In this talk, we will discuss external complete lattice structures, a new concept that was initially introduced in metric spaces as externally hyperconvex sets by Aronszajn and Panitchpakdi in their fundamental paper on hyperconvexity. This idea developed from the original work of A. Quilliot who introduced the concept of generalized metric structures to show that metric hyperconvexity is in fact similar to complete lattice structure for ordered sets. In this fashion, Tarski's fixed point theorem becomes Sine and Soardi's fixed point theorem for hyperconvex metric spaces.
Our main result yields the fact that an order preserving set-valued mapping of a complete lattice set, taking externally complete lattice values, always has a single valued selection which is order preserving. Several fixed point theorems are also obtained.