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Rigidity of unary algebras and its application to the $${\mathcal {HS} = \mathcal {SH}}$$ problem. (English) Zbl 1221.08004
H. P. Gumm and T. Schröder [Algebra Univers. 53, No. 2–3, 229–252 (2005; Zbl 1086.08002)] showed that if a functor $$T:$$ {Set} $$\to$$ {Set} preserves preimages, then $${\mathcal{H}}{\mathcal{S}}(K)={\mathcal{S}}{\mathcal{H}}(K)$$ is valid for any class $$K$$ of $$T$$-algebras. The present author applies a construction of rigid unary algebras to prove that $${\mathcal{H}}{\mathcal{S}}\neq{\mathcal{S}}{\mathcal{H}}$$ for a class of {Set}-endofunctors not preserving non-empty preimages. For related results cf. also papers by T. Brengos and V. Trnková [Algebra Univers. 63, No. 2–3, 283–301 (2010; Zbl 1220.03011)] and V. Trnková [Commentat. Math. Univ. Carol. 10, 323–352 (1969; Zbl 0183.30401); ibid. 12, 143–174 (1971; Zbl 0232.18004)].
##### MSC:
 08A60 Unary algebras 03G30 Categorical logic, topoi 08A70 Applications of universal algebra in computer science 18A22 Special properties of functors (faithful, full, etc.) 18B05 Categories of sets, characterizations
##### Keywords:
coalgebra; preimage preservation; functor; coalgebraic logic
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##### References:
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