$C^*$-correspondences for ordinal graphs

We introduce a family of $C^*$-correspondences $X_\alpha$ naturally associated to every ordinal graph $\Lambda$. When $\Lambda$ is a directed graph, $X_0$ is isomorphic to the usual $C^*$-correspondence associated to a graph. We show that ordinal graphs satisfying a weak assumption have the property that the $C^*$-algebra of $\Lambda_{\alpha + 1}$ is isomorphic to the Cuntz-Pimsner algebra of $X_\alpha$. As a consequence, the $C^*$-algebra of $\Lambda$ may be constructed starting from $c_0(\Lambda_0)$ by iteratively applying the Cuntz-Pimsner construction and inductive limits. We apply this result to strengthen the author's previous Cuntz-Krieger uniqueness theorem.