PP-2021-05: Profiniteness and representability of spectra of Heyting algebras

PP-2021-05: Bezhanishvili, Guram and Bezhanishvili, Nick and Moraschini, Tommaso and Stronkowski, Michal (2021) Profiniteness and representability of spectra of Heyting algebras. [Pre-print]

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We prove that there exist profinite Heyting algebras that are not isomorphic to the profinite completion of any Heyting algebra. This resolves an open problem from 2009. More generally, we characterize those varieties of Heyting algebras in which profinite algebras are isomorphic to profinite completions. It turns out that there exists largest such. We give different characterizations of this variety and show that
it is finitely axiomatizable and locally finite. From this it follows that it is decidable whether in a finitely axiomatizable variety of Heyting algebras all profinite members are profinite completions. In addition, we introduce and characterize representable varieties of Heyting algebras, thus drawing connection to the classical problem of representing posets as prime spectra.

Item Type: Pre-print
Report Nr: PP-2021-05
Series Name: Prepublication (PP) Series
Year: 2021
Subjects: Logic
Depositing User: Nick Bezhanishvili
Date Deposited: 25 Feb 2021 19:33
Last Modified: 26 Feb 2021 18:44
URI: https://eprints.illc.uva.nl/id/eprint/1779

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