Skip to main content Accessibility help
×
Home
Hostname: page-component-559fc8cf4f-67gxp Total loading time: 0.28 Render date: 2021-03-07T09:30:07.264Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

On weak and strong interpolation in algebraic logics

Published online by Cambridge University Press:  12 March 2014

Gábor Sági
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest Pf. 127, H-1364, Hungary. E-mail: sagi@renyi.hu
Saharon Shelah
Affiliation:
Department of Mathematics, Hebrew University, 91904 Jerusalem, Israel. E-mail: shelah@math.huji.ac.il
Corresponding

Abstract

We show that there is a restriction, or modification of the finite-variable fragments of First Order Logic in which a weak form of Craig's Interpolation Theorem holds but a strong form of this theorem does not hold. Translating these results into Algebraic Logic we obtain a finitely axiomatizable subvariety of finite dimensional Representable Cylindric Algebras that has the Strong Amalgamation Property but does not have the Superamalgamation Property. This settles a conjecture of Pigozzi [12].

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2006

Access options

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

References

[1]Andréka, H., Németi, I., and Sain, I., Algebraic logic, Handbook of philosophical logic (Gabbay, D. M. and Guenthner, F., editors), Kluwer Academic Publishers, 2nd ed., 2001.Google Scholar
[2]Baker, K., Finite equational bases for finite algebras in a congruence-distrubutive equational class, Advances in Mathematics, vol. 24 (1977), pp. 204243.CrossRefGoogle Scholar
[3]Burris, S. and Sankappanavar, H. P., A course in universal algebra, Springer Verlag, New York, 1981.CrossRefGoogle Scholar
[4]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[5]Comer, , Classes without the amalgamation property, Pacific Journal of Mathematics, vol. 28 (1969), pp. 309318.CrossRefGoogle Scholar
[6]Henkin, L., Monk, J. D., and Tarski, A., Cylindric algebras. Part 1, North-Holland, Amsterdam, 1971.Google Scholar
[7]Henkin, L., Monk, J. D., and Tarski, A., Cylindric algebras. Part 2, North-Holland, Amsterdam, 1985.Google Scholar
[8]Hodges, W., Model theory, Cambridge University Press, 1997.Google Scholar
[9]Kiss, E. W., Márki, L., Prőhle, P., and Tholen, W., Categorical algebraic properties. A compendium on amalgamation, congruence extension, epimorphisms, residual smallness and injectivity, Studia Scientiarum Mathematicarum Hungarica, vol. 18 (1983), pp. 79141.Google Scholar
[10]Maksimova, L., Beth's property, interpolation and amalgamation in varieties of modal algebras, Doklady Akademii Nauk SSSR, vol. 319 (1991), no. 6, pp. 13091312, Russian.Google Scholar
[11]Németi, I., Beth definability property is equivalent with surjectiveness ofepis in general algebraic logic, Technical report, Mathematical Institute of Hungarian Academy of Sciences, Budapest, 1983.Google Scholar
[12]Pigozzi, D., Amalgamation, congruence extension and interpolation properties in algebras, Algebra Universalis, vol. 1 (1972), no. 3, pp. 269349.CrossRefGoogle Scholar
[13]Shelah, S., Classification theory, North-Holland, Amsterdam, 1990.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: 12 *
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.

On weak and strong interpolation in algebraic logics
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.

On weak and strong interpolation in algebraic logics
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.

On weak and strong interpolation in algebraic logics
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *