The Fixed Point Property in the Set of All Order Relations on a Finite Set

For a finite ground set X, this paper investigates properties of the set of orders with the fixed point property as a subset of the set ${\mathcal O} (X)$ of all orders on X, ordered by inclusion. In particular, it is shown that this set can have singleton components in the covering graph of ${\mathcal O} (X)$ , we identify longest possible chains of orders such that orders alternate between having and not having the fixed point property, and we give examples of nondismantlable orders with the fixed point property such that every upper cover in ${\mathcal O} (X)$ has the fixed point property, too.