A note on maximal elements for acyclic binary relations on compact topological spaces

I introduce the concept of a weakly tc-upper semicontinuous acyclic binary relation on a topological space (X,t), which appears as slightly more general than other concepts of continuity which have been introduced in the literature in connection with the problem concerning the existence of maximal elements. By using such a notion, I show that if an acyclic binary relation on a compact topological space is weakly tc-upper semicontinuous, then there exists a maximal element relative to such a binary relation. In this way I generalize existing results concerning the existence of maximal elements on compact topological spaces.