pqM-spaces; Quasi-pseudo-Menger space; Probabilistic quasi-pseudo-metric spaces (Pp,qM-space); Statistical quasi-metric space (SpqM-space)">

Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843 - 7265 (print)
 Volume 2 (2007), 123 - 143

FAMILIES OF QUASI-PSEUDO-METRICS GENERATED BY PROBABILISTIC QUASI-PSEUDO-METRIC SPACES

Mariusz T. Grabiec, Yeol Je Cho and Reza Saadati

Abstract. This paper contains a study of families of quasi-pseudo-metrics (the concept of a quasi-pseudo-metric was introduced by Wilson [22] , Albert [1] and Kelly [9]) generated by probabilistic quasi-pseudo-metric-spaces which are generalization of probabilistic metric space (PM-space shortly) [2, 3, 4, 6]. The idea of PM-spaces was introduced by Menger [11, 12], Schweizer and Sklar [18] and Serstnev [19]. Families of pseudo-metrics generated by PM-spaces and those generalizing PM-spaces have been described by Stevens [20] and Nishiure [14].

2000 Mathematics Subject Classification: 54E40.
Keywords: Families generated by PpqM-spaces; Quasi-pseudo-Menger space; Probabilistic quasi-pseudo-metric spaces (PpqM-space); Statistical quasi-metric space (SpqM-space).

Full text

References

  1. G.A. Albert, A note on quasi-metric spaces, Bull. Amer. Math. Soc. 47, 479--482 (1941). MR4104(2,320b). Zbl 0027.14203.

  2. T. Birsan, Generation of the probabilistic quasipseudometric spaces, An. Stiint. Univ. Al. I. Cuza Iasi, Sect. I a Mat. (N.S.) 28 (1982), no. 1. 35--44. MR667718 (84f:54045). Zbl 0496.54003.

  3. T. Birsan, Generation of the probabilistic quasipseudometric spaces II, An. Stiint. Univ. Al. I. Cuza Iasi, Sect. I a Mat. 28 (1982). MR717286(85h:54052a). Zbl 0522.54007.

  4. T. Birsan, Sur la decomposition des espaces metriques aleatories, An. Stiint. Univ. Al. I. Cuza Iasi, Sect. I a Mat 29 No. 1 (1983), 33-38. MR717286(85h:54052b). Zbl 0522.54008.

  5. J.B. Brown, On the relationship between Menger spaces and Wald spaces, Colloq. Math. 27, 323--330 (1973). MR331338(48 #9672). Zbl 0263.60003.

  6. M. Grabiec, Fixed points in probabilistic-quasi-metric spaces, Fixed point theory and applications 7, 95--104, (2007). MR2355756(54H25).

  7. G. Gratzer, General Lattice Theory, Akademic-Verlag, Berlin (1978). MR509213(80c:06001b). Zbl 0436.06001.

  8. I. Istratescu, Some remarks on nonarchimedean probabilistic metric spaces, Glasnik Mat. Ser. III 11 (31), 155--161 (1976). MR423314(54 #11293). Zbl 0342.60006.

  9. J.C. Kelly, Bitopological spaces, Proc. London Math. Soc. 13, 71--84 (1963). MR143169(26 #729). Zbl 0107.16401.

  10. E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer, (2000). MR1790096(2002a:03106). Zbl 0972.03002.

  11. K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. USA 28, 535--537 (1942). MR7576(4,163e). Zbl 0063.03886.

  12. K. Menger, Probabilistic theories of relations, Proc. Nat. Acad. Sci. USA 37, 178--180 (1951). MR42080(13,51a). Zbl 0042.37103.

  13. D.H. Mu\v stari, On almost sure convergence in linear spaces of random variables, Theor. Probab. Appl. 15, (1970), 337--342. MR279848(43 #5569). Zbl 0222.60005.

  14. E. Nishiura, Constructive methods in probabilistic metric spaces, Fund. Math. 67, 115--124 (1970). MR259978(41 #4607). Zbl 0201.18601.

  15. V. Radu, Some remarks on the triangle inequality in probabilistic metric spaces, Seminarul de Teoria Probabilitatior si Aplicatii, Universitatea din Timisoara, 1--9 (1986). MR857700(88j:54046). Zbl 0622.60007.

  16. B. Schweizer, A. Sklar, Associative functions and statistical triangle inequalities, Pub. Math. Debrecen 8, 169--186 (1961). MR132939(24 #A2775). Zbl 0107.12203.

  17. B. Schweizer, A. Sklar, Triangle inequalities in a class of statistical metric spaces, J. London Math. Soc. 38, 401--406 (1963). MR174031(30 #4238). Zbl 0136.39301.

  18. B. Schweizer, A. Sklar, Probabilistic Metric Spaces, Nord-Holland (1983). MR790314(86g:54045). Zbl 0546.60010.

  19. A.N. Serstnev, Triangle inequalities for random metric spaces, Kazan. Gos. Univ. Ucen. Zap. 125, 90--93 (1965). MR226691(37 #2278). Zbl 0268.60018.

  20. R.R. Stevens, Metrically generated probabilistic metric spaces, Fund. Math. 61, 259--269 (1968). MR250353(40 #3592). Zbl 0175.46504.

  21. A. Wald, On a statistical generalization of metric spaces, Proc. Nat. Acad. Sci. USA 29, 196--197 (1943). MR7950(4,220b). Zbl 0063.08119.

  22. W.A. Wilson, On quasi-metric-spaces, Amer. J. Math. 53, 675--684 (1931). MR1506845 (Contributed Item). Zbl 0002.05503.

Acknowledgment. The authors would like to thank the referees and the area editor Prof. Barnabas Bede for giving useful comments and suggestions for improving the paper.

Mariusz T. Grabiec Yeol Je Cho
Department of Operation Research, Department of Mathematics and the RINS,
al. Niepodleglosci 10, Gyeongsang National University,
60-967 Poznan, Chinju 660-701,
Poland. Korea.
e-mail: m.grabiec@poczta.onet.pl e-mail: yjcho@gsnu.ac.kr


Reza Saadati
Faculty of Sciences
University of Shomal,
Amol, P.O. Box 731,
Iran.
and
Department of Mathematics and Computer Science,
Amirkabir University of Technology,
424 Hafez Avenue, Tehran 15914,
Iran.
e-mail: rsaadati@eml.cc

http://www.utgjiu.ro/math/sma