Abstract bivariant Cuntz semigroups II

Abstract

Abstract 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 Cu {\mathrm{Cu}} -semirings, and we obtain, under mild conditions, a new characterization of solid Cu {\mathrm{Cu}} -semirings R by the condition that R ≅ 〚 R , R 〛 {R\cong\llbracket R,R\rrbracket} .