# Co-universal *C**-algebras associated to generalised graphs

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DOI: 10.1007/s11856-012-0106-0

- Cite this article as:
- Brownlowe, N., Sims, A. & Vittadello, S.T. Isr. J. Math. (2013) 193: 399. doi:10.1007/s11856-012-0106-0

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## Abstract

We introduce *P*-graphs, which are generalisations of directed graphs in which paths have a degree in a semigroup *P* rather than a length in ℕ. We focus on semigroups *P* arising as part of a quasi-lattice ordered group (*G, P*) in the sense of Nica, and on *P*-graphs which are finitely aligned in the sense of Raeburn and Sims. We show that each finitely aligned *P*-graph admits a *C**-algebra *C**_{min} (Λ) which is co-universal for partialisometric representations of Λ which admit a coaction of *G* compatible with the *P*-valued length function. We also characterise when a homomorphism induced by the co-universal property is injective. Our results combined with those of Spielberg show that every Kirchberg algebra is Morita equivalent to *C**_{min} (Λ) for some (ℕ^{2}* ℕ)-graph Λ.