On Bellissima's construction of the finitely generated free Heyting algebras, and beyond
This paper is closely related to a second paper by the same authors [``Codimension and pseudometric in co-Heyting algebras'', Algebra Univers. 64, No.~3--4, 251--282 (2010; Zbl 1217.06002)], both complementing each other. While this paper collects results that follow from \textit{F. Bellissima}'s construction in his paper ``Finitely generated free Heyting algebras'' [J. Symb. Log. 51, 152--165 (1986; Zbl 0616.03021)], the second paper analyses the structure of Heyting algebras from a more geometric point of view. The present paper studies finitely generated free Heyting algebras from a topological and from a model-theoretic point of view. After recalling Bellissima's construction, the authors analyse his Heyting algebra; they prove that it is the profinite completion of the finitely generated free Heyting algebra as well as the metric completion for a naturally defined metric. They reconstruct the Kripke model as the principal ideal spectrum and show that the Zariski topology on this spectrum is induced by the partial ordering. The Kripke model is proved to be first-order interpretable. Many other results are obtained. A collection of open problems is presented. The paper is hard to read, but well written.
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