## Product systems over right-angled Artin semigroups

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- by Neal J. Fowler and Aidan Sims PDF
- Trans. Amer. Math. Soc.
**354**(2002), 1487-1509 Request permission

## Abstract:

We build upon Mac Laneโs definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid $\mathcal G$. We show that the existing notions of product systems fit into our categorical framework, as do the $k$-graphs of Kumjian and Pask. We then specialize to product systems over right-angled Artin semigroups; these are semigroups that interpolate between free semigroups and free abelian semigroups. For such a semigroup we characterize all product systems which take values in a given tensor groupoid $\mathcal G$. In particular, we obtain necessary and sufficient conditions under which a collection of $k$ $1$-graphs form the coordinate graphs of a $k$-graph.## References

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## Additional Information

**Neal J. Fowler**- Affiliation: Department of Mathematics, University of Newcastle, NSW 2308, Australia
**Aidan Sims**- Affiliation: Department of Mathematics, University of Newcastle, NSW 2308, Australia
- MR Author ID: 671497
- Received by editor(s): December 22, 1999
- Received by editor(s) in revised form: June 28, 2001
- Published electronically: November 30, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**354**(2002), 1487-1509 - MSC (1991): Primary 20F36; Secondary 18B40, 55N20
- DOI: https://doi.org/10.1090/S0002-9947-01-02911-7
- MathSciNet review: 1873016