Extending Utility Representations of Partial Orders

摘要

The problem is considered as to whether a monotone function defined on asubset P of a Euclidean space can be strictly monotonically extended to thewhole space. It is proved that this is the case if and only if the function is{\em separably increasing}. Explicit formulas are given for a class ofextensions which involves an arbitrary bounded increasing function. Similarresults are obtained for monotone functions that represent strict partialorders on arbitrary abstract sets X. The special case where P is a Paretosubset is considered.