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 \(\mathcal K\).
The author proves that, for a locally finite discriminator variety \(\mathcal M\) of a finite type, the ordered set \(\langle{\mathcal I}{\mathcal M}_{\aleph_ 0}/\equiv,\leq\rangle\) is an upper semilattice only if a) every simple algebra \(A\in {\mathcal M}\) is quasiprimal, b) every proper subalgebra in \(A\) is one-element, and c) for any two one-element subalgebras of \(A\), there exists an automorphism of \(A\) which maps one of these subalgebras to the other. The same conditions are necessary for \(\langle {\mathcal I}{\mathcal M}_{\aleph_ 0}/\equiv, \leq\rangle\) to be a lower semilattice, but, besides that, d) every simple algebra \(A\in {\mathcal M}\) has a one-element subalgebra. Conversely, if a discriminatory variety \(\mathcal M\) of a finite type is generated by a finite set of algebras satisfying a)–c) [resp. a)–d)] then the ordered set \(\langle{\mathcal I}{\mathcal K}_{\aleph_ 0}/\equiv, \leq\rangle\) is an upper [a lower] semilattice and has rather a transparent structure.
Reviewer’s remark: The same problem was studied for non-locally finite discriminator varieties by A. G. Pinus in the paper reviewed below.