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Algebraic characterizations of some relative notions of size

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Abstract

We obtain algebraic characterizations of relative notions of size in a discrete semigroup that generalize the usual combinatorial notions of syndetic, thick, and piecewise syndetic sets. “Filtered” syndetic and piecewise syndetic sets were defined and applied earlier by Shuungula et al. (Semigroup Forum 79:531–539, 2009). Other instances of these relative notions of size have appeared explicitly (and more often implicitly) in the literature related to the algebraic structure of the Stone–Čech compactification. Building on this prior work, we observe a natural duality and demonstrate how these notions of size may be composed to characterize previous notions of size (like piecewise syndetic sets) and serve as a convenient description for new notions of size.

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References

  1. Akin, E.: Recurrence in Topological Dynamics: Furstenberg Families and Ellis Actions. The University Series in Mathematics. Springer, Boston (1997). https://doi.org/10.1007/978-1-4757-2668-8

    Book  MATH  Google Scholar 

  2. Bergelson, V.: Ultrafilters, IP sets, dynamics, and combinatorial number theory. In: Ultrafilters across mathematics, Contemporary Mathematics, vol. 530, pp. 23–47. American Mathematical Society, Providence, RI (2010). https://doi.org/10.1090/conm/530/10439

  3. Bergelson, V., Furstenberg, H., Hindman, N., Katznelson, Y.: An algebraic proof of van der Waerden’s theorem. Enseign. Math. 35(3–4), 209–215 (1989). https://doi.org/10.5169/seals-57373

    Article  MathSciNet  MATH  Google Scholar 

  4. Bergelson, V., Hindman, N., McCutcheon, R.: Notions of size and combinatorial properties of quotient sets in semigroups. Topol. Proc. 23, 23–60 (1998)

    MathSciNet  MATH  Google Scholar 

  5. Bergelson, V., Rosenblatt, J.: Mixing actions of groups. Illinois J. Math. 32(1), 65–80 (1988). https://doi.org/10.1215/ijm/1255989229

    Article  MathSciNet  MATH  Google Scholar 

  6. Berglund, J.F., Hindman, N.: Filters and the weak almost periodic compactification of a discrete semigroup. Trans. Am. Math. Soc. 284(1), 1–38 (1984). https://doi.org/10.2307/1999272

    Article  MathSciNet  MATH  Google Scholar 

  7. Blass, A., Di Nasso, M.: Finite embeddability of sets and ultrafilters. Bull. Pol. Acad. Sci. Math. 63(3), 195–206 (2016). https://doi.org/10.4064/ba8024-1-2016

    Article  MathSciNet  MATH  Google Scholar 

  8. Brown, T.: An interesting combinatorial method in the theory of locally finite semigroups. Pac. J. Math. 36(2), 285–289 (1971). https://doi.org/10.2140/pjm.1971.36.285

    Article  MathSciNet  MATH  Google Scholar 

  9. Choquet, G.: Sur les notions de filtre et de grille. C. R. Acad. Sci. Paris 224, 171–173 (1947)

    MathSciNet  MATH  Google Scholar 

  10. Davenport, D.: The minimal ideal of compact subsemigroups of \(\beta \)S. Semigroup Forum 41(2), 201–213 (1990). https://doi.org/10.1007/BF02573391

    Article  MathSciNet  MATH  Google Scholar 

  11. Davenport, D., Hindman, N.: Subprincipal closed ideals in \(\beta N\). Semigroup Forum 36(2), 223–245 (1987). https://doi.org/10.1007/BF02575018

    Article  MathSciNet  MATH  Google Scholar 

  12. Erdös, P., Turán, P.: On some sequences of integers. J. London Math. Soc. 11(4), 261–264 (1936). https://doi.org/10.1112/jlms/s1-11.4.261

    Article  MathSciNet  MATH  Google Scholar 

  13. Furstenberg, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton (1981)

    Book  Google Scholar 

  14. Grimeisen, G.: Gefilterte Summation von Filtern und iterierte Grenzprozesse. I. Math. Ann. 141(4), 318–342 (1960). https://doi.org/10.1007/BF01360766

    Article  MathSciNet  MATH  Google Scholar 

  15. Hindman, N.: Notions of size in a semigroup: an update from a historical perspective. Semigroup Forum 100(1), 52–76 (2020). https://doi.org/10.1007/s00233-019-10041-0

    Article  MathSciNet  MATH  Google Scholar 

  16. Hindman, N., Strauss, D.: Algebra in the Stone-Čech Compactification: Theory and Applications, 2nd edn. De Gruyter Textbook. De Gruyter, Berlin (2012)

    MATH  Google Scholar 

  17. Kakeya, S., Morimoto, S.: On a theorem of MM. Bandet and van der Waerden. Japan. J. Math. 7, 163–165 (1930). https://doi.org/10.4099/jjm1924.7.0_163

    Article  MATH  Google Scholar 

  18. Luperi Baglini, L.: Ultrafilters maximal for finite embeddability. J. Logic Anal. 6, Article no. 6, 1–16 (2014). https://doi.org/10.4115/jla.2014.6.6

  19. Luperi Baglini, L.: \({\cal{F}}\)-finite embeddabilities of sets and ultrafilters. Arch. Math. Logic 55(5–6), 705–734 (2016). https://doi.org/10.1007/s00153-016-0489-4

    Article  MathSciNet  MATH  Google Scholar 

  20. Moreira, J.: Piecewise syndetic sets, topological dynamics and ultrafilters (2016). https://joelmoreira.wordpress.com/2016/04/18/piecewise-syndetic-sets-topological-dynamics-and-ultrafilters/

  21. Papazyan, T.: Filters and semigroup properties. Semigroup Forum 41(3), 329–338 (1990). https://doi.org/10.1007/BF02573399

    Article  MathSciNet  MATH  Google Scholar 

  22. Protasov, I.V.: Selective survey on subset combinatorics of groups. J. Math. Sci. 174(4), 486–514 (2011). https://doi.org/10.1007/s10958-011-0314-x

    Article  MathSciNet  MATH  Google Scholar 

  23. Protasov, I., Slobodianiuk, S.: Relative size of subsets of a semigroup. (2015) https://arxiv.org/abs/1506.00112

  24. Schmidt, J.: Beiträge zur Filtertheorie. II. Math. Nachr. 10(3–4), 197–232 (1953). https://doi.org/10.1002/mana.19530100309

    Article  MathSciNet  MATH  Google Scholar 

  25. Shuungula, O., Zelenyuk, Y., Zelenyuk, Y.: The closure of the smallest ideal of an ultrafilter semigroup. Semigroup Forum 79(3), 531–539 (2009). https://doi.org/10.1007/s00233-009-9173-x

    Article  MathSciNet  MATH  Google Scholar 

  26. Soifer, A.: The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators. Springer, New York, (2009). https://doi.org/10.1007/978-0-387-74642-5

  27. Szemerédi, E.: On sets of integers containing no \(k \) elements in arithmetic progression. Acta Arith. 27, 199–245 (1975). https://doi.org/10.4064/aa-27-1-199-245

    Article  MathSciNet  MATH  Google Scholar 

  28. Tao, T.: What is good mathematics? Bull. Am. Math. Soc. (N. S.) 44(4), 623–635 (2007). https://doi.org/10.1090/s0273-0979-07-01168-8

    Article  MathSciNet  MATH  Google Scholar 

  29. van der Waerden, B.L.: Beweis einer Baudetschen Vermutung. Nieuw Arch. Wiskunde 15, 212–216 (1927)

    MATH  Google Scholar 

  30. Zucker, A.: Thick, syndetic, and piecewise syndetic subsets of Fraïssé structures. Topol. Appl. 223, 1–12 (2017). https://doi.org/10.1016/j.topol.2017.03.009

    Article  MATH  Google Scholar 

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Acknowledgements

We gratefully acknowledge and thank Florian Richter for several important discussions along with valuable feedback and suggestion on earlier drafts of this article. We thank the referee for a careful reading and several suggestions that improved the exposition. We thank Jessica Christian for feedback on drafts of the introduction. We thank Baglini, Blass, and Di Nasso for helpful conversations on their work related to finite embeddability, and we also thank Anush Tserunyan and Andy Zucker for helpful discussions and interest. Finally, we also thank Vitaly Bergelson and Neil Hindman for helpful correspondence.

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Correspondence to John H. Johnson Jr..

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Communicated by Jimmie D. Lawson.

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Christopherson, C., Johnson, J.H. Algebraic characterizations of some relative notions of size. Semigroup Forum 104, 28–44 (2022). https://doi.org/10.1007/s00233-021-10215-9

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