Some lattice Horn sentences for submodules of prime power characteristic (Q1333069)

For a unital ring \(R\) of prime power characteristic \(p^ k\) let the class, in fact the quasivariety, of lattices embeddable in the submodule lattices of \(R\)-modules be denoted by \({\mathcal L}(R)\). Let \({\mathbf W} (p^ k) = \{{\mathcal L} (R): \text{char} R = p^ k\}\). Hutchinson [\textit{G. Czédli} and \textit{G. Hutchinson}: ``Submodule lattice quasi-varieties and exact embedding functors for rings with prime power characteristic'', submitted to Algebra Univers.] gave a necessary condition, leading to interesting consequences, for the inclusion \({\mathcal L} (R_ 1) \subseteq {\mathcal L} (R_ 2)\) when \({\mathcal L} (R_ 1)\), \({\mathcal L} (R_ 2) \in {\mathbf W} (p^ k)\). However, it is not known if this condition is sufficient. Another open problem from the above-mentioned paper is whether \({\mathbf W} (p^ k)\) is closed with respect to arbitrary joins. Using certain appropriate lattice Horn sentences, the present paper shows that at least one of the above-mentioned two problems has a negative solution.