Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-66644f4456-q629f Total loading time: 0 Render date: 2025-02-13T03:27:14.589Z Has data issue: true hasContentIssue false

Bounded super real closed rings

Published online by Cambridge University Press:  01 March 2011

By
Marcus Tressl
Edited by
Françoise Delon ,
Ulrich Kohlenbach ,
Penelope Maddy  and
Frank Stephan
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
Get access

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 : An → A 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 : An → A 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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×