Rigidity of unary algebras and its application to the HS = SH{mathcal {HS} = mathcal {SH}} problem

摘要

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 HS(sf K) = SH(sf K){{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 HS ¹ SH{{mathcal {HS}} neq {mathcal {SH}}} for a class of Set-endofunctors not preserving non-empty preimages; these functors have not been considered previously.