Maximal elements of quasi upper semicontinuous preorders on compact spaces

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