Upper semicontinuous representability of maximal elements for nontransitive preferences (Q2420813)

It is known that for every maximal element relative to a preorder on any nonempty set, White's Theorem, see [\textit{D. J. White}, Eur. J. Oper. Res. 4, 426--427 (1980; Zbl 0437.90006)] guarantees the existence of an order-preserving function attaining its maximum precisely at that maximal element. Note that the existence of an order-preserving function for a (reflexive) binary relation does not necessarily imply transitivity. In [\textit{P. Bevilacqua}, \textit{G. Bosi} and \textit{M. Zuanon}, Far East J. Math. Sci. (FJMS) 103, No. 1, 213---221 (2018), Theorem 3.1] White's Theorem was generalized in the sense that, for every maximal element relative to a preorder on a nonempty set, there is a weak utility (for the strict part of the preorder) attaining its maximum at that maximal element, provided that a weak utility for the strict part exists. \par In this article, the authors generalize White's theorem in the sense that they present necessary and sufficient conditions on a preorder on a topological space under which, for every maximal element relative to a preorder, there exists an upper semicontinuous order-preserving function attaining its maximum at such maximal element, provided that an upper semicontinuous order-preserving function exists (see Theorem 3.2). \par Finally, the authors show that the above results cannot be generalized to the case of a quasi-preorder (i.e., a reflexive and Suzumura consistent binary relation). In particular, they show that an order-preserving function for a quasi-preorder cannot attain its maximum at any maximal element which is not at the same time a maximal element for its transitive closure (see Proposition 2.5 and Example 2.1).