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SOME PROPERTIES OF LORENZEN IDEAL SYSTEMS

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*A. Kalapodi, A. Kontolatou and J. Mockor*

**Address.** Aleka Kalapodi, Department of Mathematics, University of
Patras, 26500 Patras, GREECE
Angeliki Kontolatou, Department of Mathematics, University of Patras,
26500 Patras, GREECE

Jiri Mockor, Department of Mathematics, University of Ostrava, CZ-702
00 Ostrava, Brafova 7, CZECH REPUBLIC

**E-mail:** Kalapodi@math.upatras.gr, angelika@math.upatras.gr, Mockor@osu.cz

**Abstract.** 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 $(R(G),\oplus,\le)$ 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
$po$-groups with special morphisms into the category of abelian ordered
semigroups.

**AMSclassification.** 06F05, 06F20

**Keywords.** $r$-ideal, $r_a$-system, system of finite character