Capacity functor on the category of ultrametric spaces

O.Hubal'

The set of all capacities with compact support on an ultrametric space can be endowed with a natural ultrametric. We show that in that way we obtain a functor in the category of ultrametric spaces and nonexpanding maps. This functor contains the inclusion hyperspace functor as a subfunctor.

1. G. Choquet, Theory of capacities, Annales de l'Institut Fourier, 5:131-295, 1953.

2. R. Engelking, General topology. Warsaw: PWN, 1977, 626p.

3. M. Grabisch, T. Murofushi, and M. Sugeno. Fuzzy Measures and Integrals. Theory and Applications (edited volume). Studies in Fuzziness. Physica Verlag, 2000.

4. O. B. Hubal', M. M. Zarichnyi, Probability measure monad on the category of ultrametric spaces, Appl. Gen. Topol. 9 (2008), no. 2, 229-237.

5. O. Hubal', M. Zarichnyi, Idempotent probability measures on ultrametric spaces, J. Math. Anal. Appl. 343 (2008), no. 2, 1052--1060.

6. S. Mac Lane, Categories for the working mathematician, 2nd ed., Springer-Verlag, N.Y., 1998, 314 p.

7. E. V. Moiseev, Spaces of closed hyperspaces of growth and inclusion, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1988), no. 3, 54--57; (Russian) translation in Moscow Univ. Math. Bull. 43 (1988), no. 3, 48--51

8. T. N. Radul, The monad of inclusion hyperspaces and its algebras, Ukrain. Mat. Zh. 42 (1990), no. 6, 806--811; (Russian) translation in Ukrainian Math. J. 42 (1990), no. 6, 712--716 (1991)

9. E.P. de Vink and J.J.M.M. Rutten, Bisimulation for probabilistic transition systems: a coalgebraic approach, Theoretical Computer Science, 221 (1999), (1/2) 271–-293.

10. M. M. Zarichnyi, O. R. Nikiforchin, Capacity functor in the category of compacta, Mat. Sb. 199 (2008), no. 2, 3--26; (Russian) translation in Sb. Math. 199 (2008), no. 1-2, 159--184.

11. Lin Zhou, Integral representation of continuous comonotonically additive functionals, Trans. Amer. Math. Soc., 350 (1998), no. 5, 1811--1822.

	Capacity functor on the category of ultrametric spaces
Author	O.Hubal' Ivan Franko National University of L'viv
Abstract	The set of all capacities with compact support on an ultrametric space can be endowed with a natural ultrametric. We show that in that way we obtain a functor in the category of ultrametric spaces and nonexpanding maps. This functor contains the inclusion hyperspace functor as a subfunctor.
Keywords	capacity functor, category, ultrametric space, compact support
DOI	doi:10.30970/ms.32.2.132-138
Reference	1. G. Choquet, Theory of capacities, Annales de l'Institut Fourier, 5:131-295, 1953. 2. R. Engelking, General topology. Warsaw: PWN, 1977, 626p. 3. M. Grabisch, T. Murofushi, and M. Sugeno. Fuzzy Measures and Integrals. Theory and Applications (edited volume). Studies in Fuzziness. Physica Verlag, 2000. 4. O. B. Hubal', M. M. Zarichnyi, Probability measure monad on the category of ultrametric spaces, Appl. Gen. Topol. 9 (2008), no. 2, 229-237. 5. O. Hubal', M. Zarichnyi, Idempotent probability measures on ultrametric spaces, J. Math. Anal. Appl. 343 (2008), no. 2, 1052--1060. 6. S. Mac Lane, Categories for the working mathematician, 2nd ed., Springer-Verlag, N.Y., 1998, 314 p. 7. E. V. Moiseev, Spaces of closed hyperspaces of growth and inclusion, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1988), no. 3, 54--57; (Russian) translation in Moscow Univ. Math. Bull. 43 (1988), no. 3, 48--51 8. T. N. Radul, The monad of inclusion hyperspaces and its algebras, Ukrain. Mat. Zh. 42 (1990), no. 6, 806--811; (Russian) translation in Ukrainian Math. J. 42 (1990), no. 6, 712--716 (1991) 9. E.P. de Vink and J.J.M.M. Rutten, Bisimulation for probabilistic transition systems: a coalgebraic approach, Theoretical Computer Science, 221 (1999), (1/2) 271–-293. 10. M. M. Zarichnyi, O. R. Nikiforchin, Capacity functor in the category of compacta, Mat. Sb. 199 (2008), no. 2, 3--26; (Russian) translation in Sb. Math. 199 (2008), no. 1-2, 159--184. 11. Lin Zhou, Integral representation of continuous comonotonically additive functionals, Trans. Amer. Math. Soc., 350 (1998), no. 5, 1811--1822.
Pages	132-138
Volume	32
Issue	2
Year	2009
Journal	Matematychni Studii
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