We consider strategy-proof rules operating on a rich domain of preference profiles in a set up where multiple private goods have to be assigned to a set of agents with entitlements where preferences display satiation. We show that if the rule is in addition "desirable", in that it is tops-only, continuous, same-sided and individually rational with respect to the entitlements, then the preferences in the domain have to satisfy a variant of single-peakedness (referred to as semilattice single-peakedness). We also provide a converse of this main finding. It turns out that this domain coincides with the one already identified in a general set up with a public good. Finally, we relate semilattice single-peaked domains to well-known restricted domains under which strategy-proof and desirable rules do exist.