Transfer Method for Characterizing the Existence of Maximal Elements of Binary Relations on Compact or Noncompact Sets

Abstract

This paper systematically studies the existence of maximal elements for unordered binary relations on compact or noncompact sets in a general topological space. This is done by developing a method, called transfer method, to derive various necessary and sufficient conditions that characterize the existence of maximal elements for a binary relation in terms of: (1) (generalized) transitivity conditions under certain topological assumptions; (2) topological conditions under certain (generalized) transitivity assumptions; and (3) (generalized) convexity conditions under certain topological assumptions. There are two basic approaches in the literature to prove the existence by providing sufficient conditions. One assumes certain convexity and continuity conditions for a topological vector space and the other assumes certain weakened transitivity and continuity conditions for a general topological space. The results unify those two aproaches and generalize almost all of the existing results in the literature.