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On weak and strong interpolation in algebraic logics

  • Gábor Sági (a1) and Saharon Shelah (a2)

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].

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[2]Baker, K., Finite equational bases for finite algebras in a congruence-distrubutive equational class, Advances in Mathematics, vol. 24 (1977), pp. 204243.
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[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.
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[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.
[12]Pigozzi, D., Amalgamation, congruence extension and interpolation properties in algebras, Algebra Universalis, vol. 1 (1972), no. 3, pp. 269349.
[13]Shelah, S., Classification theory, North-Holland, Amsterdam, 1990.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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