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

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