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## Archive of Issues

Russia Yekaterinburg
Year
2014
Issue
1
Pages
87-101
 Section Mathematics Title Some ultrafilter properties connected with extension constructions Author(-s) Chentsov A.G.a Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa Abstract General properties of ultrafilters of $\pi$-systems with zero and unit used under extension constructing for abstract attainability problems with the aim of estimation for attraction sets in topological space are considered. Possibilities of employment of the above-mentioned ultrafilters as general elements are considered. Among them, elements admissible with respect to constraints of asymptotic character of the initial problem are selected. Under very general conditions, the goal operator of the given problem extends to the continuous mapping that takes each ultrafilter of $\pi$-system to the limit of corresponding image. The basic attraction set (an asymptotic analog of the attainability domain) is estimated from below by the continuous image of an analogous auxiliary set in the space of ultrafilters. In the particular case of realization of the Stone space (when the used $\pi$-system is an algebra of sets) the above-mentioned estimate is an equality connecting a desired attraction set and an auxiliary one; for the latter a sufficiently simple representation is given. The variant of application (in estimating goals) of the Wallman extension is discussed. Keywords attraction set, topology, ultrafilter UDC 519.6 MSC 28A33 DOI 10.20537/vm140108 Received 15 January 2014 Language Russian Citation Chentsov A.G. Some ultrafilter properties connected with extension constructions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 1, pp. 87-101. References Warga J. Optimal'noe upravlenie differentsial'nymi i funktsional'nymi uravneniyami (Optimal control of differential and functional equations), Moscow: Nauka, 1977, 624 p. Duffin R.J. Infinite programs, Linear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton Univ. Press, 1956, pp. 157-170. Gol'shtein E.G. Teoriya dvoistvennosti v matematicheskom programmirovanii i ee prilozheniya (Duality theory in mathematical programming and its applications), Moscow: Nauka, 1971, 352 p. Chentsov A.G. Certain constructions of asymptotic analysis related to the Stone-Ccech compactification, Journal of Mathematical Sciences, 2007, vol. 140, issue 6, pp. 873-904. Chentsov A.G. Finitely additive measures and extensions of abstract control problems, Journal of Mathematical Sciences, 2006, vol. 133, no. 2, pp. 1045-1206. Krasovskii N.N. Teoriya upravleniya dvizheniem (Theory of motion control), Moscow: Nauka, 1968, 475 p. Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial'nye igry (Positional differential games), Moscow: Nauka, 1974, 456 p. Bulinskii A.V., Shiryaev A.N. Teoriya sluchainykh protsessov (Theory of stochastic processes), Moscow: Fizmatlit, 2005, 402 p. Chentsov A.G. Filters and ultrafilters in the constructions of attraction sets, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 1, pp. 113-142 (in Russian). Chentsov A.G. Ultrafilters in the constructions of attraction sets: Problem of compliance to constraints of asymptotic character, Differential Equations, 2011, vol. 47, issue 7, pp. 1059-1076. Chentsov A.G. Attraction sets in abstract attainability problems: Equivalent representations and basic properties, Russian Mathematics, 2013, vol. 57, issue 11, pp. 28-44. Chentsov A.G. Tiered operators and conversion based on ultrafilters, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2012, vol. 18, no. 4, pp. 298-314 (in Russian). Chentsov A.G., Morina S.I. Extensions and relaxations, Dordrecht-Boston-London: Kluwer Academic Publishers, 2002. Chentsov A.G. Asymptotic attainability, Dordrecht-Boston-London: Kluwer Academic Publishers, 1997, 322 p. Burbaki N. Obshchaya topologiya. Osnovnye struktury (General topology. The basic structures), Moscow: Nauka, 1968, 272 p. Kelli J.L. Obshchaya topologiya (General topology), Moscow: Nauka, 1981, 433 p. Chentsov A.G. Ultrafilters of measurable spaces as generalized solutions to abstract problems of attainability, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2011, vol. 17, no. 1, pp. 268-293 (in Russian). Gryzlov A.A., Bastrykov E.S., Golovastov R.A. About points of compactification of $\mathbb N$, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2010, no. 3, pp. 10-17 (in Russian). Gryzlov A.A., Golovastov R.A. The Stone spaces of Boolean algebras, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2013, no. 1, pp. 11-16 (in Russian). Chentsov A.G. The transformation of ultrafilters and their application in constructions of attraction sets, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2012, no. 3, pp. 85-102 (in Russian). Full text