The lattice of strict completions of a poset

For a given finite poset (P,⩽), a strict completion of P is a finite lattice L such that the set of join-irreducible elements of L is isomorphic to P. The family MP of strict completions of P turns out to be itself a lattice, which is lower bounded and lower semimodular. We study the cases where this lattice is atomistic, distributive or boolean. We relate some other properties of lattice MP to properties of our given poset P, and in particular we characterize the posets P for which ∣MP∣ ⩽ 2.