Abstract
H. P. Gumm and T. Schröder stated a hypothesis that the preservation of preimages by a functor T for which |T1| = 1 is equivalent to the satisfaction of the class equality \({{\mathcal {HS}}(\tt {K}) = {\mathcal {SH}}(\tt {K})}\) for any class K of T-coalgebras. Although we were not able to find a full solution, our paper gives a positive answer to this problem for a very wide class of Set-endofunctors.
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Presented by H. P. Gumm.
The first author was supported by the Grant of Warsaw University of Technology no. 504G 1120 0112 000 and acknowledges the hospitality of the Faculty of Mathematics and Physics, Charles University, during his visit to Prague in September 2008, when a part of the work on the paper was completed. The second author gratefully acknowledges the support of MSM 0021620839, a project of the Czech Ministry of Education, and of the grant 201/06/0664 by the Grant Agency of Czech Republic.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Brengos, T., Trnková, V. The \({{\mathcal {HS}} = {\mathcal {SH}}}\) problem for coalgebras. Algebra Univers. 63, 283–301 (2010). https://doi.org/10.1007/s00012-010-0079-y
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DOI: https://doi.org/10.1007/s00012-010-0079-y