Optimal Matchings in Posets

We want to match (order) ideals of posets P and Q with respect to a relation that associates with every element of P an ideal of Q and conversely. The general theory for this distributive analog of classical matching theory is investigated and the analogs of the classical matching theorems are obtained. The collection of matchable ideals of P gives rise to a distributive supermatroid whose lattice of closed ideals is representable in the lattice of subspaces of a projective geometry. It is shown that with respect to order reversing weightings on P and Q, optimal matchings may be constructed according to the greedy algorithm for posets. The theory of integral vector linkings is discussed within this context.