Mathematical Logic Quarterly 55 (2):177-184 (2009)

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Abstract
There are several ways for defining the notion submodel for Kripke models of intuitionistic first-order logic. In our approach a Kripke model A is a submodel of a Kripke model B if they have the same frame and for each two corresponding worlds Aα and Bα of them, Aα is a subset of Bα and forcing of atomic formulas with parameters in the smaller one, in A and B, are the same. In this case, B is called an extension of A. We characterize theories that are preserved under taking submodels and also those that are preserved under taking extensions as universal and existential theories, respectively. We also study the notion elementary submodel defined in the same style and give some results concerning this notion. In particular, we prove that the relation between each two corresponding worlds of finite Kripke models A ≤ B is elementary extension
Keywords Kripke model  intuitionistic first‐order logic  universal sentence  submodel  existential sentence  elementary submodel
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DOI 10.1002/malq.200710085
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References found in this work BETA

Submodels of Kripke Models.Albert Visser - 2001 - Archive for Mathematical Logic 40 (4):277-295.
Finite Kripke Models of HA Are Locally PA.E. C. W. Krabbe - 1986 - Notre Dame Journal of Formal Logic 27:528-532.

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