  Defining P* to be the complete lattice of upsets (ordered by reverse
inclusion) of a poset P we give necessary and sufficient conditions on a subset
S of P* for P to admit a meet-completion e from P to Q where e preserves the
infimum of an upwardly closed set from P if and only if it is in S. We show
that given S satisfying these conditions the set M of these completions forms a
topped weakly lower semimodular lattice. In particular, when P is finite M is a
lower semimodular lattice, and a lower bounded homomorphic image of a free
lattice. We provide an example where M does not have a bottom element.
