Advertisement

Ukrainian Mathematical Journal

, Volume 67, Issue 3, pp 347–356 | Cite as

Scattered Subsets of Groups

  • T. O. Banakh
  • I. V. Protasov
  • S. V. Slobodianiuk
Article

We define scattered subsets of a group as asymptotic counterparts of the scattered subspaces of a topological space and prove that a subset A of a group G is scattered if and only if A does not contain any piecewise shifted IP -subsets. For an amenable group G and a scattered subspace A of G, we show that μ(A) = 0 for each left invariant Banach measure μ on G. It is also shown that every infinite group can be split into ℵ0 scattered subsets.

Keywords

Cayley Graph Amenable Group Countable Group Injective Sequence Geodesic Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Hindman and D. Strauss, Algebra in the Stone–Čech Compactification, 2nd edn., de Gruyter, Berlin (2012).zbMATHGoogle Scholar
  2. 2.
    S. Todorcevic, Introduction to Ramsey Spaces, Princeton Univ. Press (2010).Google Scholar
  3. 3.
    M. Filali and I. Protasov, “Ultrafilters and topologies on groups,” Math. Stud. Monogr. Ser., VNTL Publ., Lviv, 13 (2010).Google Scholar
  4. 4.
    Y. Zelenyuk, Ultrafilters and Topologies on Groups, de Gruyter, Berlin (2012).Google Scholar
  5. 5.
    H. Dales, A. Lau, and D. Strauss, “Banach algebras on semigroups and their compactifications,” Mem. Amer. Math. Soc., 2005 (2010).Google Scholar
  6. 6.
    I. Protasov and S. Slobodianiuk, “Ultracompanions of subsets of groups,” Comment. Math. Univ. Carolin., 55, No. 2, 257–265 (2014).zbMATHMathSciNetGoogle Scholar
  7. 7.
    I. V. Protasov, “Ultrafilters on metric spaces,” Top. Appl., 164, 207–214 (2014).zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Bella and V. Malykhin, “On certain subsets of a group,” Questions Answers Gen. Top., 17, 183–197(1999).zbMATHMathSciNetGoogle Scholar
  9. 9.
    C. Chou, “On the size of the set of left invariant means on a semigroup,” Proc. Amer. Math. Soc., 23, 199–205 (1969).zbMATHMathSciNetGoogle Scholar
  10. 10.
    I. Lutsenko and I. V. Protasov, “Sparse, thin, and other subsets of groups,” Int. J. Algebra Comput., 19, 491–510 (2009).zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    I. Lutsenko and I. V. Protasov, “Relatively thin and sparse subsets of groups,” Ukr. Math. J., 63, No. 2, 216–225 (2011).MathSciNetCrossRefGoogle Scholar
  12. 12.
    I. Lutsenko and I. V. Protasov, “Thin subsets of balleans,” Appl. Gen. Top., 11, No. 2, 89–93 (2010).zbMATHMathSciNetGoogle Scholar
  13. 13.
    I. Lutsenko, “Thin systems of generators of groups,” Algebra Discrete Math., 9, 108–114 (2010).zbMATHMathSciNetGoogle Scholar
  14. 14.
    O. Petrenko and I. V. Protasov, “Thin ultrafilters,” Note Dome J. Formal Logic, 53, 79–88 (2012).zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    I. V. Protasov, “Partitions of groups into thin subsets,” Algebra Discrete Math., 11, 88–92 (2011).MathSciNetGoogle Scholar
  16. 16.
    I. V. Protasov, “Selective survey on subset combinatorics of groups,” Ukr. Math. Bull., 7, 220–257 (2011).MathSciNetGoogle Scholar
  17. 17.
    I. V. Protasov, “Thin subsets of topological groups,” Top. Appl., 160, 1083–1087 (2013).zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    I. V. Protasov and S. Slobodianiuk, “Thin subsets of groups,” Ukr. Math. J., 65, No. 9, 1237–1245 (2013).MathSciNetGoogle Scholar
  19. 19.
    T. Banakh and N. Lyaskova, “On thin complete ideals of subsets of groups,” Ukr. Math. J., 63, No. 6, 741–754 (2011).CrossRefGoogle Scholar
  20. 20.
    I. V. Protasov, “Sparse and thin metric spaces,” Math. Stud., 41, No. 1, 92–100 (2014).zbMATHMathSciNetGoogle Scholar
  21. 21.
    M. Filali, I. Lutsenko, and I. Protasov, “Boolean group ideals and the ideal structure of βG,Math. Stud., 30, 1–10 (2008).Google Scholar
  22. 22.
    I. V. Protasov, “Partitions of groups into sparse subsets,” Algebra Discrete Math., 13, No. 1, 107–110 (2012).zbMATHMathSciNetGoogle Scholar
  23. 23.
    T. Banakh and I. Zarichnyi, “Characterizing the Cantor bicube in asymptotic categories,” Groups, Geometry, and Dynamics, 5, 691–728 (2011).zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    I. Protasov and M. Zarichnyi, “General asymptology,” Math. Stud. Monogr. Ser., VNTL Publ., Lviv, 12 (2007).Google Scholar
  25. 25.
    N. Hindman, “Ultrafilters and combinatorial number theory,” Lect. Notes Math., 751, 119–184 (1979).MathSciNetCrossRefGoogle Scholar
  26. 26.
    A. Dranishnikov and M. Zarichnyi, “Universal spaces for asymptotic dimension,” Top. Appl., 140, 203–225 (2004).zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    I. V. Protasov, “Small systems of generators of groups,” Math. Notes, 76, 420–426 (2004).MathSciNetCrossRefGoogle Scholar
  28. 28.
    E. Munarini and M. Salvi, “Scattered subsets,” Discrete Math., 267, 213–228 (2003).zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • T. O. Banakh
    • 1
  • I. V. Protasov
    • 2
  • S. V. Slobodianiuk
    • 2
  1. 1.Franko Lviv National UniversityLvivUkraine
  2. 2.Shevchenko Kyiv National UniversityKyivUkraine

Personalised recommendations