The lattice of completions of an ordered set
The join dense completions of an ordered set \(P\) form a complete lattice \(K(P)\). Its least element is \(\mathcal O(P)\), the lattice of all order ideals of \(P\), and its greatest element is \(\mathcal M(P)\), the Dedekind-MacNeille completion of \(P\). It is isomorphic to an ideal of the lattice of all closure operators on \(\mathcal O(P)\). Some local structural properties of lattices of closure operators on complete lattices are proved. These are inherited by \(K(P)\) as well. In particular, if \(K(P)\) is finite, then it is an upper semimodular lattice and an upper bounded homomorphic image of a free lattice, and hence meet semidistributive.
scientific article; zbMATH DE number 1067849