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On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond

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Abstract

We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima’s representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in the Heyting algebra, from which several model theoretic and algebraic properties are derived. In particular, we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators.

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Authors and Affiliations

  1. Département de Mathématiques, Faculté des Sciences, Université d’Angers, 2 bd Lavoisier, 49045, Angers, France

    Luck Darnière

  2. Mathematisches Institut, Abteilung für Mathematische Logik, Albert–Ludwigs–Universität Freiburg, Eckerstraße 1, 79102, Freiburg, Germany

    Markus Junker

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  1. Luck Darnière
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  2. Markus Junker
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Correspondence to Markus Junker.

Additional information

L. Darnière would like to thank the Universität Freiburg for inviting him in July 2008.

M. Junker would like to thank the Université d’Angers for supporting him as an invited professor in march 2005, when main parts of this work were done.

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Darnière, L., Junker, M. On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond. Arch. Math. Logic 49, 743–771 (2010). https://doi.org/10.1007/s00153-010-0194-7

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