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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Ordered rings over which output sets are recursively enumerable sets


Author: Christian Michaux
Journal: Proc. Amer. Math. Soc. 112 (1991), 569-575
MSC: Primary 03C57; Secondary 03D10, 03D75
Corrigendum: Proc. Amer. Math. Soc. 117 (1993), null.
MathSciNet review: 1041016
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Abstract: In a recent paper [BSS], L. Blum, M. Shub, and S. Smale developed a theory of computation over the reals and over commutative ordered rings; in $ \S9$ of [BSS] they showed that over the reals (and over any real closed field) the class of recursively enumerable sets and the class of output sets are the same; it is a question (Problem 9.1 in [BSS]) to characterize ordered rings with this property (abbreviated by O = R.E. here). In this paper we prove essentially that in the class of (linearly) ordered rings of infinite transcendence degree over $ \mathbb{Q}$, that are dense (for the order) in their real closures, only real closed fields have property O = R.E.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1041016-9
PII: S 0002-9939(1991)1041016-9
Article copyright: © Copyright 1991 American Mathematical Society