Embeddability skeletons of non-locally finite discriminator varieties with a poor algebra (Q1346931)

For a class of algebras \(\mathcal K\), let \({\mathcal I}{\mathcal K}_{\aleph_ 0}\) denote the class of isomorphism types of finite or countable algebras from \(\mathcal K\). For \(a,b\in {\mathcal I}{\mathcal K}_{\aleph_ 0}\) being the isomorphism types of the algebras \(A,B\in {\mathcal K}\) respectively, let be \(a\leq b\) whenever \(A\) is isomorphic to a subalgebra of \(B\) and \(a\equiv b\) whenever \(a\leq b\) and \(b\leq a\). The quasiordered set \(\langle {\mathcal I}{\mathcal K}_{\aleph_ 0},\leq \rangle\) is called the countable embeddability skeleton of the class \(\mathcal K\). \textit{Ya. L. Mordvinov} [in the paper reviewed above] has described locally finite discriminator varieties \(\mathcal M\) such that the ordered set \(\langle{\mathcal I}{\mathcal M}_{\aleph_ 0}/\equiv, \leq \rangle\) is either an upper or a lower semilattice. In the paper under review the author proves that if a discriminator variety \(\mathcal M\) contains an infinite finitely generated simple algebra \(A\) such that the Boolean algebra of solutions of equation systems over \(A\) is superatomic, then the ordered set \(\langle{\mathcal I}{\mathcal M}_{\aleph_ 0}/\equiv, \leq\rangle\) is neither an upper nor a lower semilattice.