A fixed point theorem for commuting families of relational homomorphisms. Applications to metric spaces, ordered sets and oriented graphs

Abstract

We extend the notion of compact normal structure to binary relational systems. The notion was introduced by J.P. Penot for metric spaces. We prove that for involutive and reflexive binary relational systems, every commuting family of relational homomorphisms has a common fixed point. The proof is based upon the clever argument that J.B. Baillon discovered in order to show that a similar conclusion holds for bounded hyperconvex metric spaces. This was refined by the first author to metric spaces with a compact normal structure. Since non-expansive mappings are relational homomorphisms, our result includes those of T.C. Lim, J.B. Baillon and the first author. We show that it extends Tarski's fixed point theorem to graphs which are retracts of reflexive oriented zigzags of bounded length. In doing so, we illustrate the fact that the study of binary relational systems and of generalized metric spaces are equivalent.