We define notions of homomorphism, submodel, and sandwich of Kripke models, and
we define two syntactic operators analogous to universal and existential
closure. Then we prove an intuitionistic analogue of the generalized (dual of
the) Lyndon-Łoś-Tarski Theorem, which characterizes the
sentences preserved under inverse images of homomorphisms of Kripke models, an
intuitionistic analogue of the generalized Łoś-Tarski
Theorem, which characterizes the sentences preserved under submodels of Kripke
models, and an intuitionistic analogue of the generalized Keisler Sandwich
Theorem, which characterizes the sentences preserved under sandwiches of Kripke
models. We also define several intuitionistic formula hierarchies analogous to
the classical formula hierarchies $\forall_n (= \Pi^0_n)$ and $\exists_n (=\Sigma^0_n)$ , and we show how our generalized syntactic preservation theorems
specialize to these hierarchies. Each of these theorems implies the
corresponding classical theorem in the case where the Kripke models force
classical logic.