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Epimorphism Surjectivity in Varieties of Heyting Algebras

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    0532809 - ÚI 2021 RIV NL eng J - Journal Article
    Moraschini, Tommaso - Wannenburg, J. J.
    Epimorphism Surjectivity in Varieties of Heyting Algebras.
    Annals of Pure and Applied Logic. Roč. 171, č. 9 (2020), č. článku 102824. ISSN 0168-0072. E-ISSN 1873-2461
    R&D Projects: GA MŠk(CZ) EF17_050/0008361
    Institutional support: RVO:67985807
    Keywords : Epimorphism * Heyting algebra * Esakia space * Intuitionistic logic * Intermediate logic * Beth definability
    OECD category: Pure mathematics
    Impact factor: 0.678, year: 2020
    Method of publishing: Limited access
    http://dx.doi.org/10.1016/j.apal.2020.102824

    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.
    Permanent Link: http://hdl.handle.net/11104/0311209

     
     
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