Ordinal graphs and their $\mathrm{C}^*$-algebras

We introduce a class of left cancellative categories we call ordinal graphs for which there is a functor $d:\Lambda\rightarrow\mathrm{Ord}$ by which morphisms of $\Lambda$ factor. We use generators and relations to study the Cuntz-Krieger algebra $\mathcal{O}\left(\Lambda\right)$ defined by Spielberg. In particular, we construct a $\mathrm{C}^{*}$-correspondence $X_{\alpha}$ for each $\alpha\in\mathrm{Ord}$ in order to apply Ery\"uzl\"u and Tomforde's condition (S) and prove a Cuntz-Krieger uniqueness theorem for ordinal graphs.