by Tomasz Kowalski and Tadeusz Litak
Reference:
Tomasz Kowalski and Tadeusz Litak: Completions of GBL-algebras: negative results, In Algebra Universalis, 58(4), pp. 373–384, 2008. An exercise in substructural logic,
motivated partially by my earlier results in modal logic. We show that
many varieties related to many-valued logics are not closed under any
kind of completions - not just a specific one, like canonical
completions. A significant part of the paper has been incorporated into
this book. The result turned out to be important for an emerging field of algebraic proof theory (Kazushige Terui, Nick Galatos, Agata Ciabattoni) who showed this implies non-existence of well-behaved, strongly analytical, cut-free sequent calculi for logics in question. In his invited talk at OAL2.0 in 2011, Kazushige Terui likened the significance of our result for algebraic proof theory to that of the Gödel results for the Hilbert program, but it was a hyperbole - algebraic proof theory is alive and well. Neither he nor ourselves realized at that time that the restriction of our result to MV-algebras had been carefully hidden in the 2004 Math. Scand. paper by Gehrke and Jonsson. Look for it yourself (and note that you have the advantage of this precise bibliographical reference)... I only found it by accident in 2016.
Bibtex Entry:
@article{KowalskiL08:au,
year={2008},
issn={0002-5240},
journal={Algebra Universalis},
volume={58},
number={4},
doi={10.1007/s00012-008-2056-2},
title={Completions of {GBL}-algebras: negative results},
url={http://dx.doi.org/10.1007/s00012-008-2056-2},
publisher={Birkh\"{a}user Verlag Basel},
keywords={06F05; 06F15; 06D35; 03G10; Completions; canonicity; MV-algebras; BL-algebras; $\ell$-groups},
author={Kowalski, Tomasz and Litak, Tadeusz},
pages={373--384},
language={English},
comment={<div class="textleft">An exercise in substructural logic,
motivated partially by my earlier results in modal logic. We show that
many varieties related to many-valued logics are not closed under any
kind of completions - not just a specific one, like canonical
completions. A significant part of the paper has been incorporated into
<a href="http://www.amazon.ca/Residuated-Lattices-Algebraic-Glimpse-Substructural/dp/0444521410/ref=sr_1_1/702-3093866-3176006?ie=UTF8&s=books&qid=1185542660&sr=1-1">this book</a>. The result turned out to be important for an emerging field of algebraic proof theory (Kazushige Terui, Nick Galatos, Agata Ciabattoni) who showed this implies non-existence of well-behaved, strongly analytical, cut-free sequent calculi for logics in question. In his invited talk at <a href="http://2oal.tcs.uj.edu.pl/">OAL2.0 in 2011</a>, Kazushige Terui likened the significance of our result for algebraic proof theory to that of the Gödel results for the Hilbert program, but it was a hyperbole - algebraic proof theory is alive and well. Neither he nor ourselves realized at that time that the restriction of our result to MV-algebras had been carefully hidden in the 2004 Math. Scand. paper by Gehrke and Jonsson. Look for it yourself (and note that you have the advantage of this precise bibliographical reference)... I only found it by accident in 2016. </div>}
}