Rigidity of unary algebras and its application to the \({\mathcal {HS} = \mathcal {SH}}\) problem
\textit{H.~P. Gumm} and \textit{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 \textit{T. Brengos} and \textit{V. Trnková} [Algebra Univers. 63, No.~2--3, 283--301 (2010; Zbl 1220.03011)] and \textit{V. Trnková} [Commentat. Math. Univ. Carol. 10, 323--352 (1969; Zbl 0183.30401); ibid. 12, 143--174 (1971; Zbl 0232.18004)].
scientific article; zbMATH DE number 5886751