Maximal elements of quasi upper semicontinuous preorders on compact spaces

Abstract

We introduce the concept of quasi upper semicontinuity of a not necessarily\ud total preorder on a topological space and we prove that there exists a maximal\ud element for a preorder on a compact topological space provided that it is quasi upper\ud semicontinuous. In this way, we generalize many classical and well known results in\ud the literature. We compare the concept of quasi upper semicontinuity with the other\ud semicontinuity concepts to arrive at the conclusion that our definition can be viewed\ud as the most appropriate and natural when dealing with maximal elements of preorders\ud on compact spaces

Representing preorders with injective monotones

“…In fact, it was only recently in Minguzzi (2013) where strict monotone multi-utilities were introduced and later in Alcantud et al (2013Alcantud et al ( , 2016 where they were further studied. The relation of these ideas with optimization and the existence of maximal elements is also present in the literature (Bevilacqua et al, 2018a, b;Bosi & Zuanon, 2017;Bosi et al, 2018;White, 1980). Countable multiutilities where studied particularly in Alcantud et al (2016), Bevilacqua et al (2018c), while finite multi-utility representations were notably advanced in Kaminski (2007) and Ok (2002) and, in vector optimization, in Jahn (2009.…”

Existence of Order-Preserving Functions for Nontotal Fuzzy Preference Relations under Decisiveness

“…Let us recall the concept of a maximal element for a (quasi-)preorder (see e.g., Bosi and Zuanon [20]). …”

“…The distinction between the concept of a -order-preserving function and that of a weak utility for the strict part of proved to be useful in connection with the existence of maximal elements of a preorder (see Bosi and Zuanon [20]). …”

Pareto Optimality on Compact Spaces in a Preference-Based Setting under Incompleteness

“…In particular, we refer to the hypothesis of weak upper semicontinuity of the individual preorders i . Such an assumption appears in connection with the existence of an upper semicontinuous order-preserving function for a not necessarily total preorder on a topological space (see e.g., Bosi and Herden 12,13 Pareto Optimality on Compact Spaces 241 and Bosi and Zuanon 14 ). Recall that a preorder on a topological space (X, τ ) is said to be weakly upper semicontinuous if for every pair (x, y) ∈ ≺ there exists some open decreasing subset O x,y of X such that x ∈ O x,y and y ∈ X\O x,y .…”