On Höft's characterization of weak model classes (Q1902531)

In 1973, \textit{H. Höft} [ibid. 3, 203-215 (1973; Zbl 0287.08003)]\ characterized weak model classes of partial algebras. The authors show a counterexample to this characterization and propose a corrected version of this characterization: Theorem. Let \({\mathcal K}\) be a class of partial algebras of the same type. \({\mathcal K}\) is a weak equational class iff \({\mathcal K}\) is inductive, weakly equationally faithful and closed under weak subalgebras, closed homomorphic images and direct products.