A new approach to the lattice of convex sublattices of a lattice (Q1906525)

Let \(L\) be a lattice and \(\text{CS} (L)\) the set of all the (nonempty) convex sublattices of \(L\). The corresponding lattice \(\text{CS}(L) \cup \{\emptyset\}\) ordered by the inclusion is extensively studied. The authors consider another order relation on \(\text{CS}(L)\): \(A \leq B \Leftrightarrow\) 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 \(b \geq a\). This seems to be a lattice more closely related to the lattice \(L\). \(L\) and \(\text{CS}(L)\) belong to the same equational class. \(\text{CS}(L)\) is characterized by an embedding theorem in the direct product of the lattice I\((L)\) of all the ideals of \(L\) and the lattice \(\text{D} (L)\) of all the filters of \(L\). Relative completeness, pseudocompleteness and completeness of this lattice are also studied. An isomorphism between the lattice of all the congruences of \(\text{CS} (L)\) and the lattice of all the congruences of the above mentioned direct product is performed.