Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T07:35:26.401Z Has data issue: false hasContentIssue false
  • English
  • Français

Subsemigroups of Stone-Čech compactifications

Published online by Cambridge University Press:  24 October 2008

Talin Budak ,
Nilgün Işik  and
John Pym
Talin Budak
Affiliation:
Mathematik Bölümü, Bogaziçi Üniversitesi, 80815 Bebek, İstanbul, Turkey
Nilgün Işik
Affiliation:
Mathematik Bölümü, Bogaziçi Üniversitesi, 80815 Bebek, İstanbul, Turkey
John Pym
Affiliation:
Mathematics and Statistics, The University, Sheffield S3 7RH
Get access Rights & Permissions [Opens in a new window]

Extract

The Stone–Čech compactification βℕ of the discrete space ℕ of positive integer is a very large topological space; for example, any countable discrete subspace of the growth ℕ* = βℕ/ℕ has a closure which is homeomorphic to βℕ itself ([23], §3·5] Now ℕ, while hardly inspiring as a discrete topological space, has a rich algebrai structure. That βℕ also has a semigroup structure which extends that of (ℕ, +) and in which multiplication is continuous in one variable has been apparent for about 30 years. (Civin and Yood [3] showed that βG was a semigroup for each discrete group G, and any mathematician could then have spotted that βℕ was a subsemigroup of βℕ.) The question which now appears natural was explicitly raised by van Douwen[6] in 1978 (in spite of the recent publication date of his paper), namely, does ℕ* contain subspaces simultaneously algebraically isomorphic and homeomorphic to βℕ? Progress on this question was slight until Strauss [22] solved it in a spectacular fashion: the image of any continuous homomorphism from βℕ into ℕ* must be finite, and so the homomorphism cannot be injective. This dramatic advance is not the end of the story. It is still not known whether that image can contain more than one point. Indeed, what appears to be one of the most difficult questions about the algebraic structure of βℕ is whether it contains any non-trivial finite subgroups

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 1 , July 1994 , pp. 99 - 118
Copyright
Copyright © Cambridge Philosophical Society 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Aminpour, A. M.. A subsemigroup of some Stone–Čech compactifications. Math. Nachr. 158 (1992), 207–218.CrossRefGoogle Scholar
[2]Berglund, J. F., Junghenn, H. D. and Milnes, P.. Analysis on Semigroups (Wiley, 1989).Google Scholar
[3]Civin, P. and Yood, B.. The second conjugate space of a Banach algebra as an algebra. Pacific J. Math. 11 (1961), 847–870.CrossRefGoogle Scholar
[4]Clifford, A. H. and Preston, G. B.. The Algebraic Theory of Semigroups I. Math Surveys 7. American Math. Soc. (Providence, 1961).Google Scholar
[5]Comfort, W. W. and Negrepontis, S.. The Theory of Ultrafilters (Springer, 1974).CrossRefGoogle Scholar
[6]van Douwen, E. K.. The Čech-Stone compactification of a discrete groupoid. Topology Appl. 39 (1991), 43–60.CrossRefGoogle Scholar
[7]Ellis, R.. Distal transformation groups. Pacific J. Math. 9 (1958), 401–405.CrossRefGoogle Scholar
[8]Ellis, R.. Lectures on Topological Dynamics (Benjamin, 1969).Google Scholar
[9]Fuchs, L.. Abelian Groups. Hungarian Acad. Sci., Budapest, 1958.Google Scholar
[10]Hewitt, E. and Ross, K. A.. Abstract Harmonic Analysis I (Springer, 1963).Google Scholar
[11]Hindman, N. and Pym, J.. Free groups and semigroups in βℕ. Semigroup Forum 30 (1984), 177–193.CrossRefGoogle Scholar
[12]Kunen, K. and Vaughan, J. E. (eds). Handbook of Set-Theoretic Topology (North-Holland, 1984).Google Scholar
[13]Lau, A. T. and Pym, J.. The topological centre of a compactification of a locally compact group. To appear.Google Scholar
[14]Milnes, P.. Compactifications of topological semigroups. J. Austral. Math. Soc. 15 (1973), 488–503.CrossRefGoogle Scholar
[15]Namioka, I.. Right topological groups, distal flows and a fixed point theorem. Math. Systems Theory 6 (1972), 193–209.CrossRefGoogle Scholar
[16]Papazyan, T.. Oids, finite sums ana tne structure of the stone–Ċech compactmcation of a discrete semigroup. Semigroup Forum 42 (1991), 265–277.CrossRefGoogle Scholar
[17]Pym, J.. Semigroup structure in Stone–Čech compactifications. J. London Math. Soc. 36 (1988), 421–428.Google Scholar
[18]Pym, J.. Compact semigroups with one-sided continuity. In The Analytical and Topological Theory of Semigroups. By Hofmann, K. H. et al. (de Gruyter, 1990), 197–217.CrossRefGoogle Scholar
[19]Ruppert, W.. Rechtstopologische Halbgruppen. J. reine Angew. Math. 261 (1973), 123–133.Google Scholar
[20]Ruppert, W.. On semigroup compactifications of topological groups. Proc. Roy. Irish Acad. Sect. A 79 (1979), 179–200.Google Scholar
[21]Steauss, D.. Semigroup structures on βℕ. Semigroup Forum 44 (1992), 238–244.CrossRefGoogle Scholar
[22]Steauss, D.. ℕ* does not contain an algebraic and topological copy of βℕ. J. London Math. Soc. 46 (1992), 463–470.CrossRefGoogle Scholar
[23]Walker, R. C.. The Stone–Čech Compactification (Springer, 1974).CrossRefGoogle Scholar