On Bellissima's construction of the finitely generated free Heyting algebras, and beyond
type: Item
instance of: scholarly article
title: On Bellissima's construction of the finitely generated free Heyting algebras, and beyond
zbMATH Open document ID: 1215.06004
author: L. Darnière
author: Markus Junker
published in: Archive for Mathematical Logic
publication date: 2010-10-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/0812.2027
review text: 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.
reviewed by: Afrodita Iorgulescu
Mathematics Subject Classification ID: 06D20
Mathematics Subject Classification ID: 03C64
Mathematics Subject Classification ID: 06B23
Mathematics Subject Classification ID: 06B30
Mathematics Subject Classification ID: 08B20
zbMATH DE Number: 5806723
zbMATH Keywords: free Heyting algebra
zbMATH Keywords: finitely generated Heyting algebra
zbMATH Keywords: completion
zbMATH Keywords: irreducible elements
zbMATH Keywords: spectrum
zbMATH Keywords: Kripke model
zbMATH Keywords: automorphism group
MaRDI profile type: Publication
OpenAlex ID: W2028822621
arXiv ID: 0812.2027
cites work: Finitely generated free Heyting algebras
cites work: Profinite completions and canonical extensions of Heyting algebras
cites work: Modal logic
cites work: Profinite Heyting algebras
cites work: Codimension and pseudometric in co-Heyting algebras
cites work: Q3837730
cites work: Q5582318
cites work: Q4277651
cites work: Q4873970
cites work: Model completions and r-Heyting categories
cites work: Q4836217
cites work: Q4003410
cites work: Q3987718
cites work: On closed elements in closure algebras
cites work: On an interpretation of second order quantification in first order intuitionistic propositional logic
cites work: Elementary theories of free topo-Boolean and pseudo-Boolean algebras
cites work: Free Heyting algebras
DOI: 10.1007/S00153-010-0194-7
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description: scientific article