The Category Cv_(D_(0), N)-Cmod and MP

This paper deals with the halfing morphism of $Cv_{D_0, N}$-convex modules, where $D_0$ is the semiring of non-negative dyadic rationals, and its use in the explicit construction of the left adjoint to the functor $O_{D_0, N}:D_0\,$-{\bf Mod} $\longrightarrow Cv_{D_0, N}$- {\bf Cmod} that assigns to each $D_0$-module its canonical $Cv_{D_0, N}$-convex module. Furthermore it is shown that $Cv_{D_0, N}$-{\bf Cmod} is isomorphic to the category {\bf MP} of midpoint algebras