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Journal of Mathematical Sciences

, Volume 234, Issue 1, pp 49–60 | Cite as

Recent progress in subset combinatorics of groups

  • Igor V. Protasov
  • Ksenia D. Protasova
Article
  • 12 Downloads

Abstract

We systematize and analyze some results obtained in the subset combinatorics of G groups presented in previous surveys [1, 2, 3, 4]. The main topics: the dynamical and descriptive characterizations of subsets of a group relatively to their combinatorial size, Ramsey-product subsets in connection with some general concept of recurrence in G-spaces, new ideals in the Boolean algebra \( {\mathcal{P}}_G \) of all subsets of a group G and in the Stone– Čech compactification βG of G, and the combinatorial derivation.

Keywords

Large, small, thin, thick, sparse, and scattered subsets of groups descriptive complexity Boolean algebra of subsets of a group Stone– Čech compactification ultracompanion Ramsey-product subset of a group recurrence combinatorial derivation 

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References

  1. 1.
    I. Protasov, “Selective survey on subset combinatorics of groups,” J. Math. Sci., 174, 486–514 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    I. Protasov and S. Slobodianiuk, “On the subset combinatorics of G-spaces,” Alg. Discrete Math., 17, 98–109 (2011).MathSciNetzbMATHGoogle Scholar
  3. 3.
    I. Protasov and S. Slobodianiuk, “Partitions of groups,” Math. Stud., 42, 115–128 (2014).MathSciNetzbMATHGoogle Scholar
  4. 4.
    T. Banakh, I. Protasov, and S. Slobodianiuk, “Densities, submeasures and partitions of groups,” Alg. Discrete Math., 17, 193–221 (2014).MathSciNetzbMATHGoogle Scholar
  5. 5.
    N. Hindman and D. Strauss, Algebra in the Stone– Čech Compactification: Theory and Applications, Walter de Gruyter, Berlin, New York (1998).CrossRefzbMATHGoogle Scholar
  6. 6.
    I. Protasov and S. Slobodianiuk, “Ultracompanions of subsets of a group,” Comment. Math. Univ. Carolin., 55, 257–265 (2014).MathSciNetzbMATHGoogle Scholar
  7. 7.
    T. Banakh, I. Protasov, and S. Slobodianiuk, “Scattered subsets of groups,” Ukr. Math. J., 67, No. 3, 347–356 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ie. Lutsenko and I. Protasov, “Sparse, thin and other subsets of groups,” Intern. J. Algebra Comp., 19, 491–510 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. Kechris, Classical Descriptive Set Theory, Springer, Berlin (1995).CrossRefzbMATHGoogle Scholar
  10. 10.
    T. Banakh and N. Lyaskovska, “Weakly P-small not P-small subsets in groups,” Intern. J. Algebra Comput., 19, 1–6 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    I. Protasov and K. Protasova, “Around P-small subsets of groups,” Carpath. Math. Publ., 6, 337–341 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    T. Banakh and N. Lyaskovska, “On thin-complete ideals of subsets of groups,” Ukr. Math. J., 63, No. 6, 216–225 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M. Filali, Ie. Lutsenko, and I. Protasov, “Boolean group ideals and the ideal structure of βG,” Math. Stud., 30, 1–10 (2008).Google Scholar
  14. 14.
    T. Banakh, I. Protasov, and K. Protasova, “Descriptive complexity of the sizes of subsets of groups,” Ukr. Mat. J., 69, No. 9, 1280–1283 (2017).MathSciNetGoogle Scholar
  15. 15.
    I. Protasov and S. Slobodianiuk, “The dynamical look at the subsets of a group,” Appl. Gen. Topol., 16, No. 2, 217–224 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    H. Furstenberg, “Poincaré recurrence and number theory,” Bull. Amer. Math. Soc., 5, No. 3, 211–234 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    N. Hindman, “Ultrafilters and combinatorial number theory,” In: M. B. Nathanson (Ed.), Number Theory, Carbondale 1979. Proceedings, Lecture Notes in Math., Vol. 571, (1979), pp. 119–184.Google Scholar
  18. 18.
    V. Bergelson and N. Hindman, “Quotient sets and density recurrent sets,” Trans. Amer. Math. Soc., 364, 4495–4531 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    I. Protasov, “Filters and topologies on groups,” Math. Stud., 3, 15–28 (1994).zbMATHGoogle Scholar
  20. 20.
    I. Protasov and K. Protasova, “On recurrence in G-spaces,” Alg. Discrete Math., 23, No. 2, 80–85 (2017).MathSciNetzbMATHGoogle Scholar
  21. 21.
    T. Banakh, I. Protasov, and K. Protasova, “Ramsey-product subsets of a group,” Math. Stud., 47, 145–149 (2017).MathSciNetGoogle Scholar
  22. 22.
    Ie. Lutsenko and I. Protasov, “Thin subsets of balleans,” Appl. Gen. Topology, 11, 89–93 (2010).MathSciNetzbMATHGoogle Scholar
  23. 23.
    I. Protasov and S. Slobodianiuk, “Thin subsets of groups,” Ukr. Math. J., 65, 1384–1393 (2013).MathSciNetCrossRefGoogle Scholar
  24. 24.
    I. Protasov and T. Banakh, Ball Structures and Colorings of Graphs and Groups, VNTL, Lviv (2003).zbMATHGoogle Scholar
  25. 25.
    I. Protasov and K. Protasova, “Ideals in PG and βG,” Topology Appl., 238, 24–31 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    I. Protasov, “The combinatorial derivation,” Appl. Gen. Topology, 14, 171–178 (2013).MathSciNetzbMATHGoogle Scholar
  27. 27.
    I. Protasov, “The combinatorial derivation and its inverse mapping,” Central Europ. J. Math., 11, 1276–1281 (2013).MathSciNetzbMATHGoogle Scholar
  28. 28.
    J. Erde, “A note on combinatorial derivation,” ArXiv: 1210.7622.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Computer Science and CyberneticsTaras Shevchenko National University of KyivKyivUkraine

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