Weighted join semilattices and transversal matroids

We investigate join-semilattices in which each element is assigned a nonnegative weight in a strictly increasing way. A join-subsemilattice of a Boolean lattice is weighted by cardinality, and we give a characterization of these in terms of the notion of a spread. The collection of flats with no coloops (isthmuses) of a matroid or pregeometry, partially ordered by set-theoretic inclusion, forms a join-semilattice which is weighted by rank. For transversal matroids these join-semilattices are isomorphic to join-subsemilattices of Boolean lattices. Using a previously obtained characterization of transversal matroids and results on weighted join-semilattices, we obtain another characterization of transversal matroids. The problem of constructing a transversal matroid whose join-semilattice of flats is isomorphic to a given join-subsemilattice of a Boolean lattice is then investigated.