Mathematical Logic Quarterly 49 (5):479-484 (2003)

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Abstract
We study the relations of being substructure and elementary substructure between Kripke models of intuitionistic predicate logic with the same arbitrary frame. We prove analogues of Tarski's test and Löwenheim-Skolem's theorems as determined by our definitions. The relations between corresponding worlds of two Kripke models [MATHEMATICAL SCRIPT CAPITAL K] ⪯ [MATHEMATICAL SCRIPT CAPITAL K]′ are studied
Keywords Kripke model  intuitionistic logic  substructure  elementary substructure
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DOI 10.1002/malq.200310052
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Preservation Theorems for Kripke Models.Morteza Moniri & Mostafa Zaare - 2009 - Mathematical Logic Quarterly 55 (2):177-184.
Some Preservation Theorems in an Intermediate Logic.Seyed M. Bagheri - 2006 - Mathematical Logic Quarterly 52 (2):125-133.
Back and Forth Between First-Order Kripke Models.Tomasz Połacik - 2008 - Logic Journal of the IGPL 16 (4):335-355.
Homomorphisms and Chains of Kripke Models.Morteza Moniri & Mostafa Zaare - 2011 - Archive for Mathematical Logic 50 (3-4):431-443.

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