Multicontact semilattices

We characterize those join semilattices with a multicontact relation which can be embedded into (the reduct of) a Boolean algebra with multicontact given by overlap, equivalently, into a distributive lattice with additive multicontact, still equivalently, into the multicontact semilattice associated to some topological space. A similar characterization is presented for embeddability into Boolean algebras or distributive lattices with a not necessarily overlap nor additive multicontact.
Comment: v2, various improvements