Weakening partial orderings to lattice and semilattice orderings (Q1078586)

It is shown that a partial ordering on a set P can be weakened to an upper or lower semilattice ordering if and only if P is filtered in the appropriate direction, and to a lattice ordering if and only if it is filtered in both directions. Questions are raised and some counterexamples given for analogous questions asking for semilattice and lattice orderings satisfying completeness conditions.