Existence of Richter-Peleg Representation for General Preferences
We characterize the binary relations that admit a Richter-Peleg representation, imposing neither completeness nor transitivity. A relation admits such a representation if and only if it is strongly acyclic and its transitive closure is separable, where separability means embeddability in a preorder possessing a countable separating stratification-a countable family of pairwise disjoint subsets, none containing a strictly ranked pair, that resolves every strict comparison. Separating stratifications generalize Debreu's order-density condition, and for complete preorders our theorem reduces to Debreu's. The embedding clause is the exact price of the stratification structure: indispensable in general, as a Richter-Peleg representable partial order admitting no countable separating stratification shows, but unnecessary whenever the representation is continuous and the lower contour sets are connected, which covers many standard incomplete preferences in economics. By providing existence, our results operationalize the optimization theory of White (1980), which we also extend by delivering the set of maximal elements of each subset with a single Richter-Peleg representation and no completeness assumption. As a second application, we show that the impossibility theorem of Basu and Mitra (2003) on intergenerational equity is exactly a failure of separability.