Finite Orders and Their Minimal Strict Completion Lattices

Whereas the Dedekind-MacNeille completion D(P ) of a poset P is the minimal lattice L such that every element of L is a join of elements of P, the minimal strict completion D(P )∗ is the minimal lattice L such that the poset of join-irreducible elements of L is isomorphic to P. (These two completions are the same if every element of P is joinirreducible). In this paper we study lattices which are minimal strict completions of finite orders. Such lattices are in one-to-one correspondence with finite posets. Among other results we show that, for every finite poset P, D(P )∗ is always generated by its doubly-irreducible elements. Furthermore, we characterize the posets P for which D(P )∗ is a lower semimodular lattice and, equivalently, a modular lattice.