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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

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

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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. 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