Fixed points of products and the strong fixed point property (Q1106249)

The paper is motivated by the well known open problem: If ordered sets X and Y both have the fixed point property (fpp), will their product XY also have the fixed point property? The authors introduce what they call the strong fixed point property: An ordered set X has the strong fixed point property if there is an order preserving map \(\Phi\) of \(X^ X\) to X such that \(\Phi\) (f) is a fixed point of f for all \(f\in X^ X\). Such a map \(\Phi\) is also called a selection map for X. Examples of four classes of ordered sets with the strong fixed point property are given, and questions are raised about the extent of the strong fixed point property within the collection of ordered sets with the fixed point property. Question: Does the fixed point property imply the strong fixed point property in general? In the case of finite ordered sets? What is the relation between the strong fixed point property and the relational fixed point property (every order preserving multifunction has a fixed point)? Is there an example of a finite ordered set with the fpp that is not dismantlable? Some results: Let X and Y be ordered sets with the fixed point property. If at least one of X and Y has the strong fixed point property, then XY has the fpp. If both X and Y have the strong fixed point property, then XY has the strong fixed point property.