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Epimorphism Surjectivity in Varieties of Heyting Algebras
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SYSNO ASEP 0532809 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title Epimorphism Surjectivity in Varieties of Heyting Algebras Author(s) Moraschini, Tommaso (UIVT-O) SAI, RID
Wannenburg, J. J. (ZA)Number of authors 2 Article number 102824 Source Title Annals of Pure and Applied Logic. - : Elsevier - ISSN 0168-0072
Roč. 171, č. 9 (2020)Number of pages 31 s. Publication form Print - P Language eng - English Country NL - Netherlands Keywords Epimorphism ; Heyting algebra ; Esakia space ; Intuitionistic logic ; Intermediate logic ; Beth definability Subject RIV BA - General Mathematics OECD category Pure mathematics R&D Projects EF17_050/0008361 GA MŠMT - Ministry of Education, Youth and Sports (MEYS) Method of publishing Limited access Institutional support UIVT-O - RVO:67985807 UT WOS 000553439500003 EID SCOPUS 85084860827 DOI 10.1016/j.apal.2020.102824 Annotation It was shown recently that epimorphisms need not be surjective in a variety K of Heyting algebras, but only one counter-example was exhibited in the literature until now. Here, a continuum of such examples is identified, viz. the variety generated by the Rieger-Nishimura lattice, and all of its (locally finite) subvarieties that contain the original counter-example K. It is known that, whenever a variety of Heyting algebras has finite depth, then it has surjective epimorphisms. In contrast, we show that for every integer n⩾2, the variety of all Heyting algebras of width at most n has a non-surjective epimorphism. Within the so-called Kuznetsov-Gerčiu variety (i.e., the variety generated by finite linear sums of one-generated Heyting algebras), we describe exactly the subvarieties that have surjective epimorphisms. This yields new positive examples, and an alternative proof of epimorphism surjectivity for all varieties of Gödel algebras. The results settle natural questions about Beth-style definability for a range of intermediate logics. Workplace Institute of Computer Science Contact Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 Year of Publishing 2021 Electronic address http://dx.doi.org/10.1016/j.apal.2020.102824
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