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Functor of Idempotent Probability Measures with Compact Support and Open Mappings

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Abstract

In this paper, we show that the functor of idempotent probability measures with compact support acting in the category of Tikhonov spaces and their continuous mappings is normal. It is found that this functor is monodic. Further, it is proved that the functor of idempotent probability measures with compact support preserves the openness of continuous mappings of Tikhonov spaces.

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References

  1. Yu. Al’-Kassas, “Metrizability and paracompactness of spaces of probability measures,” Thesis, MSU, Moscow (1991).

  2. J. P. Aubin, Dynamic Economic Theory: A Viability Approach, Springer–Verlag, Berlin–Heidelberg (1997).

    Book  Google Scholar 

  3. A. Ch. Chigogidze, “On extension of normal functors,” Vestn. Mosk. Un-ta. Ser. 1. Mat. Mekh., No. 6, 23–26 (1984).

  4. S. Eilenberg and J. Moore, “Adjoint functors and triples,” Ill. J. Math., 9, No. 3, 381–398 (1965).

    MathSciNet  Google Scholar 

  5. V. V. Fedorchuk, “Multivalued retractions and characterization of n-soft mappings,” Tr. Mosk. Mat. Ob-va, 51, 169–207 (1988).

    Google Scholar 

  6. V. V. Fedorchuk, “Probability measures in topology,” Usp. Mat. Nauk, 46, No. 1, 41–80 (1991).

    MathSciNet  Google Scholar 

  7. V. V. Fedorchuk and Yu. V. Sadovnichiy, “On some topological and categorical properties of alternating measures,” Fundam. i Prikl. Mat., 5, No. 2, 597–618 (1999).

    MathSciNet  Google Scholar 

  8. V. V. Fedorchuk and V. V. Filippov, General Topology. Basic Constructions [in Russian], MGU, Moscow (1988).

  9. A. Ya. Ishmetov, “On the functor of compactly supported idempotent probability measures,” Uzb. Mat. Zh., No. 1, 72–80 (2010).

  10. V. N. Kolokol’tsov, “Idempotent structures in optimization,” Itogi Nauki i Tekhn. Ser. Sovrem. Mat. i Ee Pril., 65, 118–174 (1999).

  11. V. N. Kolokol’tsov and V. P. Maslov, “Idempotent analysis as an apparatus of control theory. I,” Funkts. Analiz i Ego Prilozh., 23, No. 1, 1–14 (1989).

  12. V. N. Kolokol’tsov and V. P. Maslov, “Idempotent analysis as an apparatus of control theory and optimal synthesis. II,” Funkts. Analiz i Ego Prilozh., 23, No. 4, 53–62 (1989).

  13. G. L. Litvinov, V. P. Maslov, and G. B. Shpiz, “Idempotent functional analysis. Algebraic approach,” Mat. Zametki, 69, No. 5, 758–797 (2001).

    MathSciNet  Google Scholar 

  14. V. P. Maslov and V. N. Kolokol’tsov, Idempotent Analysis and Its Application in Optimal Control [in Russian], Fizmatlit, Moscow (1994).

  15. T. N. Radul, “On the functor of order-preserving functionals,” Comment. Math. Univ. Carolin., 39, No. 3, 609–615 (1998).

    MathSciNet  Google Scholar 

  16. T. Radul, “Idempotent measures: absolute retracts and soft maps,” ArXiv, 1810.09140v1 [math.GN] (2018).

  17. E. V. Shchepin, “Functors and uncountable powers of compacta,” Usp. Mat. Nauk, 36, No. 3, 3–62 (1981).

    Google Scholar 

  18. A. D. Taymanov, “On closed mappings,” Mat. Sb., 46, 349–352 (1955).

    Google Scholar 

  19. A. A. Zaitov, “On extension of weakly additive functionals,” Dokl. AN RUz, No. 5, 3–7 (2005).

    Google Scholar 

  20. A. A. Zaitov, “An open mapping theorem for spaces of weakly additive homogeneous functionals,” Mat. Zametki, 88, No. 5, 683–688 (2010).

    MathSciNet  Google Scholar 

  21. A. A. Zaitov, “Geometric and topological properties of the subspace Pf (X) of probability measures,” Izv. Vuzov. Ser. Mat., No. 10, 28–37 (2019).

  22. A. A. Zaitov, “Order-preserving variants of the basic principles of functional analysis,” Fundam. J. Math. Appl., 2, No. 1, 10–17 (2019).

    Google Scholar 

  23. A. A. Zaitov, “On a metric on the space of idempotent probability measures,” Appl. Gen. Topol., 21, No. 1, 35–51 (2020).

    Article  MathSciNet  Google Scholar 

  24. A. A. Zaitov and A. Ya. Ishmetov, “On the monad generated by the functor Iβ,” Vestn. NUUz, No. 2, 61–64 (2013).

    Google Scholar 

  25. A. A. Zaitov and A. Ya. Ishmetov, “Homotopy properties of the space If (X) of idempotent probability measures,” Mat. Zametki, 106, No. 4, 531–542 (2019).

    Article  Google Scholar 

  26. A. A. Zaitov and Kh. F. Kholturaev, “On the relationship of functors P of probability measures and I idempotent probability measures,” Uzb. Mat. Zh., No. 4, 36–45 (2014).

  27. A. A. Zaitov and I. I. Tozhiev, “Functional representations of closed subsets of a compactum,” Uzb. Mat. Zh., No. 1, 53–63 (2010).

  28. M. M. Zarichnyi, “Spaces and mappings of idempotent measures,” Izv. RAN. Ser. Mat., 74, No. 3, 45–64 (2010).

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Tashkent Institute of Architecture and Civil Engineering, Tashkent, Uzbekistan

    A. Ya. Ishmetov

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  1. A. Ya. Ishmetov
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Correspondence to A. Ya. Ishmetov.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 67, No. 4, Science — Technology — Education — Mathematics — Medicine, 2022.

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Ishmetov, A.Y. Functor of Idempotent Probability Measures with Compact Support and Open Mappings. J Math Sci 278, 463–475 (2024). https://doi.org/10.1007/s10958-024-06933-4

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