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The strongest t-norm for fuzzy metric spaces. (English) Zbl 1264.54020
Summary: In this paper, we prove that for a given positive continuous t-norm there is a fuzzy metric space in the sense of George and Veeramani, for which the given t-norm is the strongest one. For the opposite problem, we obtain that there is a fuzzy metric space for which there is no strongest t-norm. As an application of the main results, it is shown that there are infinite non-isometric fuzzy metrics on an infinite set.

54A40 Fuzzy topology
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[1] Dombi, J.: A general class of fuzzy operators, the demorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy Set and Systems 8 (1982), 149-163. · Zbl 0494.04005
[2] George, A., Veeramani, P.: On some results in fuzzy metric spaces. Fuzzy Set and Systems 64 (1994), 395-399. · Zbl 0843.54014
[3] Gregori, V., Romaguera, S.: On completion of fuzzy metrics paces. Fuzzy Set and Systems 130 (2002), 399-404. · Zbl 1010.54002
[4] Gregori, V., Morillas, S., Sapena, A.: Examples of fuzzy metrics and applications. Fuzzy Set and Systems 170 (2011), 95-111. · Zbl 1210.94016
[5] Klement, E. P., Mesiar, R., Pap, E.: Triangular norms. Kluwer Academic, Dordrecht 2000. · Zbl 1087.20041
[6] Kramosil, I., Michálek, J.: Fuzzy metric and statistical metric spaces. Kybernetika 11 (1975), 326-334.
[7] Menger, K.: Statistical metrics. Proc. Nat. Acad. Sci. USA 28 (1942), 535-537. · Zbl 0063.03886
[8] Sapena, A.: A contribution to the study of fuzzy metric spaces. Appl. Gen. Topology 2 (2001), 63-76. · Zbl 1249.76040
[9] Schweizer, B., Sklar, A.: Statistical metric spaces. Pacific J. Math. 10 (1960), 314-334. · Zbl 0136.39301
[10] Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North Holland, Amsterdam 1983. · Zbl 0546.60010
[11] Thorp, E.: Best possible triangle inequalities for statistical metric spaces. Proc. Amer. Math. Soc. 11 (1960), 734-740. · Zbl 0125.37002
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