Epimorphism surjectivity in varieties of Heyting algebras

0522847 - ÚI 2020 US eng V - Research Report
Moraschini, Tommaso - Wannenburg, J. J.
Epimorphism surjectivity in varieties of Heyting algebras.
Cornell University, 2019. 36 s. arXiv.org e-Print archive, arXiv:1908.00287 [math.LO].
Institutional support: RVO:67985807
Keywords : Epimorphism * Heyting algebra * Brouwerian algebra * Esakia space * intuitionistic logic * intermediate logic * Beth definability
OECD category: Pure mathematics
https://arxiv.org/abs/1908.00287

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 greater or equal than 2, the variety of all Heyting algebras of width at most n has a non-surjective epimorphism. Within the so-called Kuznetsov-Gerciu 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 Goedel algebras. The results settle natural questions about Beth-style definability for a range of intermediate logics.
Permanent Link: http://hdl.handle.net/11104/0307269