Divisibility in β N and *N

• Autorzy: Šobot Boris

Identyfikatory

DOI

10.4467/20842589RM.19.003.10651

• Języki publikacji: EN

Abstrakty

EN

The paper first covers several properties of the extension of the divisibility relation to a set ∗ N of nonstandard integers, including an analogue of the basic theorem of arithmetic. After that, a connection is established with the divisibility in the Stone-Čech compactification βN, proving that the divisibility of ultrafilters introduced by the author is equivalent to divisibility of some elements belonging to their respective monads in an enlargement. Some earlier results on ultrafilters on lower levels on the divisibility hierarchy are illuminated by nonstandard methods. Using limits by ultrafilters we obtain results on ultrafilters above these finite levels, showing that for them a distribution by levels is not possible.

Słowa kluczowe

EN

divisibility; nonstandard integer; Stone-Čech compactification; ultrafilter

• Wydawca: Wydawnictwo Uniwersytetu Jagiellońskiego
• Czasopismo: Reports on Mathematical Logic
• Rocznik: 2019
• Tom: Vol. 54
• Strony: 65--82
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Šobot Boris

• Department of Mathematics and Informatics, Faculty of Sciences University of Novi Sad Trg Dositeja Obradovica 4, 21000 Novi Sad, Serbia

Bibliografia

• [1] V. Benci, M. Di Nasso, M. Forti, Hausdorff nonstandard extensions, Bol. Soc. Parana. Mat. 20 (2002), 9–20.
• [2] M. Di Nasso, On the foundations of nonstandard mathematics, Sci. Math. Jpn. 50:1 (1999), 131–160.
• [3] M. Di Nasso, M. Forti, Topological and nonstandard extensions, Monatsh. Math. 144 (2005), 89–112.
• [4] M. Di Nasso, M. Forti, Hausdorff ultrafilters, Proc. Amer. Math. Soc. 134:6 (2006), 1809–1818.
• [5] C.W. Henson, Foundations of nonstandard analysis, In: Nonstandard Analysis: Theory and Applications, L.O. Arkeryd et al. (eds.), Kluwer Academic Publishers, 1997.
• [6] N. Hindman, D. Strauss, Algebra in the Stone-Čech compactification, theory and applications, 2nd revised and extended edition, De Gruyter, 2012.
• [7] W.A.J. Luxemburg, A general theory of monads, In: Applications of Model Theory to Algebra, Analysis and Probability, W.A.J. Luxemburg (ed.), Holt, Rinehart and Winston, 1968, pp. 18–86.
• [8] S.-A. Ng, H. Render, The Puritz order and its relationship to the Rudin-Keisler order, In: Reuniting the antipodes – Constructive and nonstandard views of the continuum, P. Schuster, U. Berger, H. Osswald (eds.), Kluwer Academic Publishers, 2001, pp. 157–166.
• [9] C. Puritz, Skies, constellations and monads, in: Contributions to non-standard analysis, W.A.J. Luxemburg, A. Robinson (eds.), North Holland, 1972, pp. 215–243.
• [10] B. Šobot, Divisibility in the Stone- Čech compactification, Rep. Math. Logic 50 (2015), 53–66.
• [11] B. Šobot, ¬| -divisibility of ultrafilters, submitted, arXiv: 1703.05999.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).