Geodesic
Finite orders and their minimal strict completion lattices
Discussiones Mathematicae. General Algebra and Applications, Tome 23 (2003) no. 2, pp. 85-100

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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 join-irreducible). 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.
Keywords: atomistic lattice, join-irreducible element, distributive lattice, modular lattice, lower semimodular lattice, Dedekind-MacNeille completion, strict completion, weak order. 
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Bordalo, Gabriela; Monjardet, Bernard. Finite orders and their minimal strict completion lattices. Discussiones Mathematicae. General Algebra and Applications, Tome 23 (2003) no. 2, pp. 85-100. http://geodesic.mathdoc.fr/item/DMGAA_2003_23_2_a0/