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Bounded super real closed rings

Published online by Cambridge University Press:  01 March 2011

Françoise Delon
Affiliation:
UFR de Mathématiques
Ulrich Kohlenbach
Affiliation:
Technische Universität, Darmstadt, Germany
Penelope Maddy
Affiliation:
University of California, Irvine
Frank Stephan
Affiliation:
National University of Singapore
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Summary

Introduction. This note is a complement to the paper, where super real closed rings are introduced and studied. A super real closed ring A is a commutative unital ring A together with an operation FA : AnA for every continuous map F : ℝn → ℝ, n ∈ ℕ, so that all term equalities between the F's remain valid for the FA's. For example if C(X) is the ring of real valued continuous functions on a topological space X, then C(X) carries a natural super real closed ring structure, where FC(X) is composition with F. Super real closed rings provide a natural framework for the algebra and model theory of rings of continuous functions.

A bounded super real closed ring A is a commutative unital ring A together with an operation FA : AnA for every bounded continuous map F : ℝn → ℝ, n ∈ ℕ, so that all term equalities between the F's remain valid for the FA's (cf. 2.7 below).

In particular every super real closed ring is a bounded super real closed ring by forgetting the operation of the unbounded functions. An example of a bounded super real closed ring, which is not a super real closed ring, is the ring Cpol(ℝn) of all polynomially bounded continuous functions ℝn → ℝ.

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Logic Colloquium 2007 , pp. 220 - 237
Publisher: Cambridge University Press
Print publication year: 2010

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References

[Schw] N., Schwartz, The basic theory of real closed spaces, Memoirs of the American Mathematical Society, vol. 77 (1989), no. 397, pp. viii+122.Google Scholar
[Tr1] M., Tressl, Computation of the z-radical in C(X), Advances in Geometry, vol. 6 (2006), no. 1, pp. 139–175.Google Scholar
[Tr2] M., Tressl, Super real closed rings, Fundamenta Mathematicae, vol. 194 (2007), no. 2, pp. 121–177.Google Scholar

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