Sketch of group balleans

I.V.Protasov; O.I.Protasova

A ballean is a set $X$ endowed with some family of subset of $X$ which are called the balls. We postulate the properties of the family of balls in such a way that the balleans with the appropriate morphisms can be considered as the asymptotic counterparts of the uniform topological spaces. The purpose of the paper is to find the asymptotic analogues for topological groups.

1. Budak T., Isik N., Pym J. Subsemigroup of Stone- Cech compactification, Math. Proc.Camb. Phil. Soc., 116 (1994), 99–118.

2. Dranishnikov A. Asymptotic topology, Russian Math. Surveys, 55 (2000), 71–116.

3. Fillali M., Protasov I. Slowly oscillating function on locally compact groups, Appl. Gener. Topology (submitted).

4. Frider V., Zarichnyi M. On coarse anti-Lawson semilattices, Mat. Stud. 21 (2004), no.1, 3–12.

5. Higman G., Neumann B., Heumann H. Embedding theorems for groups, J. London Math. Soc., 24 (1949), 247–254.

6. Hindman N., Strauss D. Algebra in the Stone-Cech Compactification, de Grueter Exposition in Math., V.27, 1998.

7. Markov A.A. On unconditionally closed sets, Topology and Topol. Algebra. Translations Series 1, 8 (1962), 273–304.

8. Mitchener P.D. Coarse Homology Theories, Algebr. Geom. Topology, 1 (2001), 271–294.

9. Ольшанский А.Ю. Геометрия определяющих соотношений, М.: Наука, 1984.

10. Protasov I.V. Combinatorics of Numbers, Math. Stud. Monogr. Ser., V.2, VNTL, Lviv, 1997.

11. Protasov I.V., Zelenyuk E.G. Topologies on Groups Determined by Sequences, Math. Stud. Monogr. Ser., V.4, VNTL, Lviv, 1999.

12. Protasov I., Banakh T. Ball Structures and Colorings of Groups and Graphs, Mat. Stud. Monogr. Ser., V.11, VNTL, Lviv, 2003.

13. Protasov I.V. Maximal topologies on groups, Sib. Math. J., 39 (1998), 1368-1381.

14. Protasov I.V. Metrizable ball structures, Algebra and Discrete Math. (2002), no.1, 129–141.

15. Protasov I.V. Uniform ball structures, Algebra and Discrete Math. (2003), no.1, 93–102.

16. Protasov I.V. Normal ball structures, Math. Stud. 20 (2003), no.1, 3–16.

17. Protasov I.V. Morphisms of ball structures of groups and graphs, Ukr. Math. J. 54 (2002), 847-855.

18. Protasov I.V. Binary coronas of balleans, Algebra and Discrete Math. (2003), no.4, 50–65.

19. Protasov I.V. Coronas of balleans, Topol. Appl. (to appear).

20. Protasov I.V. Resolvability of ball structures, Appl. Gener. Topology (to appear).

	Sketch of group balleans
Author	I.V.Protasov, O.I.Protasova kseniya@profit.net.ua Kyiv National University
Abstract	A ballean is a set $X$ endowed with some family of subset of $X$ which are called the balls. We postulate the properties of the family of balls in such a way that the balleans with the appropriate morphisms can be considered as the asymptotic counterparts of the uniform topological spaces. The purpose of the paper is to find the asymptotic analogues for topological groups.
Keywords	ballean, uniform topological space, topological group
DOI	doi:10.30970/ms.22.1.10-20
Reference	1. Budak T., Isik N., Pym J. Subsemigroup of Stone- Cech compactification, Math. Proc.Camb. Phil. Soc., 116 (1994), 99–118. 2. Dranishnikov A. Asymptotic topology, Russian Math. Surveys, 55 (2000), 71–116. 3. Fillali M., Protasov I. Slowly oscillating function on locally compact groups, Appl. Gener. Topology (submitted). 4. Frider V., Zarichnyi M. On coarse anti-Lawson semilattices, Mat. Stud. 21 (2004), no.1, 3–12. 5. Higman G., Neumann B., Heumann H. Embedding theorems for groups, J. London Math. Soc., 24 (1949), 247–254. 6. Hindman N., Strauss D. Algebra in the Stone-Cech Compactification, de Grueter Exposition in Math., V.27, 1998. 7. Markov A.A. On unconditionally closed sets, Topology and Topol. Algebra. Translations Series 1, 8 (1962), 273–304. 8. Mitchener P.D. Coarse Homology Theories, Algebr. Geom. Topology, 1 (2001), 271–294. 9. Ольшанский А.Ю. Геометрия определяющих соотношений, М.: Наука, 1984. 10. Protasov I.V. Combinatorics of Numbers, Math. Stud. Monogr. Ser., V.2, VNTL, Lviv, 1997. 11. Protasov I.V., Zelenyuk E.G. Topologies on Groups Determined by Sequences, Math. Stud. Monogr. Ser., V.4, VNTL, Lviv, 1999. 12. Protasov I., Banakh T. Ball Structures and Colorings of Groups and Graphs, Mat. Stud. Monogr. Ser., V.11, VNTL, Lviv, 2003. 13. Protasov I.V. Maximal topologies on groups, Sib. Math. J., 39 (1998), 1368-1381. 14. Protasov I.V. Metrizable ball structures, Algebra and Discrete Math. (2002), no.1, 129–141. 15. Protasov I.V. Uniform ball structures, Algebra and Discrete Math. (2003), no.1, 93–102. 16. Protasov I.V. Normal ball structures, Math. Stud. 20 (2003), no.1, 3–16. 17. Protasov I.V. Morphisms of ball structures of groups and graphs, Ukr. Math. J. 54 (2002), 847-855. 18. Protasov I.V. Binary coronas of balleans, Algebra and Discrete Math. (2003), no.4, 50–65. 19. Protasov I.V. Coronas of balleans, Topol. Appl. (to appear). 20. Protasov I.V. Resolvability of ball structures, Appl. Gener. Topology (to appear).
Pages	10-20
Volume	22
Issue	1
Year	2004
Journal	Matematychni Studii
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