Skip to main content Accessibility help
×
Home

Effective presentability of Boolean algebras of Cantor-Bendixson rank 1

Published online by Cambridge University Press:  12 March 2014

Rod Downey
Affiliation:
Department of Mathematics, Victoria University of Wellington, Department of Mathematics, Wellington, New Zealand E-mail: rod.downey@vuw.ac.nz
Carl G. Jockusch
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Il 61801–2917, USA E-mail: jockusch@math.uiuc.edu
Corresponding

Abstract

We show that there is a computable Boolean algebra and a computably enumerable ideal I of such that the quotient algebra /I is of Cantor-Bendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even though Feiner's construction yields a Boolean algebra of infinite Cantor-Bendixson rank.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1999

Access options

Get access to the full version of this content by using one of the access options below.

References

[1]Downey, R., On presentations of algebraic structures, Complexity, logic and recursion theory (Sorbi, A., editor), Lecture Notes in Pure and Applied Mathematics, vol. 197, Marcel Dekker, 1997, pp. 157–206.Google Scholar
[2]Downey, R. and Jockusch, C., Every low Boolean algebra is isomorphic to a recursive one, Proceedings of the American Mathematical Society, vol. 122 (1994), pp. 871–880.CrossRefGoogle Scholar
[3]Feiner, L., Hierarchies of Boolean algebras, this Journal, vol. 35 (1970), pp. 365–374.Google Scholar
[4]Feiner, L., Degrees of nonrecursive presentability, Proceedings of the American Mathematical Society, vol. 38 (1973), pp. 621–624.CrossRefGoogle Scholar
[5]Jockusch, C. and Soare, R., Boolean algebras, Stone spaces, and the iterated Turing jump, this Journal, vol. 59 (1994), pp. 1121–1138.Google Scholar
[6]Ketonen, J., The structure of countable Boolean algebras, Annals of Mathematics, vol. 108 (1978), pp. 41–89.CrossRefGoogle Scholar
[7]Koppelberg, Sabine, General theory of Boolean algebras, Handbook of Boolean algebras (Monk, J. D. and Bonnet, R., editors), vol. 1, North-Holland, Amsterdam, New York, Oxford, Tokyo, 1989.Google Scholar
[8]Pierce, R. S., Countable Boolean algebras, Handbook of Boolean algebras (Monk, J. D. and Bonnet, R., editors), vol. 3, North-Holland, Amsterdam, New York, Oxford, Tokyo, 1989, pp. 775–876.Google Scholar
[9]Remmel, J., Recursive Boolean algebras, Handbook of Boolean algebras (Monk, J. D. and Bonnet, R., editors), vol. 3, North-Holland, Amsterdam, New York, Oxford, Tokyo, 1989, pp. 1097–1166.Google Scholar
[10]Soare, R. I., Recursively enumerable sets and degrees; A study of computable functions and computably generated sets, Springer-Verlag, Heidelberg, 1987.CrossRefGoogle Scholar
[11]Soare, R. I., Computability and recursion, Bulletin of Symbolic Logic, vol. 2 (1996), pp. 284–321.CrossRefGoogle Scholar
[12]Thurber, J., Recursive and r.e. quotient Boolean algebras, Archive for Mathematical Logic, vol. 33 (1994), pp. 121–129.CrossRefGoogle Scholar
[13]Thurber, J., Every low2 Boolean algebra has a recursive copy, Proceedings of the American Mathematical Society, vol. 123 (1995), pp. 3859–3966.Google Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 7 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 26th January 2021. This data will be updated every 24 hours.

Hostname: page-component-898fc554b-sztd2 Total loading time: 0.189 Render date: 2021-01-26T13:14:17.169Z Query parameters: { "hasAccess": "0", "openAccess": "0", "isLogged": "0", "lang": "en" } Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false }

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org 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 sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent 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.

Effective presentability of Boolean algebras of Cantor-Bendixson rank 1
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and 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 <service> account. Find out more about sending content to Dropbox.

Effective presentability of Boolean algebras of Cantor-Bendixson rank 1
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and 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 <service> account. Find out more about sending content to Google Drive.

Effective presentability of Boolean algebras of Cantor-Bendixson rank 1
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *