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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
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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

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