Skip to main content Accessibility help
×
Home
Hostname: page-component-559fc8cf4f-s65px Total loading time: 0.344 Render date: 2021-03-07T09:59:51.769Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Degree spectra of intrinsically c.e. relations

Published online by Cambridge University Press:  12 March 2014

Denis R. Hirschfeldt
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
Corresponding
E-mail address:

Abstract

We show that for every c.e. degree a > 0 there exists an intrinsically c.e. relation on the domain of a computable structure whose degree spectrum is {0, a}. This result can be extended in two directions. First we show that for every uniformly c.e. collection of sets S there exists an intrinsically c.e. relation on the domain of a computable structure whose degree spectrum is the set of degrees of elements of S. Then we show that if αω ∪ {ω} then for any α-c.e. degree a > 0 there exists an intrinsically α-c.e. relation on the domain of a computable structure whose degree spectrum {0, a}. All of these results also hold for m-degree spectra of relations.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2001

Access options

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

References

[1]Ash, C. J., Cholak, P., and Knight, J. F., Permitting, forcing, and copying of a given recursive relation, Annals of Pure and Applied Logic, vol. 86 (1997), pp. 219236.CrossRefGoogle Scholar
[2]Ash, C. J. and Nerode, A., Intrinsically recursive relations, Aspects of effective algebra (Clayton, 1979) (Crossley, J. N., editor), Upside Down A Book Company, Yarra Glen, Australia, 1981, pp. 2641.Google Scholar
[3]Barker, E., Intrinsically relations, Annals of Pure and Applied Logic, vol. 39 (1988), pp. 105130.CrossRefGoogle Scholar
[4]Cholak, P., Goncharov, S. S., Khoussainov, B., and Shore, R. A., Computably categorical structures and expansions by constants, this Journal, vol. 64 (1999), pp. 1337.Google Scholar
[5]Cooper, S. B., Harrington, L., Lachlan, A. H., Lempp, S., and Soare, R. I., The d.r.e. degrees are not dense, Annals of Pure and Applied Logic, vol. 55 (1991), pp. 125151.CrossRefGoogle Scholar
[6]Epstein, R. L., Haas, R., and Kramer, R. L., Hierarchies of sets and degrees below 0′, Logic year 1979–80 (Proceedings, Seminars and Conferences in Mathematical Logic, University of Connecticut, Storrs, Connecticut, 1979/80) (Lerman, M., Schmerl, J. H., and Soare, R. I., editors), Lecture Notes in Mathematics, no. 859, Springer-Verag, Heidelberg, 1981, pp. 3248.Google Scholar
[7]Ershov, Yu. L., Goncharov, S. S., Nerode, A., and Remmel, J.B. (editors), Handbook of recursive mathematics, Studies in Logic and Foundational Mathematics, Elsevier Science, Amsterdam, 1998.Google Scholar
[8]Goncharov, S. S., Autostability of models and abelian groups, Algebra and Logic, vol. 19 (1980), pp. 1327.CrossRefGoogle Scholar
[9]Goncharov, S. S., Computable single-valued numerations, Algebra and Logic, vol. 19 (1980), pp. 325356.CrossRefGoogle Scholar
[10]Goncharov, S. S., Problem of the number of non-self-equivalent constructivizations, Algebra and Logic, vol. 19 (1980), pp. 401414.CrossRefGoogle Scholar
[11]Goncharov, S. S. and Khoussainov, B., On the spectrum of degrees of decidable relations, Doklady mathematics, vol. 55 (1997), pp. 5557, research announcement.Google Scholar
[12]Harizanov, V. S., The possible turing degree of the nonzero member in a two element degree spectrum, Annals of Pure and Applied Logic, vol. 60 (1993), pp. 130.CrossRefGoogle Scholar
[13]Hirschfeldt, D. R., Khoussainov, B., Shore, R. A., and Slinko, A. M., Degree spectra and computable dimension in algebraic structures, to appear.Google Scholar
[14]Khoussainov, B. and Shore, R. A., Computable isomorphisms, degree spectra of relations, and scott families, Annals of Pure and Applied Logic, vol. 93 (1998), pp. 153193.CrossRefGoogle Scholar
[15]Soare, R. I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, Heidelberg, 1987.CrossRefGoogle 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: 10 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 7th March 2021. This data will be updated every 24 hours.

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.

Degree spectra of intrinsically c.e. relations
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.

Degree spectra of intrinsically c.e. relations
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.

Degree spectra of intrinsically c.e. relations
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *