Preservation theorems for Kripke models

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 A^α and B^α of them, A^α is a subset of B^α 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 (in the classical sense) (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)