Subsets of the square with the continuous and order preserving fixed point property

This study characterizes the convex sets whose complements in the unit square exhibit the fixed point property for mappings which are jointly continuous and order preserving. Hence, one can readily construct simple sets with this fixed point property, but which neither have the fixed point property individually for continuous mappings nor for order preserving mappings. This is the first characterization of any non-trivial set with this property.