Profiniteness and representability of spectra of Heyting algebras

0545611 - ÚI 2022 RIV US eng J - Journal Article
Bezhanishvili, G. - Bezhanishvili, N. - Moraschini, T. - Stronkowski, Michał
Profiniteness and representability of spectra of Heyting algebras.
Advances in Mathematics. Roč. 391, 19 November 2021 (2021), č. článku 107959. ISSN 0001-8708. E-ISSN 1090-2082
Institutional support: RVO:67985807
Keywords : Heyting algebra * Profinite algebra * Profinite completion * Representation Problem * Priestley space * Esakia space
OECD category: Pure mathematics
Impact factor: 1.675, year: 2021
Method of publishing: Open access
http://dx.doi.org/10.1016/j.aim.2021.107959

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.
Permanent Link: http://hdl.handle.net/11104/0322290