Let $G$ be a partially ordered abelian group ($po$-group). The construction of the Lorenzen ideal $r_a$-system in $G$ is investigated and the functorial properties of this construction with respect to the semigroup $\left(R\right(G),\oplus ,\le )$ of all $r$-ideal systems defined on $G$ are derived, where for $r,s\in R\left(G\right)$ and a lower bounded subset $X\subseteq G$, $X_{r\oplus s}=X_r\cap X_s$. It is proved that Lorenzen construction is the natural transformation between two functors from the category of $po$-groups with special morphisms into the category of abelian ordered semigroups.