Exact and extendible formulas in intuitionistic propositional calculus

It is well-known that any substitution in intuitionistic propositional logic can be seen as an endomorphism between free Heyting algebras. For a given substitution σ : F (n)→ F (m), (where F (n) is the n-generated free Heyting algebra, n ∈ ω), the theory of σ is the filter σ−1(>). The theory of σ is finitely axiomatizable if σ−1(>) is a principal filter. The formula which generates this filter (axiomatizes the theory of σ) is called an exact formula. It is an immediate consequence of Pitts’ uniform interpolation theorem, that for any substitution σ, the theory of σ is finitely axiomatizable, and hence axiomatized by an exact formula. Exact formulas can be characterized purely semantically (in terms of finite Kripke models). We say that a formula φ has the extension property if any disjoint union of finite rooted Kripke models validating φ can be extended to a Kripke model validating φ by adding a new root to the disjoint union. It was proved by Visser (see e.g. [1]) that if φ is exact, then it has the extension property. The more difficult converse has been proved by Ghilardi (see [4]). We investigate n-universal models, the so-called dual models of n-generated free Heyting algebras. n-universal models are, in a sense, the smallest models in which all finite n-models can be p-morphically embedded. A nice description of the formulas with the extension property can be given in these models, especially in the two-variable case. This, by Visser’s and Ghilardi’s theorems, leads to a characterization of (the infinitely many) exact formulas in two variables. (Note, that as it was shown by de Jongh [2], there are only five exact formulas in one variable.) Using the structure of the n-universal model we will give a relatively simple proof of Ghilardi’s (and Visser’s) theorems and show that there are exactly n! automorphisms of the n-generated free Heyting algebra.