Uniqueness of monads for the capacity functor and its subfunctors

O.R.Nykyforchyn

It is proved that the capacity functor in the category of compacta and two its subfunctors, namely the functors of cap-capacities (also called necessity measures) and cup-capacities (also known as sup-measures or possibility measures) are functorial parts of unique monads that admit a special kind of automorphisms.

1. G.L. O'Brien, W. Verwaat, How subsadditive are subadditive capacities?, Comment. Math. Univ. Carolinae, 35 (1994), №2, 311-324.

2. G. Choquet, Theory of Capacity, Ann. l'Institute Fourier, 5 (1953-1954), 131-295.

3. I.D. Hlushak, O.R. Nykyforchyn, Submonads of the capacity monad, Carpathian J.\ of Math. 24 (2008), №1, 56-67.

4. S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer-Verlag, N.Y., 1998.

5. O.R. Nykyforchyn, On the axiom of continuity, openness and bicommutativity, Meth. Func. Anal. and Top., 4 (1998), 82-85.

6. Nykyforchyn O.R., A monad for the inclusion hyperspace functor is unique, Mat. Studii, 27 (2007), 3-18.

7. O.R. Nykyforchyn, Capacities with values in compact Hausdorff lattices, Applied Categorical Structures, 15 (2008), №3, 243-257.

8. D. Schmeidler, Subjective probability and expected utility without additivity, Econometrica, 57 (1989), 571-587.

9. M.M. Zarichnyi, O.R. Nykyforchyn, Capacity functor in the category of compacta, Sb. Math., 199 (2008), №2, 159-184.

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

11. М.М. Заричный, Монада суперрасширения и ее алгебры, Укр. матем. журнал, 39 (1987), №3, 303-309.

12. Т.Н. Радул, Монада гиперпространств включения и ее алгебры, Укр. матем. журнал, 42 (1990), №6, 806-812.

13. В.В. Федорчук, Мягкие отображения, многозначные ретракции и функторы, Успехи матем. наук, 41 (1986), №6, 121-159.

14. В.В. Федорчук, Вероятностные меры в топологии, Успехи матем. наук, 46 (1991), №1, 41-80.

15. В.В. Федорчук, В.В. Филиппов, Общая топология: основные конструкции, М.: Изд-во МГУ, 1988.

	Uniqueness of monads for the capacity functor and its subfunctors
Author	O.R.Nykyforchyn Vasyl' Stefanyk Precarpathian National University ,Department of Mathematics and Computer Science Shevchenka,57, Ivano-Frankivsk, Ukraine
Abstract	It is proved that the capacity functor in the category of compacta and two its subfunctors, namely the functors of cap-capacities (also called necessity measures) and cup-capacities (also known as sup-measures or possibility measures) are functorial parts of unique monads that admit a special kind of automorphisms.
Keywords	functor, category, compacta, cap-capacity, necessity measure, cup-capacity, sup-measures, possibility measure
DOI	doi:10.30970/ms.33.1.11-16
Reference	1. G.L. O'Brien, W. Verwaat, How subsadditive are subadditive capacities?, Comment. Math. Univ. Carolinae, 35 (1994), №2, 311-324. 2. G. Choquet, Theory of Capacity, Ann. l'Institute Fourier, 5 (1953-1954), 131-295. 3. I.D. Hlushak, O.R. Nykyforchyn, Submonads of the capacity monad, Carpathian J.\ of Math. 24 (2008), №1, 56-67. 4. S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer-Verlag, N.Y., 1998. 5. O.R. Nykyforchyn, On the axiom of continuity, openness and bicommutativity, Meth. Func. Anal. and Top., 4 (1998), 82-85. 6. Nykyforchyn O.R., A monad for the inclusion hyperspace functor is unique, Mat. Studii, 27 (2007), 3-18. 7. O.R. Nykyforchyn, Capacities with values in compact Hausdorff lattices, Applied Categorical Structures, 15 (2008), №3, 243-257. 8. D. Schmeidler, Subjective probability and expected utility without additivity, Econometrica, 57 (1989), 571-587. 9. M.M. Zarichnyi, O.R. Nykyforchyn, Capacity functor in the category of compacta, Sb. Math., 199 (2008), №2, 159-184. 10. Lin Zhou, Integral representation of continuous comonotonically additive functionals, Trans. Amer. Math. Soc. 350 (1998), №5, 1811-1822. 11. М.М. Заричный, Монада суперрасширения и ее алгебры, Укр. матем. журнал, 39 (1987), №3, 303-309. 12. Т.Н. Радул, Монада гиперпространств включения и ее алгебры, Укр. матем. журнал, 42 (1990), №6, 806-812. 13. В.В. Федорчук, Мягкие отображения, многозначные ретракции и функторы, Успехи матем. наук, 41 (1986), №6, 121-159. 14. В.В. Федорчук, Вероятностные меры в топологии, Успехи матем. наук, 46 (1991), №1, 41-80. 15. В.В. Федорчук, В.В. Филиппов, Общая топология: основные конструкции, М.: Изд-во МГУ, 1988.
Pages	11-16
Volume	33
Issue	1
Year	2010
Journal	Matematychni Studii
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