Coloring chains (Q1264160)

The problem is to characterize those chains (linear orderings) C having a first and second element and with all principal final segments isomorphic to C. A description is obtained in the countable scattered case. For nonscattered countable chains, the problem is the same as the following: Characterize the colorings of the rational chain \({\mathbb{Q}}\) such that every open principal final segment of \({\mathbb{Q}}\) is color-isomorphic to \({\mathbb{Q}}\).