Document Type: Research Paper
Authors
- Ildar Sadeqi ^{1}
- Farnaz Yaqub Azari ^{} ^{2}
^{1} Department of Mathematics, Faculty of Science, Sahand University of Technology, Tabriz, Iran.
^{2} University of Payame noor, Tabriz, Iran.
Abstract
In this paper, we formalize the Menger probabilistic normed space as a category in which its objects are the Menger probabilistic normed spaces and its morphisms are fuzzy continuous operators. Then, we show that the category of probabilistic normed spaces is isomorphicly a subcategory of the category of topological vector spaces. So, we can easily apply the results of topological vector spaces in probabilistic normed spaces.
Keywords
- Category of probabilistic normed space
- Category of topological vector space
- Fuzzy continuous operator
Main Subjects
[1] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, Fuzzy Math. 11 (2003) 687-705.
[2] G. Constantin and I. Istratfescu, Elements of probabilistic analysis, Kluwer Academic Publishers, 1989.
[3] P. Freyd, Abelian categories, An Introduction to the theory of functors, Happer & Row, New York, Evanston & London and John Weatherhill, INC, Tokyo, 1964.
[4] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64 (1994) 395-399.
[5] U. Hohle, A note on the hyporgraph functor, Fuzzy Sets and Systems, 131 (2002) 353-356.
[6] O. Kaleva and S. Seikala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984) 143-154.
[7] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika, 11 (1975) 326-334.
[8] K. Menger, Statistical metrics. Proc. Nat. Acad. Sci, USA, 28 (1942) 53-57.
[9] S.E. Rodabaugh, Fuzzy addition in the L-fuzzy real line, Fuzzy Sets and Systems, 8 (1982) 39-51.
[10] S.E. Rodabaugh, Complete fuzzy topological hyperelds and fuzzy multiplication in the fuzzy real lines, Fuzzy Sets and Systems, 15 (1985) 285-310.
[11] S.E. Rodabaugh, A theory of fuzzy uniformities with applications to the fuzzy real lines, J. Math. Anal. Appl., 129 (1988) 37-70.
[12] W. Rudin, Functional Analysis, Tata McGraw-Hill Publishing Company, 1990.
[13] I. Sadeqi and F. Solaty kia, Fuzzy normed linear space and it's topological structure, Chaos, fractal, solution & Fractals, 40 (2007) 2576-2589.
[14] I. Sadeqi and F. Solaty kia, The category of fuzzy normed linear space, The journal of fuzzy mathematics, 3 (2010) 733-742.
[15] I. Sadeqi, F. Solaty kia and F. Yaqub azari, Menger probabilistic normed linear spaces and its topological structure, Fuzzy Inteligent and Systems, (On published data).
[16] E.S. Santos, Topology versus fuzzy topology, preprint, Youngstown State University, 1977.
[17] B. Schweizer and A. Sklar, Probablisitic metric spaces, North-Holand, Amesterdam, 1983.