Fixed Point Theory in Ordered Sets from the Metric Point of View

In 1983 A. Quilliot published his original work on graphs and ordered sets viewed as metric spaces. His approach was revolutionary. It was the first time that metric ideas and concepts could be defined in discrete sets. In particular one can show that graphs or order preserving maps are exactly the class of nonexpansive mappings defined on metric spaces. Pouzet and his students Jawhari and Misane were able to build on Quilliot’s ideas to establish some new insights into absolute retracts in ordered sets. For example it was amazing that the metric results discovered by Aronszajn and Panitchpakdi (Pac. J. Math. 6:405–439, 1956), the work of Isbell (Comment. Math. Helv. 39:439–447, 1964), and the fixed point theorems of Sine (Nonlinear Anal. 3:885–890, 1979) and Soardi (Proc. Am. Math. Soc. 73:25–29 1979) are exactly the Banaschewski–Bruns theorem (Archiv. Math. Basel 18:369–377, 1967), the MacNeille completion (Trans. Am. Soc. 42:416–460, 1937) and the famous Tarski fixed point theorem (Pac. J. Math. 5:285–309, 1955). Recently Abu-Sbeih and Khamsi used the same ideas to define a concept similar to externally hyperconvex metric sets introduced by Aronszajn and Panitchpakdi in their original work in ordered sets. They also proved a an intersection property similar to the one discovered by Baillon in metric spaces to show a common fixed point result. In conclusion this approach supports the idea that certain concepts of infinistic nature, like those which inspired metric spaces, can easily translate into discrete structures like ordered sets and graphs. In this chapter, we will only focus on ordered sets.