ON INTERRELATIONSHIPS BETWEEN FUZZY METRIC STRUCTURES

Document Type: Research Paper

Authors

1 Department of Statistics and Operations Research, University of Jaen, Campus Las Lagunillas, s/n, E-23071, Jaen, Spain

2 Department of Mathematics, University of Jaen, Campus Las Lagunillas, s/n, E-23071, Jaen, Spain

3 Department of Statistics and Operations Research, University of Granada, Campus Fuentenueva s/n, E-18071, Granada, Spain

Abstract

Considering the increasing interest in fuzzy theory and possible applications,
the concept of fuzzy metric space concept has been introduced by several
authors from different perspectives. This paper interprets the theory in terms
of metrics evaluated on fuzzy numbers and defines a strong Hausdorff topology.
We study interrelationships between this theory and other fuzzy theories such
as intuitionistic fuzzy metric spaces, Kramosil and Michalek's spaces, Kaleva
and Seikkala's spaces, probabilistic metric spaces, probabilistic
metric co-spaces, Menger spaces and intuitionistic probabilistic metric
spaces, determining their position in the framework of theses different theories.

Keywords


[1] H. Adibi, Y. J. Cho, D. O'Regan and R. Saadati, Common xed point theorems in L-fuzzy
metric spaces, Appl. Math. Comput., 182 (2006), 820{828.
[2] C. Alaca, D. Turkoglu and C. Yildiz, Fixed points in intuitionistic fuzzy metric spaces, Chaos
Soliton Fract, 29 (2006), 1073{1078.
[3] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), 87{96.
[4] F. Castro-Company and S. Romaguera, Experimental results for information system based
on accesses locality via intuicionist fuzzy metrics, Open Cybern. Syst. J., 2 (2008), 158{172.
[5] Y. J. Cho, M. T. Grabiec and V. Radu, On nonsymmetric topological and probabilistic struc-
tures, Nova Science Publishers, Inc., New York, 2006.
[6] Z. Deng, Fuzzy pseudo metric spaces, J. Math. Anal. Appl., 86 (1982), 74{95.
[7] G. Deschrijver and E. E. Kerre, On the position of intuitionistic fuzzy set theory in the
framework of theories modelling imprecision, Information Sciences, 177 (2007), 1860{1866.
[8] D. Dubois and H. Prade, Operations on fuzzy numbers, Int. J. Syst. Sci., 9(6) (1978), 613{
626.
[9] M. S. El Naschie, On the veri cations of heteritic strings theory and (1) theory, Chaos
Soliton Fract, 11(2) (2000), 2397{2407.
[10] M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl., 69 (1979), 205{230.
[11] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems,
64 (1994), 395{399.
[12] D. Gomez, J. Montero and J. Ya~nez, A coloring fuzzy graph approach for image classi cation,
Information Sciences, 176(24) (2006), 3645{3657.
[13] M. T. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 64 (1994),
395{399.
[14] M. T. Grabiec, Y. J. Cho and R. Saadati, Families of quasi-pseudo-metrics generated by
probabilistic quasi-pseudo-metric spaces, Surveys in Mathematics and its Applications, 2
(2007), 123{143.
[15] V. Gregori, S. Morillas and A. Sapena, Examples of fuzzy metrics and applications, Fuzzy
Sets and Systems, 170 (2011), 95{111.
[16] V. Gregori, S. Romaguera and P. Veereamani, A note on intuitionistic fuzzy metric spaces,
Chaos Soliton Fract, 28 (2006), 902{905.
[17] H. L. Huang and F. G. Shi, L-fuzzy numbers and their properties, Information Sciences, 178
(2008), 1141{1151.
[18] H. Huang and C. Wu, On the triangle inequalities in fuzzy metric spaces, Information Sciences,
177(4) (2007), 1063{1072.
[19] M. Imdad, J. Ali and M. Hasan, Common xed point theorems in modi ed intuitionistic fuzzy
metric spaces, Iranian Journal of Fuzzy Systems, 9(5) (2012), 77-92.
[20] O. Kaleva, On the convergence of fuzzy sets, Fuzzy Sets Syst., 17(1985), 53{65.
[21] O. Kaleva, A comment on the completion of fuzzy metric spaces, Fuzzy Sets and Systems,
159 (2008), 2190{2192.
[22] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984),
215{229.
[23] I. Kramosil and J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika 11
(1975), 336{344.
[24] H. Y. Li, CLM-Fuzzy topological spaces, Iranian Journal of Fuzzy Systems, to appear.

[25] J. Martnez-Moreno, A. Roldan and C. Roldan, A note on the L-fuzzy Banach's contraction
principle, Chaos Soliton Fract, 41(5) (2009), 2399{2400.
[26] J. Martnez-Moreno, A. Roldan and C. Roldan, KM-Fuzzy approach space, Proceedings of
the International Fuzzy Systems Association World Conference, (2009), 1702{1705.
[27] K. Menger, Statistical metrics, Proc National Acad Sci of the United States of America, 28
(1942), 535{537.
[28] N. N. Morsi, On fuzzy pseudo-normed vector spaces, Fuzzy Sets and Systems, 27 (1988),
351{372.
[29] E. Pap, Pseudo-analysis and nonlinear equations, Soft Comput., 6 (2002), 21{32.
[30] J. H. Park, Intuitionistic fuzzy metric spaces, Chaos Soliton Fract, 22 (2004), 1039{1046.
[31] S. Romaguera and P. Tirado, Contraction maps on IFQM-spaces with application to recur-
rence equations of quicksort, Electronic Notes in Theoretical Computer Science, 225 (2009),
269{279.
[32] R. Saadati, Notes to the paper xed points in intuitionistic fuzzy metric spaces" and its
generalization to L-fuzzy metric spaces, Chaos Soliton Fract, 35 (2008), 176{180.
[33] R. Saadati and J. H. Park, On the intuitionistic fuzzy topological spaces, Chaos Soliton Fract,
27 (2006), 331{44.
[34] R. Saadati, S. Mansour Vaezpour and Y. J. Cho, Quicksort algorithm: application of a xed
point theorem in intuitionistic fuzzy quasi-metric spaces at a domain of words, J. Comput.
Appl. Math., 228 (2009), 219{225.
[35] R. Saadati, A. Razani and H. Adibi, A common xed point theorem in L-fuzzy metric spaces,
Chaos Soliton Fract, 33 (2007), 358{363.
[36] R. Saadati, S. Sedghi and N. Shobe, Modi ed intuitionistic fuzzy metric spaces and some
xed point theorems, Chaos Soliton Fract, 38 (2008), 36{47.
[37] B. Schweizer and A. Sklar, Probabilistic metric spaces, Dover Publications, New York, 2005.
[38] F. G. Shi, (L;M)-Fuzzy metric spaces, Indian Journal of Mathematics, 52(2) (2010), 231{
250.
[39] F. G. Shi, Regularity and normality of (L;M)-Fuzzy topological spaces, Fuzzy Sets and Systems,
182 (2011), 37{52.
[40] L. H. Son, B. C. Cuong, P. L. Lanzi and N. T. Thong, A novel intuitionistic fuzzy clustering
method for geo-demographic analysis, Expert Syst. Appl., doi:10.1016/j.eswa.2012.02.167,
(2012).
[41] P. Tirado, On compactness and G-completeness in fuzzy metric spaces, Iranian Journal of
Fuzzy Systems, 9(4) (2012), 151-158.
[42] Z. S. Xu, J. Chen and J. Wu, Clustering algorithm for intuitionistic fuzzy sets, Information
Sciences, 178 (2008), 3775{3790.
[43] Z. Xu, A method based on distance measure for interval-valued intuitionistic fuzzy group
decision making, Information Sciences, 180 (2010), 181{190.
[44] Y. Xu and H. Wang, The induced generalized aggregation operators for intuitionistic fuzzy
sets and their application in group decision making, Appl. Soft. Comput., 12 (2012), 1168{
1179.
[45] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{353.
[46] D. Zhang, A natural topology for fuzzy numbers, J. Math. Anal. Appl., 264(2) (2001), 344{
353.
[47] S. F. Zhang and S. Y. Liu, A GRA-based intuitionistic fuzzy multi-criteria group decision
making method for personnel selection, Expert Syst. Appl., 38 (2011), 11401{11405.
[48] Z. Zhanga, J. Yanga, Y. Yea, Y. Huc and Q. Zhang, A type of score function on intuitionistic
fuzzy sets with double parameters and its application to pattern recognition and medical
diagnosis, Procedia Engineering, 29 (2012), 4336{4342.
[49] Z. Zhao and C. Wu, The equivalence of convergences of sequences of fuzzy numbers and
its applications to the characterization of compact sets, Information Sciences, 179 (2009),
3018{3025.