Numerical representation of intransitive preferences on a countable set (Q1054247)

Let \(\succ\) be a preference relation on a countable set \(X\). The author proves that if \(\succ\) is acyclic (that is, has irreflexive transitive-closure), then there exists a map \(u\colon X\to\mathbb R\) such that \(x\succ y\) entails \(u(x)\succ u(y)\). He also extends, and gives a simple proof of, a theorem of Fishburn which provides representations of an interval order.