The paper generalizes theorems of Ky Fan and Hugo Sonnenschein on the existence of maximal elements for non-transitive relations. I used these results to show that a binary relation could be constructed whose maximal element must be a competitive equilibrium. Thus proving the existence of competitive equilibrium under somewhat more general conditions than had been done previously. In 1975, I thought this was a useful extension of the Gale Mas Collel existence theorem. Journal referees then didn't agree with me, so I let it ripen in my desk for 15 years. I still think it is worth looking at if you are interested in the existence of competitive equilibrium or in maximization of funny preference orderings.