The Lattice of Completions of an Ordered Set J. B. Nation and Alex Pogel

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 of P. The latticeK(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 nite, then it is an upper semimodular lattice and an upper bounded homomorphic image of a free lattice, and hence meet semidistributive. A join dense completion of an ordered set P is an order embedding " : P! L ofP into a complete lattice L such that "(P) is join dense in L, i.e., for every x 2 L, x = W f"(p) : "(p) xg. It is well known that the embedding p 7! #p of P into the complete lattice O(P) of order ideals of P is a join dense completion of P. Likewise, so is the natural embedding of P into the lattice of normal ideals of P, which is the Dedekind-MacNeille completion. Each join dense completion " : P ! L induces a closure operator " on P such that "(fpg) = #p for all p 2 P, given by "(S) = fp 2 P : "(p) W "(S)g. Moreover, L is isomorphic to the lattice of closed sets of " via the mapping h : x 7! fp 2 P : "(p) xg. Conversely, let be a closure operator on P such that (fpg) = #p for all p 2 P, and let C denote the lattice of -closed sets, ordered by inclusion. Then the map : P ! C given by (p) = #p is a join dense completion of P. Moreover, if = ", then the join dense completions " and are isomorphic in the sense that h : L = C and h" = . It is naturally more convenient to work with this set of closure operators on P than to work with the class of all join dense completions. Let K(P) be the set of closure operators on P such that (fpg) = #p for all p 2 P, ordered by if (S) (S) for all S P. It is not hard to see that K(P) is a complete lattice, since it has a largest element and is closed under arbitrary nonempty intersections. The closure operator corresponding to the completion : P ! O(P) is its smallest element, and : P! M(P) induces the greatest element of K(P). With the lattice K(P) in mind, let us consider more generally the lattice of all closure operators on a complete lattice. 1991 Mathematics Subject Classi cation. 06A23, 06A15.