Finite distributive lattices which are isomorphic to direct products of chains (Q1272126)

For a lattice \(L\), let \(CS(L)\) be the set of all convex sublattices of \(L\) ordered by the order relation \(A\leq B\) iff for each \(a\in A\) there exists a \(b\in B\) with \(a\leq b\) and for each \(b\in B\) there exists an \(a\in A\) such that \(b\geq a\). The lattice \(CS(L)\) was also studied by \textit{G. Dorfer} [``Lattice-extensions by means of convex sublattices'', Contributions to general algebra 9, 127-132 (1995; Zbl 0884.06004)]. The authors describe the class \(\mathcal K\) of all lattices \(L\) for which every quotient lattice is a sublattice of \(CS(L)\). They show that \(\mathcal K\) does not form a variety and they characterize distributive members of \(\mathcal K\). It is proved that a finite distributive lattice \(L\) belongs to \(\mathcal K\) iff \(L\) is isomorphic to a direct product of chains.