The Lattice of Completions of an Ordered Set

For any ordered set P, the join dense completions of P form a complete lattice K(P) with least element O(P), the lattice of order ideals of P, and greatest element M(P), the Dedekind–MacNeille completion P. The lattice K(P) is isomorphic to an ideal of the lattice of all closure operators on the lattice O(P). Thus it inherits some local structural properties which hold in the lattice of closure operators on any complete lattice. 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.