Lexicographic behaviour of chains

This paper introduces a different approach to the study of the existence of numerical representations of totally ordered sets (chains). We pay attention to the properties of non-representable chains showing that, under certain conditions, those chains must have a sort of lexicographic behaviour similar to that of the lexicographic plane. We prove that a countably bounded connected chain (Z, ?) admits a lexicographic decomposition as a subset of the lexicographic product R x Z. Then we apply our approach to state both a sufficient and a necessary condition for the lack of utility functions. The concept of planar chain is also introduced.