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The Boolean space of orderings of a field


Author: Thomas C. Craven
Journal: Trans. Amer. Math. Soc. 209 (1975), 225-235
MSC: Primary 12D15; Secondary 10C05
DOI: https://doi.org/10.1090/S0002-9947-1975-0379448-X
MathSciNet review: 0379448
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Abstract: It has been pointed out by Knebusch, Rosenberg and Ware that the set $ X$ of all orderings on a formally real field can be topologized to make a Boolean space (compact, Hausdorff and totally disconnected). They have called the sets of orderings $ W(a) = \{ < {\text{ in }}X\vert a < 0\} $ the Harrison subbasis of $ X$. This subbasis is closed under symmetric difference and complementation. In this paper it is proved that, given any Boolean space $ X$, there exists a formally real field $ F$ such that $ X$ is homeomorphic to the space of orderings on $ F$. Also, an example is given of a Boolean space and a basis of clopen sets closed under symmetric difference and complementation which cannot be the Harrison subbasis of any formally real field.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0379448-X
Keywords: Witt ring, formally real field, orderings of a field, Boolean space, Boolean algebra, Harrison subbasis, strong approximation property
Article copyright: © Copyright 1975 American Mathematical Society

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