On P-dominions of continuous algebras (Q1076704)

For each subset system Z and each, possibly infinitary, type of algebras, a sufficient condition is presented for a Z-continuous homomorphism \(f: A\to B\) to be an epi. This is related to the problem whether, for a given Z-continuous algebra A, the cardinality of B is bounded (i.e., whether the category of Z-continuous algebras is co-wellpowered) investigated by the reviewer and \textit{V. Koubek}, \textit{E. Nelson} and \textit{J. Reiterman} [Trans. Am. Math. Soc. 291, 681-699 (1985; Zbl 0578.08005)]. The answer is negative for infinitary algebras in case \(Z=all\) countably directed subsets.