We previously showed that abstract Cuntz semigroups form a closed symmetric monoidal category. This automatically provides additional structure in the category, such as a composition and an external tensor product, for which we give concrete constructions in order to be used in applications. We further analyze the structure of not necessarily commutative -semirings, and we obtain, under mild conditions, a new characterization of solid -semirings R by the condition that .
Funding source: Ministerio de Economía y Competitividad
Award Identifier / Grant number: MTM2014-53644-P
Award Identifier / Grant number: MTM2017-83487-P
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: SFB 878 Groups
Award Identifier / Grant number: Geometry & Actions
Funding statement: The two first named authors were partially supported by MINECO (grants MTM2014-53644-P and MTM2017-83487-P), and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. The third named author was partially supported by the Deutsche Forschungsgemeinschaft (SFB 878 Groups, Geometry & Actions).
This work was initiated during a research in pairs (RiP) stay at the Oberwolfach Research Institute for Mathematics (MFO) in March 2015. The authors would like to thank the MFO for financial support and for providing inspiring working conditions. Part of this research was conducted while the third named author was visiting the Universitat Autònoma de Barcelona (UAB) in September 2015 and June 2016, and while the first and second named authors visited Münster Universität in June 2015 and 2016. Part of the work was also completed while the second and third named authors were attending the Mittag-Leffler institute during the 2016 program on Classification of Operator Algebras: Complexity, Rigidity, and Dynamics. They would like to thank all the involved institutions for their kind hospitality. The authors also thank the anonymous referee for her or his careful reading of the paper.
 R. Antoine, M. Dadarlat, F. Perera and L. Santiago, Recovering the Elliott invariant from the Cuntz semigroup, Trans. Amer. Math. Soc. 366 (2014), no. 6, 2907–2922. 10.1090/S0002-9947-2014-05833-9Search in Google Scholar
 B. Blackadar, K-theory for Operator Algebras, 2nd ed., Math. Sci. Res. Inst. Publ. 5, Cambridge University, Cambridge, 1998. Search in Google Scholar
 N. P. Brown, F. Perera and A. S. Toms, The Cuntz semigroup, the Elliott conjecture, and dimension functions on -algebras, J. Reine Angew. Math. 621 (2008), 191–211. 10.1515/CRELLE.2008.062Search in Google Scholar
 G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott, Continuous Lattices and Domains, Encyclopedia Math. Appl. 93, Cambridge University, Cambridge, 2003. 10.1017/CBO9780511542725Search in Google Scholar
 G. M. Kelly, Basic concepts of enriched category theory, Repr. Theory Appl. Categ. 2005 (2005), no. 10, 1–136. Search in Google Scholar
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