Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-20T13:22:17.887Z Has data issue: false hasContentIssue false
  • English
  • Français

On weak and strong interpolation in algebraic logics

Published online by Cambridge University Press:  12 March 2014

Gábor Sági  and
Saharon Shelah
Gábor Sági
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest Pf. 127, H-1364, Hungary. E-mail: [email protected]
Saharon Shelah
Affiliation:
Department of Mathematics, Hebrew University, 91904 Jerusalem, Israel. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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].

Keywords

Key words and phrases. Craig InterpolationStrong AmalgamationSuperamalgamationVarieties of Cylindric Algebras
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 1 , March 2006 , pp. 104 - 118
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. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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. 204–243.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. 309–318.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. 79–141.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. 1309–1312, 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. 269–349.CrossRefGoogle Scholar
[13]Shelah, S., Classification theory, North-Holland, Amsterdam, 1990.Google Scholar