Rigidity of unary algebras and its application to the HS = SH problem
AbstractH. 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.
|Journal series||Algebra universalis, ISSN 0002-5240, 1420-8911, (0 pkt)|
|Keywords in English||algebra, coalgebra, coalgebraic logic, functor, preimage preservation, Primary: 03B70, Secondary: 03C99|
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