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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 L\"{o}wenheim-Skolem's theorems as determined by
our definitions. The relations between corresponding worlds of two
Kripke models $\cal{K}\preceq\cal{K'}$ are studied.
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