On fuzzy metric groups. (English) Zbl 0994.54007

Based on a modification of the concept of metric fuzziness given by I. Kramosil and J. Michálek [Kybernetica, Praha 11, 336-344 (1975; Zbl 0319.54002)], A. George and P. Veeramani [Fuzzy Sets Syst. 64, No. 3, 395-399 (1994; Zbl 0843.54014)] introduced and studied a notion of fuzzy metric space which permits to extend several well-known results on complete and compact metric spaces to the fuzzy setting. The work of V. Gregori and S. Romaguera [ibid. 115, No. 3, 485-489 (2000; Zbl 0985.54007)] further created considerable interest in this direction. In this paper the authors show that the fuzzy metric notion studied by George and Veeramani can be used as an efficient tool to obtain a satisfactory notion of a fuzzy metric group. The results proved here extend classical theorems on metric groups. To analyze some properties of the quotient subgroups of a fuzzy metric group the authors propose a weaker notion of fuzzy metric space termed as lower fuzzy metric space. Using this concept the authors obtain some interesting results for lower fuzzy metric groups.


54A40 Fuzzy topology
54H11 Topological groups (topological aspects)
Full Text: DOI


[1] Engelking, R., General Topology (1977), Polish. Sci. Publ: Polish. Sci. Publ Warsaw
[2] George, A.; Veeramani, P., On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64, 395-399 (1994) · Zbl 0843.54014
[3] George, A.; Veeramani, P., On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems, 90, 365-368 (1997) · Zbl 0917.54010
[4] Grabiec, M., Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27, 385-389 (1989) · Zbl 0664.54032
[5] Gregori, V.; Romaguera, S., Some properties of fuzzy metric spaces, Fuzzy Sets and Systems, 115, 485-489 (2000) · Zbl 0985.54007
[6] Kramosil, O.; Michalek, J., Fuzzy metric and statistical metric spaces, Kybernetika, 11, 326-334 (1975)
[7] Roelcke, W.; Dierolf, S., Uniform Structures on Topological Groups and their Quotients (1981), McGraw-Hill: McGraw-Hill New York · Zbl 0489.22001
[8] Schweirzer, B.; Sklar, A., Statistical metric spaces, Pacific J. Math., 10, 314-334 (1960) · Zbl 0091.29801
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