Upper Semicontinuous Representability of Maximal Elements for Nontransitive Preferences

We characterize the possibility of determining all the maximal elements for a preorder on a topological space by maximizing all the functions in a suitable family of upper semicontinuous order-preserving functions. We discuss the possibility of extending this result to the case of a quasi-preorder (i.e., a reflexive and Suzumura consistent binary relation) on a topological space.