Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-20T12:35:32.793Z Has data issue: false hasContentIssue false
  • English
  • Français

More on Compact Hausdorff Spaces and Finitary Duality

Published online by Cambridge University Press:  20 November 2018

B. Banaschewski
B. Banaschewski*
Affiliation:
McMaster University, Hamilton, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is an old conjecture by P. Bankston that the category CompHaus of compact Hausdorff spaces and their continuous maps is not dually equivalent to any elementary P-class of finitary algebras (taken as a category with all homomorphisms between its members as maps), where elementary means defined by first order axioms, and a P-class is one closed under arbitrary (cartesian) products. One motivation for this conjecture is the fact that such a dual equivalence would make ultracopowers of compact Hausdorff spaces correspond to ultrapowers of finitary algebras, and one might expect this to have contradictory consequences.

As a possible step towards proving his conjecture, Bankston [2] showed that no elementary SP-class of finitary algebras can be dually equivalent to CompHaus. However, it was subsequently proved in [1] that the same holds for any SP-class of finitary algebras, using an argument independent of ultrapowers.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 6 , 01 December 1984 , pp. 1113 - 1118
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Banaschewski, B., On categories of algebras equivalent to a variety, Alg. Univ. 76 (1983), 264–267.Google Scholar
2. Banaschewski, B., On lattices of continuous functions, Quaest. Math. 6 (1983), 1–12.Google Scholar
3. Bankston, P., Some obstacles to duality in topological algebra, Can. J. Math. 34 (1982), 80–90.Google Scholar
4. Chang, C. C. and Keisler, H. J., Model theory (North Holland, Amsterdam, 1973).Google Scholar
5. Duskin, J., Variations of Beck's tripleability criterion, LNM106 (Springer-Verlag 1969), 74–129.Google Scholar
6. Isbell, J. R., The unit ball of C(X) as an abstract algebra, Lectures at the Banach Centre, Warsaw, (1974).Google Scholar
7. Rosicky, J., Categories of models, Seminarberichte Mathematik Informatik FernUniversitat Hagen 79 (1984), 377–413.Google Scholar