeramani). © 2010 Elsevier B.V. All rights reserved.This research was supported by the Ministry of Science and Innovation of Spain under Grants MTM2009-12872-C02-01 and MTM2009-12872-C02-02. J. Gutierrez Garcia also acknowledges financial support from the University of the Basque Country under Grant GIU07/27.Gutiérrez García, J.; Romaguera Bonilla, S. (2011). Examples of non-strong fuzzy metrics. Fuzzy Sets and Systems. 162(1):91-93. doi:10.1016/j.fss.2010.09.017S9193162

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Examples of non-strong fuzzy metrics

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