SOME PROPERTIES OF LORENZEN IDEAL SYSTEMS

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