To question about realization of attraction elements in abstract attainability problems

A. G. Chentsov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 2, pp. 212-229

Résumé

An abstract attainability problem under constraints of asymptotic character is considered; the corresponding solution is identified with an attraction set in the class of ultrafilters of the space of ordinary solutions. The remainder of the above-mentioned set with respect to closuring the set of results supplied by precise solutions is investigated (the given notion of a precise solution conceptually corresponds to Warga scheme although it is applied to the case of more general constraints). To represent the above-mentioned (basic) attraction set, the corresponding analog (of the last set) realized in the space of generalized elements is used. For thus obtained auxiliary attraction set, the remainder is analyzed; its connection with the remainder of the basic attraction set is investigated. Conditions of identifying the remainders for basic and auxiliary attraction sets are obtained. General statements are detailed for the case when generalized elements are defined in the form of ultrafilters of widely interpreted measurable spaces where free ultrafilters are responsible for the realization of remainders. It is established that, under existence of a remainder, the set of generalized admissible elements does not coincide with closuring a set of ordinary solutions (this set does not admit standard realization).

Détail

Keywords: remainder, attraction set, ultrafilter.