On the utility of Robinson-Amitsur ultrafilters. III (Q2678440)

In the paper \textit{P. Zusmanovich} [J. Algebra 466, 370--377 (2016; Zbl 1437.03122)] the author has shown: Let \(\kappa\) be a strongly compact cardinal. Then any embedding of a \(\kappa\)-subdirectly irreducible algebra into a direct product can be factored by a \(\kappa\)-complete ultrafilter. Here the author wants to get rid of \(\kappa\) being strongly compact. This is done by replacing subdirect irreducibility by indecomposibility. He shoes the following theorem:\par Let \(A\) be a \(\kappa\)-subdirectly irreducible algebraic system with \(\kappa \geq \omega\) and \(f : A \hookrightarrow \prod_{i \in \mathbb{I}} B_i\) an indecomposable embedding. Then there exists a \(\kappa\)-complete ultrafilter \(\mathcal{U}\) on \(\mathbb{I}\) such that the composition of \(f\) with the canonical homomorphism \(\prod_{i\in \mathbb{I}}B_i \rightarrow \prod_{\mathcal{U}} B_i\) is an embedding.