A unified view of the existence of maximals

The paper examines the conditions for the existence of maximals of a relation on every nonempty compact subset of its ground set. A preliminary analysis shows that the existence of maximals of a Suzumura-consistent relation is implied by the existence of maximals of the right trace of its transitive closure. Building on this fact, various theorems of the literature are unified by identifying a common topological property of their assumptions that concerns the right trace of the transitive closure of a relation. Next, a generalization is provided so as to accommodate some cases of interest to economics. Finally, a necessary and sufficient condition is presented for the existence of maximals on every nonempty compact subset of the ground set of a relation.