Order Extensions and the Fixed Point Property

The purpose of this paper is to investigate how the fixed point property and its negation behave when a covering relation is added to the order. We prove that every finite ordered set which is not totally ordered and which is dismantlable by retractables, respectively by irreducibles, has an upper cover (in its extension lattice) which is also dismantlable by retractables, respectively by irreducibles. We also provide examples of finite ordered sets having the fixed point property so that none of their upper covers has the fixed point property. Part of this work was done while the author was visiting Brandon University. The author thanks M. Roddy for his hospitality and financial support.