Maximality with or without binariness: Transfer-type characterizations

We provide the first necessary and sufficient conditions in the literature for the existence of maximal elements in a non-binary choice framework. As an application, the characterization of the existence of maximal elements for acyclic binary relations obtained in Rodríguez-Palmero and García-Lapresta, 2002, Mathematical Social Sciences 43, 55-60 [Rodríguez-Palmero, C., García-Lapresta, J.L., 2002. Maximal elements for irreflexive binary relations on compact sets. Mathematical Social Sciences 43, 55-60], is deduced as a Corollary. Further characterizations in different settings given by Tian and Zhou, 1995, Journal of Mathematical Economics 24, 281-303 [Tian, G., Zhou, J., 1995. Transfer continuities, generalizations of the Weierstrass and maximum theorems: a full characterization. Journal of Mathematical Economics 24, 281-303], follow as well. Analogous characterization problems in the k-acyclic binary cases are solved too.