Submodels of Kripke models

A Kripke model K is a submodel of another Kripke modelM if K is ob tained by restricting the set of nodes ofM In this paper we show that the class of formulas of Intuitionistic Predicate Logic that is preserved under taking submodels of Kripke models is precisely the class of semipositive formulas This result is an analogue of the Lo s Tarski theorem for the Classical Predicate Calculus In appendix A we prove that for theories with decidable identity we can take as the embeddings between domains in Kripke models of the theory the identical embeddings This is a well known fact but we know of no correct proof in the literature In appendix B we answer negatively a question posed by Sam Buss whether there is a classical theory T such that HT is HA Here HT is the theory of all Kripke modelsM such that the structures assigned to the nodes of M all satisfy T in the sense of classical model theory