Abstract
The familiar 1940 result of M. H. Stone characterizing the rings of real-valued continuous functions on compact Hausdorff spaces as certain partially ordered rings is obtained here in its pointfree form, replacing the spaces in question by appropriate frames, which avoids the classical recourse to the choice-dependent existence of maximal ideals. The main tools for this are a direct proof that the partially ordered rings involved are f-rings, the pointfree notion of rings of real-valued continuous functions, and the representation of archimedean bounded f-rings as rings of that kind.
Thanks go to the Natural Sciences and Engineering Research Council of Canada for continuing support in the form of a research grant; the University of Florida for financial assistance in connection with the Conference on Lattice-Ordered Groups and f-Rings, February 28 to March 3, 2001; my late friend and collaborator J. J. C. Vermeulen for some stimulating discussion of the problems dealt with in Section 1; and the referees for some valuable criticism.
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Banaschewski, B. (2002). Stone’s Real Gelfand Duality in Pointfree Topology. In: Martínez, J. (eds) Ordered Algebraic Structures. Developments in Mathematics, vol 7. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3627-4_7
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DOI: https://doi.org/10.1007/978-1-4757-3627-4_7
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