Fuzzy hyperspase monad

A.Savchenko

The hyperspace of a fuzzy metric space is defined by J. Rodríguez-López and S. Romaguera. In this paper, it is shown that the hyperspace construction determines a functor on the category of fuzzy metric spaces and nonexpanding maps. We also prove that this functor determines a monad on this category and that the $G$-symmetric power functor can be extended over the Kleisli category of this monad.

1. Almanzor Sapena, A contribution to the study of fuzzy metric spaces, Applied General Topology. 2 (2001), No. 1, 63-75.

2. J.W. de Bakker, J.I. Zucker, Processes and denotational semantics of concurrency, Information and Control, 54 (1982), 70-120.

3. M. Barr, Ch. Wells, Toposes, triples and theories, Springer-Verlag, Berlin, 1985.

4. A. George, P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems, 90 (1997), 365-368.

5. V. Gregori, S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy sets and systems, 115 (2000), 485-489.

6. V. Gregori, S. Romaguera, P. Veeramani, A note on intuitionistic fuzzy metric spaces, Chaos, Solitons \& Fractals, 28 (2006), No. 4, 902-905.

7. O. Hubal, M. Zarichnyi, Idempotent probability measures on ultrametric spaces, J. Math. Anal. Appl. 343 (2008), 1052-1060.

8. J.R. Isbell, Uniform Spaces, Mathematical Surveys, V.12, Providence, R.I., 1964.

9. O. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica. 11 (1975), 326-334.

10. D. Mihe t, Fuzzy $\psi$-contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets and Systems archive. 159 (2008), No.6, 739-744.

11. Sam B. Nadler, Hyperspaces of sets. Monographs and Textbooks on Pure and Applied Mathematics, No. 49, Marcel Dekker, Inc., New York, 1978, xvi + 707 pp.

12. J.H. Park, Intuitionistic fuzzy metric spaces. Chaos, Solitons \& Fractals, 22 (2004), 1039-1046.

13. K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, 1967.

14. Mohd. Rafi Segi Rahmat, Mohd. Salmi Md. Noorani, Product of Fuzzy Metric Spaces and Fixed Point Theorems, Int. J. Contemp. Math. Sciences. 3 (2008), No.15, 703-712.

15. J. Rodríguez-López, S. Romaguera, The Hausdorff fuzzy metric on compact sets, Fuzzy Sets and Systems, 147 (2004), No.2. 273-283.

16. S. Romaguera, A. Sapena, P. Tiradoi, The Banach fixed point theorem in fuzzy quasi-metric spaces with application to the domain of words, Topology and its Applications 154 (2007), 2196-2203.

17. J. M. Harvey, Categorical characterization of uniform hyperspaces, Math. Proc. Cambridge Philos. Soc., 94 (1983), No.2, 229-233.

18. P. Veeramani, Best approximation in fuzzy metric spaces, J. Fuzzy Math. 9 (2001), 75-80.

19. V. V. Fedorchuk, Triples of infinite iterates of metrizable functors, Math. USSR Izv. 36 (1991), 411-433.

20. O. Wyler, Algebraic topology of continuous lattices, Lecture Notes in Math., 871 (1981), 390-413.

21. M.M. Zarichnyi, Characterization of the $G$-symmetric power functors and extension of functors to the Kleisli categories, Matem. zametki. 52 (1992), No.5, 42-48.

	Fuzzy hyperspase monad
Author	A.Savchenko savchenko1960@rambler.ru Kherson Agrarian University
Abstract	The hyperspace of a fuzzy metric space is defined by J. Rodríguez-López and S. Romaguera. In this paper, it is shown that the hyperspace construction determines a functor on the category of fuzzy metric spaces and nonexpanding maps. We also prove that this functor determines a monad on this category and that the $G$-symmetric power functor can be extended over the Kleisli category of this monad.
Keywords	hyperspace, fuzzy metric space, functor, nonexpanding map, mona, Kleisli category
DOI	doi:10.30970/ms.33.2.192-198
Reference	1. Almanzor Sapena, A contribution to the study of fuzzy metric spaces, Applied General Topology. 2 (2001), No. 1, 63-75. 2. J.W. de Bakker, J.I. Zucker, Processes and denotational semantics of concurrency, Information and Control, 54 (1982), 70-120. 3. M. Barr, Ch. Wells, Toposes, triples and theories, Springer-Verlag, Berlin, 1985. 4. A. George, P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems, 90 (1997), 365-368. 5. V. Gregori, S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy sets and systems, 115 (2000), 485-489. 6. V. Gregori, S. Romaguera, P. Veeramani, A note on intuitionistic fuzzy metric spaces, Chaos, Solitons \& Fractals, 28 (2006), No. 4, 902-905. 7. O. Hubal, M. Zarichnyi, Idempotent probability measures on ultrametric spaces, J. Math. Anal. Appl. 343 (2008), 1052-1060. 8. J.R. Isbell, Uniform Spaces, Mathematical Surveys, V.12, Providence, R.I., 1964. 9. O. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica. 11 (1975), 326-334. 10. D. Mihe t, Fuzzy $\psi$-contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets and Systems archive. 159 (2008), No.6, 739-744. 11. Sam B. Nadler, Hyperspaces of sets. Monographs and Textbooks on Pure and Applied Mathematics, No. 49, Marcel Dekker, Inc., New York, 1978, xvi + 707 pp. 12. J.H. Park, Intuitionistic fuzzy metric spaces. Chaos, Solitons \& Fractals, 22 (2004), 1039-1046. 13. K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, 1967. 14. Mohd. Rafi Segi Rahmat, Mohd. Salmi Md. Noorani, Product of Fuzzy Metric Spaces and Fixed Point Theorems, Int. J. Contemp. Math. Sciences. 3 (2008), No.15, 703-712. 15. J. Rodríguez-López, S. Romaguera, The Hausdorff fuzzy metric on compact sets, Fuzzy Sets and Systems, 147 (2004), No.2. 273-283. 16. S. Romaguera, A. Sapena, P. Tiradoi, The Banach fixed point theorem in fuzzy quasi-metric spaces with application to the domain of words, Topology and its Applications 154 (2007), 2196-2203. 17. J. M. Harvey, Categorical characterization of uniform hyperspaces, Math. Proc. Cambridge Philos. Soc., 94 (1983), No.2, 229-233. 18. P. Veeramani, Best approximation in fuzzy metric spaces, J. Fuzzy Math. 9 (2001), 75-80. 19. V. V. Fedorchuk, Triples of infinite iterates of metrizable functors, Math. USSR Izv. 36 (1991), 411-433. 20. O. Wyler, Algebraic topology of continuous lattices, Lecture Notes in Math., 871 (1981), 390-413. 21. M.M. Zarichnyi, Characterization of the $G$-symmetric power functors and extension of functors to the Kleisli categories, Matem. zametki. 52 (1992), No.5, 42-48.
Pages	192-198
Volume	33
Issue	2
Year	2010
Journal	Matematychni Studii
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