A new ordering on the set of all sublattices of a lattice (Q1907289)

Let \(L\) be a lattice and \(\text{Sub}(L)\) the set of all the (nonempty) sublattices of \(L\). The corresponding lattice \(\text{Sub} (L) \cup \{\emptyset\}\) ordered by inclusion is extensively studied. The authors consider another order relation on \(\text{Sub}(L): A \leq B \Leftrightarrow\) for every \(a \in A\) there is a \(b \in B\) such that \(a \leq b\) and for every \(b \in B\) there is an \(a \in A\) such that \(b \geq a\). They prove that together with this partial order \(\text{Sub}(L)\) is a lattice iff \(L\) satisfies one of the following two conditions: (i) all subchains of \(L\) are finite; (ii) \(L\) is chain isomorphic to the usual ordered chain of all the negative integers. Further, if \(L\) is a join semidistributive noetherian and artinian lattice then \(\text{Sub}(L)\) is a join semidistributive lattice iff \(L\) does not contain a direct product of a 2-chain with a 3-chain as sublattice.