One week ago, I asked a question on math.stackexchange.com (<https://math.stackexchange.com/questions/209120/a-question-on-intuitionistc-propositional-logic>). But nobody answered my question. So I present it here:


In the Kripke's semantics of intuitionistic propositional logic, the frames are all partially ordered frames. Prove that:


Two finite-rooted frames are isomorphic iff they validate the same formulas in the language of intuitionistic propositional logic.


Thanks very much.



A question on intuitionistic propositional logic
Let $F,G$ be the two frames. Let $\beta$ be the frame formula of $F$ (using notation from the Chagrov and Zakharyaschev book you mention in the MSE question, $\beta=\beta^\sharp(F,\bot)$). Since $\beta$ is refutable in $F$, it is also refutable in $G$, hence there exists a generated subframe $H\subseteq G$ and a surjective p-morphism $f\colon H\to F$. A symmetric argument shows that $F$ is a p-morphic image of a generated subframe of $G$, hence $|F|\le|G|$. This in turn implies that $|H|=|G|=|F|$, and as the frames are finite, this can only happen if $H=G$ and $f$ is a bijection. Thus, $f$ is an isomorphism.


