Idempotent ultrafilters without Zorn’s Lemma
HTML articles powered by AMS MathViewer
- by Mauro Di Nasso and Eleftherios Tachtsis PDF
- Proc. Amer. Math. Soc. 146 (2018), 397-411 Request permission
Abstract:
We introduce the notion of additive filter and present a new proof of the existence of idempotent ultrafilters on $\mathbb {N}$ without using Zorn’s Lemma in its entire power, and where one only assumes the Ultrafilter Theorem for the continuum.References
- U. Andrews and I. Goldbring, Hindman’s theorem and idempotent types, arXiv:1508.03613, 2015.
- Vitaly Bergelson and Neil Hindman, Partition regular structures contained in large sets are abundant, J. Combin. Theory Ser. A 93 (2001), no. 1, 18–36. MR 1807110, DOI 10.1006/jcta.2000.3061
- Vitaly Bergelson and Neil Hindman, Quotient sets and density recurrent sets, Trans. Amer. Math. Soc. 364 (2012), no. 9, 4495–4531. MR 2922599, DOI 10.1090/S0002-9947-2012-05417-1
- W. W. Comfort, Some recent applications of ultrafilters to topology, General topology and its relations to modern analysis and algebra, IV (Proc. Fourth Prague Topological Sympos., Prague, 1976) Lecture Notes in Math., Vol. 609, Springer, Berlin, 1977, pp. 34–42. MR 0451187
- R. L. Graham and B. L. Rothschild, Ramsey’s theorem for $n$-parameter sets, Trans. Amer. Math. Soc. 159 (1971), 257–292. MR 284352, DOI 10.1090/S0002-9947-1971-0284352-8
- Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer, Ramsey theory, 2nd ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1990. A Wiley-Interscience Publication. MR 1044995
- Eric J. Hall and Kyriakos Keremedis, Čech-Stone compactifications of discrete spaces in ZF and some weak forms of the Boolean prime ideal theorem, Topology Proc. 41 (2013), 111–122. MR 2946795
- Eric J. Hall, Kyriakos Keremedis, and Eleftherios Tachtsis, The existence of free ultrafilters on $\omega$ does not imply the extension of filters on $\omega$ to ultrafilters, MLQ Math. Log. Q. 59 (2013), no. 4-5, 258–267. MR 3100753, DOI 10.1002/malq.201100092
- Horst Herrlich, Kyriakos Keremedis, and Eleftherios Tachtsis, Remarks on the Stone spaces of the integers and the reals without $\textbf {AC}$, Bull. Pol. Acad. Sci. Math. 59 (2011), no. 2, 101–114. MR 2852866, DOI 10.4064/ba59-2-1
- Neil Hindman, The existence of certain ultra-filters on $N$ and a conjecture of Graham and Rothschild, Proc. Amer. Math. Soc. 36 (1972), 341–346. MR 307926, DOI 10.1090/S0002-9939-1972-0307926-0
- Neil Hindman, Finite sums from sequences within cells of a partition of $N$, J. Combinatorial Theory Ser. A 17 (1974), 1–11. MR 349574, DOI 10.1016/0097-3165(74)90023-5
- Neil Hindman, Algebra in the Stone-Čech compactification and its applications to Ramsey theory, Sci. Math. Jpn. 62 (2005), no. 2, 321–329. MR 2179958
- Neil Hindman and Dona Strauss, Algebra in the Stone-Čech compactification, De Gruyter Textbook, Walter de Gruyter & Co., Berlin, 2012. Theory and applications; Second revised and extended edition [of MR1642231]. MR 2893605
- Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, Mathematical Surveys and Monographs, vol. 59, American Mathematical Society, Providence, RI, 1998. With 1 IBM-PC floppy disk (3.5 inch; WD). MR 1637107, DOI 10.1090/surv/059
- Kyriakos Keremedis, Tychonoff products of two-element sets and some weakenings of the Boolean prime ideal theorem, Bull. Pol. Acad. Sci. Math. 53 (2005), no. 4, 349–359. MR 2214925, DOI 10.4064/ba53-4-1
- P. Krautzberger, Idempotent Filters and Ultrafilters, Ph.D. thesis, Freie Universität, Berlin, 2009.
- Talin Papazyan, The existence of almost translation invariant ultrafilters on any semigroup, Proc. Amer. Math. Soc. 107 (1989), no. 4, 1133–1135. MR 987613, DOI 10.1090/S0002-9939-1989-0987613-9
- E. Tachtsis, On the set-theoretic strength of Ellis’ Theorem and the existence of free idempotent ultrafilters on $\omega$, submitted manuscript.
- Stevo Todorcevic, Introduction to Ramsey spaces, Annals of Mathematics Studies, vol. 174, Princeton University Press, Princeton, NJ, 2010. MR 2603812, DOI 10.1515/9781400835409
Additional Information
- Mauro Di Nasso
- Affiliation: Dipartimento di Matematica, Università di Pisa, Pisa 56126, Italy
- MR Author ID: 610241
- Email: mauro.di.nasso@unipi.it
- Eleftherios Tachtsis
- Affiliation: Department of Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece
- MR Author ID: 657401
- Email: ltah@aegean.gr
- Received by editor(s): November 25, 2016
- Received by editor(s) in revised form: February 19, 2017
- Published electronically: August 1, 2017
- Communicated by: Mirna Džamonja
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 397-411
- MSC (2010): Primary 03E25, 03E05, 54D80; Secondary 05D10
- DOI: https://doi.org/10.1090/proc/13719
- MathSciNet review: 3723149