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$ \kappa$-topologies for right topological semigroups


Authors: John Baker, Neil Hindman and John Pym
Journal: Proc. Amer. Math. Soc. 115 (1992), 251-256
MSC: Primary 22A15
MathSciNet review: 1093590
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Abstract: Given a cardinal $ \kappa $ and a right topological semigroup $ S$ with topology $ \tau $, we consider the new topology obtained by declaring any intersection of at most $ \kappa $ members of $ \tau $ to be open. Under appropriate hypotheses, we show that this process turns $ S$ into a topological semigroup. We also show that under these hypotheses the points of any subsemigroup $ T$ with card $ T \leq \kappa $ can be replaced by (new) open sets that algebraically behave like $ T$. Examples are given to demonstrate the nontriviality of these results.


References [Enhancements On Off] (What's this?)

  • [1] J. W. Baker and P. Milnes, The ideal structure of the Stone-Čech compactification of a group, Math. Proc. Cambridge Philos. Soc. 82 (1977), no. 3, 401–409. MR 0460516
  • [2] John F. Berglund, Hugo D. Junghenn, and Paul Milnes, Analysis on semigroups, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1989. Function spaces, compactifications, representations; A Wiley-Interscience Publication. MR 999922
  • [3] W. W. Comfort and S. Negrepontis, The theory of ultrafilters, Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 211. MR 0396267
  • [4] Eric K. van Douwen, The Čech-Stone compactification of a discrete groupoid, Topology Appl. 39 (1991), no. 1, 43–60. MR 1103990, 10.1016/0166-8641(91)90074-V
  • [5] Neil Hindman, The ideal structure of the space of 𝜅-uniform ultrafilters on a discrete semigroup, Rocky Mountain J. Math. 16 (1986), no. 4, 685–701. MR 871030, 10.1216/RMJ-1986-16-4-685
  • [6] -, Ultrafilters and Ramsey theory--an update, Set Theory and its Applications, (V. Steprans and S. Watson, eds.) Lecture Notes in Math., vol 1401, Springer-Verlag, New York, 1989, pp. 97-118.
  • [7] -, The semigroups $ \beta \mathbb{N}$ and its applications to number theory, The Analytical and Topological Theory of Semigroups, (K. H. Hofmann et al, eds.), de Gruyter, Berlin, 1990, pp. 347-360.
  • [8] John Pym, Footnote to a paper of J. W. Baker and P. Milnes: “The ideal structure of the Stone-Čech compactification of a group” (Math. Proc. Cambridge Philos. Soc. 82 (1977), no. 3, 401–409), Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 2, 315. MR 516090, 10.1017/S0305004100055729
  • [9] Russell C. Walker, The Stone-Čech compactification, Springer-Verlag, New York-Berlin, 1974. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83. MR 0380698

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DOI: https://doi.org/10.1090/S0002-9939-1992-1093590-5
Article copyright: © Copyright 1992 American Mathematical Society