Partitions of groups

I. Protasov; S. Slobodianiuk

We classify the subsets of a group by their sizes, formalize the basic methods of partitions and apply them to partition a group to subsets of prescribed sizes.

1. T. Banakh, Extremal densities and submeasures on groups, preprint, http://arxiv.org/abs/1211.0717.

2. T. Banakh, N. Lyaskovska, Completeness of translation-invariant ideals in groups, Ukr. Math. J., 62 (2010), 1022–1031.

3. Banakh T., Lyaskovska N. On thin complete ideals of subsets of groups, Ukr. Math. J., 62 (2011), 741–754.

4. T. Banakh, I. Protasov, Partitions of groups and matroids onto independent sets, Algebra Discrete Math., 10 (2010), 1–7.

5. T. Banakh, I. Protasov, S. Slobodianiuk, Scattered subsets of groups, preprint, http://arxiv.org/abs/ 1312.6946.

6. T. Banakh, I. Protasov, S. Slobodianiuk, Densities, submeasures and partitions of groups, Algebra Discrete Math., 17 (2014), №2, 193–221.

7. T. Banakh, I.V. Protasov, S. Slobodianiuk, Syndetic submeasures and partitions of G-spaces and groups, Intern. J. Algebra Comp., 23 (2013), 1611–1623.

8. T. Banakh, I. Protasov, S. Slobodianiuk, Cellular and ordinal balleans, preprint.

9. A. Bella, V. Malykhin, Certain subsets of a group, Questions Answers General Topology, 17 (1999), 183–187.

10. T. Carlson, N. Hindman, J. Mcleod, D. Strauss, Almost disjoint large subsets of a semigroups, Topology Appl., 155 (2008), 433–444.

11. C. Chou, On the size of the set of left invariant means on a group, Proc. Amer. Math. Soc., 23 (1969), 199–205.

12. J. Erde, A note on combinatorial derivation, preprint, http://arxiv.org/abs/1210.7622.

13. P. Erdos, S. Kakutani, On denumerable graphs, Bull. Amer. Math. Soc., 49 (1943), 457–461.

14. M. Filali, I. Protasov, Ultrafilters and topologies on groups, Math. Stud. Monogr. Ser., V.13, VNTL, Lviv, 2010.

15. M. Filali, Ie. Lutsenko, I.V. Protasov, Boolean group ideals and the ideal structure of $\beta G$, Mat. Stud., 31 (2009), №1, 19–28.

16. V. Gavrylkiv, Algebraic-topological structures in superextensions, PhD Dissertation, Lviv University, 2009.

17. N. Hindman, D. Strauss, Algebra in the Stone-Cech compactification, de Gruyter, Berlin, New York, 1998.

18. J.L. Kelley, Measures on Bollean algebras, Pacific Journal of Math., 9 (2010), 108–114.

19. Lutsenko Ie., Protasov I.V., Sparse, thin and other subsets of groups, Intern. J. Algebra Computation, 19 (2009), 491–510.

20. Ie. Lutsenko, I. Protasov, Thin subsets of balleans, Appl. Gen. Topology., 11 (2010), 89–93.

21. Ie. Lutsenko, I. Protasov, Relatively thin and sparse subsets of groups, Ukr. Math. J., 63 (2011), 216–225.

22. V.I. Malykhin, I.V. Protasov, Maximal resolvability of bounded groups, Topology Appl., 20 (1996), 1-6.

23. V.D. Mazurov, E.I. Khukhro (eds), Unsolved problems in group theory, the Kourovka notebook, 13-th augmented edition, Novosibirsk, 1995.

24. I. Protasov, Combinatorics of numbers, Math. Stud. Monogr. Ser., V.2, VNTL, Lviv, 1997.

25. I.V. Protasov, Resolvability of groups, Math. Stud., 9 (1998), 130–148.

26. I.V. Protasov, Maximal topologies on groups, Sib. Math. J., 9 (1998), 1184–1194.

27. I.V. Protasov, Partitions of groups onto large subsets, Math. Notes, 73 (2003), 271–281.

28. I.V. Protasov, Small systems of generators of groups, Math. Notes, 76 (2004), 420–426.

29. I. Protasov, Partitions of groups into thin subsets, Algebra Discrete Math., 11 (2011), 88–92.

30. I. Protasov, Partitions of groups into sparse subsets, Algebra Discrete Math., 13 (2012), 107–110.

31. I. Protasov, Selective survey on subset combinatorics of groups, Ukr. Math. Bull., 7 (2010), 220–257.

32. I. Protasov, Packings and coverings of groups: some results and open problem, Mat. Stud., 33 (2010), №2, 115–119.

33. I. Protasov, The combinatorial derivation, Appl. Gen. Topology, 14 (2013), 171–178.

34. I. Protasov, The combinatorial derivation and its inverse mapping, Central Europ. Math J., 11 (2013), 2176–2181.

35. I. Protasov, T. Banakh, Ball structures and colorings of groups and graphs, Math. Stud. Monogr. Ser., V.11, VNTL, Lviv, 2003.

36. I. Protasov, S. Slobodianiuk, Prethick subsets in partitions of groups, Algebra Discrete Math., 14 (2012), 267–275.

37. I. Protasov, S. Slobodianiuk, Thin subsets of groups, Ukr. Math. J., 65 (2013), 1245–1253.

38. I. Protasov, S. Slobodianiuk, On the subset combinatorics of $G$-spaces, Algebra Discrete Math., 17 (2014), №1, 98–109.

39. I. Protasov, S. Slobodianiuk, Ultracompanions of subsets of groups, Comment. Math. Univ Carolin, 55 (2014), №2, 257–265.

40. I. Protasov, S. Slobodianiuk, Partition of groups into large subsets, Theory of Groups (to appear).

41. I. Protasov, S. Slobodianiuk, A conjecture on partition of groups, Combinatorica (to appear).

42. I. Protasov, S. Slobodianiuk, A note on partition of groups, preprint is available at arXiv: 1408.5607.

43. I. Protasov, K. Protasova, Kaleidoscopical configurations, Ukr. Math. Bull., 11 (2014), 79–86.

44. I. Protasov, M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., V.12, VNTL, Lviv, 2007.

45. J. Roe, Lectures on coarse geometry, Amer. Math. Soc., Providence, R.I., 2003.

46. S. Solecki, Size of subsets of groups and Haar null sets, Geom. Funct. Anal., 15 (2005), 246–273.

47. B. Weiss, Minimal models for free actions, in: Dynamical Systems and Group Actions (L. Bowen, R. Grigorchuk, Ya. Vorobets eds.), Contemp. Math. 567, Amer. Math. Soc. Providence, RI, (2012), 249–264.

48. T.L. Yurchuk, Chromatic numbers of graphs and partitions of groups, Dopov, NAN Ukrain, (1997), №1, 8–10.

49. P. Zakrzewski, On the complexity of the ideal of absolute null sets, Ukr. Math. J., 64 (2012), 306–308.

	Partitions of groups
Author	I. Protasov, S. Slobodianiuk i.v.protasov@gmail.com, s.slobodianiuk@i.ua Kyiv University, Department Cybernetics; Kyiv University, Department Mechanics and Mathematics
Abstract	We classify the subsets of a group by their sizes, formalize the basic methods of partitions and apply them to partition a group to subsets of prescribed sizes.
Keywords	partitions of groups; large, thick, small, thin subsets of a group; filtration; submeasure
Reference	1. T. Banakh, Extremal densities and submeasures on groups, preprint, http://arxiv.org/abs/1211.0717. 2. T. Banakh, N. Lyaskovska, Completeness of translation-invariant ideals in groups, Ukr. Math. J., 62 (2010), 1022–1031. 3. Banakh T., Lyaskovska N. On thin complete ideals of subsets of groups, Ukr. Math. J., 62 (2011), 741–754. 4. T. Banakh, I. Protasov, Partitions of groups and matroids onto independent sets, Algebra Discrete Math., 10 (2010), 1–7. 5. T. Banakh, I. Protasov, S. Slobodianiuk, Scattered subsets of groups, preprint, http://arxiv.org/abs/ 1312.6946. 6. T. Banakh, I. Protasov, S. Slobodianiuk, Densities, submeasures and partitions of groups, Algebra Discrete Math., 17 (2014), №2, 193–221. 7. T. Banakh, I.V. Protasov, S. Slobodianiuk, Syndetic submeasures and partitions of G-spaces and groups, Intern. J. Algebra Comp., 23 (2013), 1611–1623. 8. T. Banakh, I. Protasov, S. Slobodianiuk, Cellular and ordinal balleans, preprint. 9. A. Bella, V. Malykhin, Certain subsets of a group, Questions Answers General Topology, 17 (1999), 183–187. 10. T. Carlson, N. Hindman, J. Mcleod, D. Strauss, Almost disjoint large subsets of a semigroups, Topology Appl., 155 (2008), 433–444. 11. C. Chou, On the size of the set of left invariant means on a group, Proc. Amer. Math. Soc., 23 (1969), 199–205. 12. J. Erde, A note on combinatorial derivation, preprint, http://arxiv.org/abs/1210.7622. 13. P. Erdos, S. Kakutani, On denumerable graphs, Bull. Amer. Math. Soc., 49 (1943), 457–461. 14. M. Filali, I. Protasov, Ultrafilters and topologies on groups, Math. Stud. Monogr. Ser., V.13, VNTL, Lviv, 2010. 15. M. Filali, Ie. Lutsenko, I.V. Protasov, Boolean group ideals and the ideal structure of $\beta G$, Mat. Stud., 31 (2009), №1, 19–28. 16. V. Gavrylkiv, Algebraic-topological structures in superextensions, PhD Dissertation, Lviv University, 2009. 17. N. Hindman, D. Strauss, Algebra in the Stone-Cech compactification, de Gruyter, Berlin, New York, 1998. 18. J.L. Kelley, Measures on Bollean algebras, Pacific Journal of Math., 9 (2010), 108–114. 19. Lutsenko Ie., Protasov I.V., Sparse, thin and other subsets of groups, Intern. J. Algebra Computation, 19 (2009), 491–510. 20. Ie. Lutsenko, I. Protasov, Thin subsets of balleans, Appl. Gen. Topology., 11 (2010), 89–93. 21. Ie. Lutsenko, I. Protasov, Relatively thin and sparse subsets of groups, Ukr. Math. J., 63 (2011), 216–225. 22. V.I. Malykhin, I.V. Protasov, Maximal resolvability of bounded groups, Topology Appl., 20 (1996), 1-6. 23. V.D. Mazurov, E.I. Khukhro (eds), Unsolved problems in group theory, the Kourovka notebook, 13-th augmented edition, Novosibirsk, 1995. 24. I. Protasov, Combinatorics of numbers, Math. Stud. Monogr. Ser., V.2, VNTL, Lviv, 1997. 25. I.V. Protasov, Resolvability of groups, Math. Stud., 9 (1998), 130–148. 26. I.V. Protasov, Maximal topologies on groups, Sib. Math. J., 9 (1998), 1184–1194. 27. I.V. Protasov, Partitions of groups onto large subsets, Math. Notes, 73 (2003), 271–281. 28. I.V. Protasov, Small systems of generators of groups, Math. Notes, 76 (2004), 420–426. 29. I. Protasov, Partitions of groups into thin subsets, Algebra Discrete Math., 11 (2011), 88–92. 30. I. Protasov, Partitions of groups into sparse subsets, Algebra Discrete Math., 13 (2012), 107–110. 31. I. Protasov, Selective survey on subset combinatorics of groups, Ukr. Math. Bull., 7 (2010), 220–257. 32. I. Protasov, Packings and coverings of groups: some results and open problem, Mat. Stud., 33 (2010), №2, 115–119. 33. I. Protasov, The combinatorial derivation, Appl. Gen. Topology, 14 (2013), 171–178. 34. I. Protasov, The combinatorial derivation and its inverse mapping, Central Europ. Math J., 11 (2013), 2176–2181. 35. I. Protasov, T. Banakh, Ball structures and colorings of groups and graphs, Math. Stud. Monogr. Ser., V.11, VNTL, Lviv, 2003. 36. I. Protasov, S. Slobodianiuk, Prethick subsets in partitions of groups, Algebra Discrete Math., 14 (2012), 267–275. 37. I. Protasov, S. Slobodianiuk, Thin subsets of groups, Ukr. Math. J., 65 (2013), 1245–1253. 38. I. Protasov, S. Slobodianiuk, On the subset combinatorics of $G$-spaces, Algebra Discrete Math., 17 (2014), №1, 98–109. 39. I. Protasov, S. Slobodianiuk, Ultracompanions of subsets of groups, Comment. Math. Univ Carolin, 55 (2014), №2, 257–265. 40. I. Protasov, S. Slobodianiuk, Partition of groups into large subsets, Theory of Groups (to appear). 41. I. Protasov, S. Slobodianiuk, A conjecture on partition of groups, Combinatorica (to appear). 42. I. Protasov, S. Slobodianiuk, A note on partition of groups, preprint is available at arXiv: 1408.5607. 43. I. Protasov, K. Protasova, Kaleidoscopical configurations, Ukr. Math. Bull., 11 (2014), 79–86. 44. I. Protasov, M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., V.12, VNTL, Lviv, 2007. 45. J. Roe, Lectures on coarse geometry, Amer. Math. Soc., Providence, R.I., 2003. 46. S. Solecki, Size of subsets of groups and Haar null sets, Geom. Funct. Anal., 15 (2005), 246–273. 47. B. Weiss, Minimal models for free actions, in: Dynamical Systems and Group Actions (L. Bowen, R. Grigorchuk, Ya. Vorobets eds.), Contemp. Math. 567, Amer. Math. Soc. Providence, RI, (2012), 249–264. 48. T.L. Yurchuk, Chromatic numbers of graphs and partitions of groups, Dopov, NAN Ukrain, (1997), №1, 8–10. 49. P. Zakrzewski, On the complexity of the ideal of absolute null sets, Ukr. Math. J., 64 (2012), 306–308.
Pages	115-128
Volume	42
Issue	2
Year	2014
Journal	Matematychni Studii
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Table of content of issue	html