Profiniteness and representability of spectra of Heyting algebras

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F21%3A00545611" target="_blank" >RIV/67985807:_____/21:00545611 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1016/j.aim.2021.107959" target="_blank" >http://dx.doi.org/10.1016/j.aim.2021.107959</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.aim.2021.107959" target="_blank" >10.1016/j.aim.2021.107959</a>

Result language
angličtina
Original language name
Profiniteness and representability of spectra of Heyting algebras
Original language description
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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2021
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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