## Rigidity of unary algebras and its application to the HS = SH problem

### Tomasz Brengos

#### Abstract

H. P. Gumm and T. Schröder stated a conjecture that the preservation of preimages by a functor T for which \\textbar\T1\\textbar\ = 1 is equivalent to the satisfaction of the class equality \$\\\\textbackslash\mathcal \HS\$\\\textbackslash\sf K$ = \\\textbackslash\mathcal \SH\$\\\textbackslash\sf K$\\$ for any class K of T-coalgebras. Although T. Brengos and V. Trnková gave a positive answer to this problem for a wide class of Set-endofunctors, they were unable to find the full solution. Using a construction of a rigid unary algebra we prove \$\\\\textbackslash\mathcal \HS\\ \\textbackslash\neq \\\textbackslash\mathcal \SH\\\\$ for a class of Set-endofunctors not preserving non-empty preimages; these functors have not been considered previously.
 Author Tomasz Brengos (FMIS / DAC) Tomasz Brengos,, - Department of Algebra and Combinatorics Journal series Algebra universalis, ISSN 0002-5240, 1420-8911, (0 pkt) Issue year 2011 Vol 65 No 1 Pages 73-89 Keywords in English algebra, coalgebra, coalgebraic logic, functor, preimage preservation, Primary: 03B70, Secondary: 03C99 DOI DOI:10.1007/s00012-011-0118-3 URL http://link.springer.com/article/10.1007/s00012-011-0118-3 Language en angielski Score (nominal) 0 Score source journalList Publication indicators WoS Citations = 0 Citation count*
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