Ordinal graphs and their \(C^*\)-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 \(\mathcal{O}(\Lambda)\) defined by Spielberg. In particular, we construct a \(\mathrm{C}^*\)-correspondence \(X_\alpha\) for each \(\alpha \in \mathrm{Ord}\) in order to apply Eryüzlü and Tomforde’s condition (S) and prove a Cuntz-Krieger uniqueness theorem for ordinal graphs.

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