Finite Embeddability of Sets and Ultrafilters

• Autorzy: Blass A. , Di Nasso M.

Identyfikatory

DOI

10.4064/ba8024-1-2016

• Języki publikacji: EN

Abstrakty

EN

A set A of natural numbers is finitely embeddable in another such set B if every finite subset of A has a rightward translate that is a subset of B. This notion of finite embeddability arose in combinatorial number theory, but in this paper we study it in its own right. We also study a related notion of finite embeddability of ultrafilters on the natural numbers. Among other results, we obtain connections between finite embeddability and the algebraic and topological structure of the Stone-Čech compactification of the discrete space of natural numbers. We also obtain connections with nonstandard models of arithmetic.

Słowa kluczowe

EN

ultrafilter; nonstandard models; shift map

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2015
• Tom: Vol. 63, no. 3
• Strony: 195--206
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Blass A.

• Mathematics Department, University of Michigan, Ann Arbor, MI 48109, U.S.A.

autor

Di Nasso M.

• Dipartimento di Matematica, Università di Pisa, 56127 Pisa, Italy

Bibliografia

• [1] M. Beiglböck, An ultrafilter approach to Jin's theorem, Israel J. Math. 185 (2011), 369-374.
• [2] C. C. Chang and H. J. Keisler, Model Theory, 3rd ed., North-Holland, 1990.
• [3] M. Davis, Applied Nonstandard Analysis, Wiley, 1977.
• [4] M. Di Nasso, Embeddability properties of difference sets, Integers 14 (2014), no. A27.
• [5] N. Hindman and D. Strauss, Algebra in the Stone-Čech Compactification. Theory and Applications, 2nd ed., de Gruyter, 2012.
• [6] P. Krautzberger, Idempotent filters and ultrafilters, Ph.D. thesis, Freie Univ. Berlin, 2009.
• [7] L. Luperi Baglini, Hyperintegers and nonstandard techniques in combinatorics of numbers, Ph.D. thesis, Univ. di Siena, 2012.
• [8] L. Luperi Baglini, Ultrafilters maximal for finite embeddability, J. Logic Anal. 6 (2014), no. 6, 1-16.
• [9] I. Z. Ruzsa, On difference sets, Studia Sci. Math. Hungar. 13 (1978), 319-326.