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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 joinirreducible). In this paper we study lattices which are minimal strict completions of flnite orders. Such lattices are in one-to-one correspondence with flnite posets. Among other results we show that, for every flnite poset P; D(P) ⁄ is always generated by its doubly-irreducible elements. Furthermore, we characterize the posets P for which D(P) ⁄

Finite orders and their minimal strict completion lattices