For any TU game and any ranking of players, the set of all preimputations compatible with the ranking, equipped with the Lorenz order, is a bounded join semi-lattice. Furthermore, the set admits as sublattice the S-Lorenz core intersected with the region compatible with the ranking. This result uncovers a new property about the structure of the S-Lorenz core. As immediate corollaries, we obtain complementary results to the findings of Dutta and Ray :403–422, 1991), by showing that any S-constrained egalitarian allocation (...) is the Lorenz greatest element of the S-Lorenz core on the rank-preserving region the allocation belongs to. Besides, our results suggest that the comparison between W- and S-constrained egalitarian allocations is more puzzling than what is usually admitted in the literature. (shrink)