Ordered semirings and subadditive morphisms

An ordered semiring is a commutative semiring equipped with a compatible preorder. Ordered semirings generalise both distributive lattices and commutative rings, and provide a convenient framework to unify certain aspects of lattice theory and ring theory. The ideals of an ordered semiring $A$ form a commutative integral quantale $\mathrm{I}\mathrm{d}\mathrm{l}(A)$, and similarly, the radical ideals of $A$ form a (spatial) frame $\mathrm{R}\mathrm{a}\mathrm{d}(A)$. We characterise $\mathrm{I}\mathrm{d}\mathrm{l}$ and $\mathrm{R}\mathrm{a}\mathrm{d}$ as the left adjoints of the (non-full) inclusion functors from the categories of commutative integral quantales and of frames, respectively, to that of ordered semirings and subadditive morphisms between them. The (sober) topological space $\mathrm{p}\mathrm{t}(\mathrm{R}\mathrm{a}\mathrm{d}(A))$ corresponding to $\mathrm{R}\mathrm{a}\mathrm{d}(A)$ is homeomorphic to the space $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(A)$ of prime ideals of $A$.