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.

We show the embedding clause is indispensable by exhibiting a partial order that is Richter-Peleg representable yet admits no countable separating stratification.

Two corollaries specialize the result to preorders and to countable domains, and we sharpen White's (1980) optimization theory by representing the maximal set of any subset with a single Richter-Peleg representation.