General Conditions for Existence of Maximal Elements via the Uncovered Set

This paper disentangles the topological assumptions of classical results (e.g., Walker (1977)) on existence of maximal elements from rationality conditions. It is known from the social choice literature that under the standard topological conditions-with no other restrictions on preferences-there is an element such that the upper section of strict preference at that element is minimal in terms of set inclusion, i.e., the uncovered set is non-empty. Adding a condition that weakens known acyclicity and convexity assumptions, each such uncovered alternative is indeed maximal. A corollary is a result that weakens the semi-convexity condition of Yannelis and Prabhakar (1983).