Extending Utility Representations of Partial Orders

The problem is considered as to whether a monotone function defined on a subset of Euclidean space can be strictly monotonically extended to . It is proved that this is the case if and only if the function is Explicit formulas are given for a class of extensions which involves an arbitrary function. Similar results are obtained for utility functions that represent strict partial orders on abstract sets The special case where is a Pareto subset of (or of ) is considered