Rigidity of unary algebras and its application to the $${\mathcal {HS} = \mathcal {SH}}$$ problem |
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Authors: | Tomasz Brengos |
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Institution: | 1.Faculty of Mathematics and Information Science,Warsaw University of Technology,Warszawa,Poland |
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Abstract: | H. P. Gumm and T. Schröder stated a conjecture that the preservation of preimages by a functor T for which |T1| = 1 is equivalent to the satisfaction of the class equality \({{\mathcal {HS}}({\sf K}) = {\mathcal {SH}}({\sf K})}\) 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 \({{\mathcal {HS}} \neq {\mathcal {SH}}}\) for a class of Set-endofunctors not preserving non-empty preimages; these functors have not been considered previously. |
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