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Footnote to a paper of Baker and Mines

Published online by Cambridge University Press:  24 October 2008

John Pym
John Pym
Affiliation:
University of Sheffield
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Extract

Let S be a semigroup with a compact topology in which multiplication is continuous on the left (i.e. xi→x implies xiy→xy for each y in S). Then S has a minimal left ideal L which is compact; each idempotent e in L is a right identity for L (xe = xfor each x ∈ L)and L = Se; Ge = eL is a group and L is the union of all such groups; and if f is a second idempotent in L, the canonical map x ↦ fx of Ge to Gf is an algebraic isomorphism (see Ruppert(2) for these facts). Baker and Milnes(1), §4(A), have observed that, in the case in which S is the Stone–Cech compactification of a discrete abelian group, the canonical map from Ge to Gf may not be a homeomorphism. (This contrasts with the situation in compact semigroups with separately continuous multiplication.) We present a simple proof of a more definitive result.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 2 , March 1979 , pp. 315
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

(1)Baker, J. W. and Milnes, P.The ideal structure of the Stone-Cech compactification of a group. Math. Proc. Cambridge Philos. Soc. 82 (1977), 401–409.CrossRefGoogle Scholar
(2)Ruppert, W.Rechtstopologische Halbgruppen. J. Reine Angew. Math. 261 (1973), 123–133.Google Scholar