Lexicographic behaviour of chains
The problem of representability of ordered sets by means of real-valued order-preserving functions is very natural in the theory of ordered sets and is important for applications. It is known that a chain is representable if and only if it is perfectly separable. The authors of the paper study more general cases of numerical representations of chains. They prove among others that any countably bounded connected chain \(Z\) admits a lexicographic decomposition as a subset of the lexicographic product \(\mathbb{R}\times Z\). Both necessary and sufficient conditions for the lack of utility functions are also discussed there.
scientific article; zbMATH DE number 1295485