Let $G$ be a partially ordered abelian group ( $p o$ -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 $(R (G), \oplus, \leq)$ of all $r$ -ideal systems defined on $G$ are derived, where for $r, s \in R (G)$ 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 $p o$ -groups with special morphisms into the category of abelian ordered semigroups.

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Algebraic and categorical properties of r -ideal systems.

The theory of quasi-divisors on cartesian products

Some properties of Lorenzen ideal systems

Algebraic and categorical properties of $r$ -ideal systems.